Substance

A Computer Program about Markowitz Portfolio Selection, Second Edition

A Computer Program about Markowitz Portfolio Selection, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the small example about Markowitz portfolio selection [3] on pp. 888-890 of Li and Tsai [2].

Minimize    .0108 * X(1) ^ 2 + .0584 * X(2) ^ 2 + .0942 * X(3) ^ 2 + .0248 * X(1) * X(2) + .0262 * X(1) * X(3) + .1108 * X(2) * X(3)

subject to    1.089 * X(1) + 1.214 * X(2) + 1.235 * X(3) >= 115

X(1) + X(2) + X(3) = 100

where X(1), X(2), and X(3) are integers.

Below X(1) through X(3) are integer variables, and X(4) is a slack variable.

Whereas line 128 of the earlier edition is 128 FOR I = 1 TO 10000, here line 128 is 128 FOR I = 1 TO 8000.

0 REM DEFDBL A-Z

2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 3
65 A(J44) = RND * 10

66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 8000

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 J = 1 + FIX(RND * 3)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

223 X(1) = 100 - X(2) - X(3)

224 FOR J33 = 1 TO 3

226 X(J33) = INT(X(J33))

227 NEXT J33

229 REM X(5) = 1
233 X(4) = -115 + 1.089 * X(1) + 1.214 * X(2) + 1.235 * X(3)

281 FOR J44 = 1 TO 3

283 IF X(J44) < 0 THEN 1670

284 IF X(J44) > 100 THEN 1670

285 NEXT J44

331 FOR J44 = 4 TO 4

332 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0
333 NEXT J44
403 POBA2 = -(.0108 * X(1) ^ 2 + .0584 * X(2) ^ 2 + .0942 * X(3) ^ 2 + .0248 * X(1) * X(2) + .0262 * X(1) * X(3) + .1108 * X(2) * X(3))

411 POBA = POBA2 - 1000000 * (PS(4))

459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -224.647 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)

1904 PRINT M, JJJJ, PPOBA2
1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [5]. The complete output through JJJJ=-16563 is shown below:

53       35       12       .027
-224.6452       -29251       -224.6452

53       36       11       .006
-223.8916       -19156       -223.8916

53       35       12       .027
-224.6452       -18228       -224.6452

53       36       11       .006
-223.8916       -16563       -223.8916

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [5], the wall-clock time for obtaining the output through JJJJ=-16563 was 3 minutes and 20 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp.882-891.

[3] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[4] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[5] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[6] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/

A general-purpose computer program for solving nonlinear systems of equations, second edition

A general-purpose computer program for solving nonlinear systems of equations, second edition
Jsun Yui Wong
Similar to the computer program in Wong [104], the computer program listed below seeks to solve the following system of 100 nonlinear equations based on the system of 50 nonlinear equations in Sharma and Gupta [80, p. 7, Problem 5]:
X(i)^2 * X(i+1) - 1 = 0, (i=1, 2,..., 99)
X(100)^2* X(1) - 1 = 0.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
116 FOR J44 = 1 TO 100
118 A(J44) = INT(0 + RND * 10)

119 NEXT J44

128 FOR i = 1 TO 20

129 FOR KKQQ = 1 TO 100
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 100)

140 B = 1 + FIX(RND * 100)
144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

165 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

231 FOR J44 = 2 TO 100
234 X(J44) = 1 / (X(J44 - 1)) ^ 2
239 NEXT J44

311 X(1) = 1 / X(100) ^ 2

334 FOR J44 = 1 TO 100
335 IF X(J44) < 0 THEN 1670
337 NEXT J44
463 PD1 = -ABS(X(100) ^ 2 * X(1) - 1)

466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 100

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i
1777 IF M <> 0 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), A(19), A(20), A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), A(30), A(31), A(32), A(33), A(34), A(35), A(36), A(37), A(38), A(39), A(40), A(41), A(42), A(43), A(44), A(45), A(46), A(47), A(48), A(49), A(50)
1907 PRINT A(51), A(52), A(53), A(54), A(55), A(56), A(57), A(58), A(59), A(60), A(61), A(62), A(63), A(64), A(65), A(66), A(67), A(68), A(69), A(70), A(71), A(72), A(73), A(74), A(75), A(76), A(77), A(78), A(79), A(80), A(81), A(82), A(83), A(84), A(85), A(86), A(87), A(88), A(89), A(90), A(91), A(92), A(93), A(94), A(95), A(96), A(97), A(98), A(99), A(100), M, JJJJ
1922 REM PRINT M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through
JJJJ= -31939 is shown below:

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31988

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31939

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31939 was 2 seconds, counting from "Starting program...".

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
[3] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[4] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[5] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[6] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[7] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.
[11] Hirak Basumatary (1 January 2019). Solve system of equations and inequalities with multiple solutions?
https://www.mathworks.com/matlabcentral/answers/437815-solve-system-of-equations-and-inequalities-with-multiple-solutions?_tid=prof_contriblnk
[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton--Approximate nonlinear equations by sequence of linear equations--lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[14] F. Bazikar, M. Saraj (2018), MathLAB Journal, vol. 1 no. 3, 2018.
http://purkh.com/index.php/mathlab

[15] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, Inc., 1968.

[16] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.

[17] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.

[18] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values.
https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute_values#Numerical_ Example

[19] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.

[20] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

[21] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.

[22] H. W. Corley, E. O. Dwobeng (2020). Relating optimization problems to systems of inequalities and equalities, American Journal of Operations Research, 2020, 10, 284-298. https://www.scirp.org/journal/ajor.

[23] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[24] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[25] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000). Mechanical component design for multi objectives using elitist non-dominated sorting GA. Proceedings of the Parallel Probl.Solv. Nat. PPSN VI Conference, Paris, France, pp. 859-868 (2000). (Please see pp. 1-10 in Technical Report No. 200002 via Google search.)

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.

[27] Wassila Drici, Mustapha Moulai (2019): An exact method for solving multi-objective integer indefinite quadratic programs, Optimization Methods and Software.

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.
[34] Google. This paper's system of nonlinear equations was certainly obtained from Google search.
[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8...

[37] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[38] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[39] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems, Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263
[40] C. H. Huang, H. Y. Kao. (2009). An effective linear approximation method for geometric programming problems, IEEE Publications, 1743-1746.
[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[42] R. Israel, A Karush-Kuhn-Tucker Example
https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

[43] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.

[44] N. K. Jain, V. K. Jain, K. Deb (2007). Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms. International Journal of Machine Tools and Manufacture 47 (2007), 900-919.

[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016).
https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.

[48] Ram Keval, Constrained Geometric Programming Problem.
[from Gooogle Search and Youtube].

[49] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[50] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[54] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.

[55] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming
[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Subhash C. Narula, H. Roland Weistroffer (1989). A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] Rashmi Ranjan Ota, jayanta kumar dASH, A. K. Ojha (2014). A hybrid optimization techniques for solving non-linear multi-objective optimization problem. amo-advanced modelling and optimization, volume 16,number 1, 2014.

[69] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[70] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[71] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.

[72] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statitiscal Computation and Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370.
[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[74] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.

[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.
[78] Shafiullah, Irfan Ali, Abdul Bari (2015). Fuzzy geometric programming approach in multi-objective multivariate stratified sample surveys in presence of non-respnse, International Journal of Operations Research, Vol. 12, No. 2, pp. 021-035 (2015).
[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.

[80] J. R. Sharma, Puneet Gupta (2014). An efficient family of Traub-Steffensen-Type methods for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.

[81] Vikas Sharma, Puneet Gupta (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[82] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[83] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[84] Stanford University, https://www.google.com/search?q=simultaneous+equations+from+stanford+universitytest
[85] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.

[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.

[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529
One can get a Google view of this article. www.mdpi.com/journal/symmetry.

[92] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

[93] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.

[94] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

[95] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.

[96] Rahul Varshney, Najmussehar, M. J. Ahsan (2012). An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

[97] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.

[98] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

[99] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.

[100] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

[101] Eric W. Weisstein, "Euler's Sum of Powers Conjecture." https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

[102] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[103] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002.

[104] Jsun Yui Wong (04/03/2024). A general-purpose computer program for solving nonlinear systems of equations. https://myblogsubstance.typepad.com/substance/2024/04/Index.html

[105] Helen Wu, (2015), Geometric Programming
https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.
[107] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.
[108] www.youtube.com/watch?v=7xttRiA6ZVc

Computer program to solve nonlinear/linear fractional programming problems: another illustration

Computer program to solve nonlinear/linear fractional programming problems: another illustration
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following fractional programming problem from Raouf and Hezam [73, Problem f1, page 4 of 10; free access]:

Maximize (4 * X(1) + 2 * X(2) + 10) / (X(1) + 2 * X(2) + 5)
subject to
X(1) + 3 * X(2) <= 30
-X(1) + 2 * X(2) <= 5
X(1) >= 1
X(2) >= 1.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 2
121 A(J44) = 0 + (RND * 3)
126 NEXT J44
127 FOR I = 1 TO 5000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(RND * 2)
156 IF RND < .5 THEN 160 ELSE GOTO 167
160 r = (1 - RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * r
166 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
186 FOR J44 = 1 TO 2
188 REM X(J44) = INT(X(J44))

189 NEXT J44
422 FOR J44 = 1 TO 2
435 REM
436 REM

455 NEXT J44

457 IF X(1) < 0 THEN 1670
459 IF X(2) < 0 THEN 1670
462 IF X(1) + 3 * X(2) > 30 THEN 1670

463 IF -X(1) + 2 * X(2) > 5 THEN 1670
619 P = (4 * X(1) + 2 * X(2) + 10) / (X(1) + 2 * X(2) + 5)
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M < -999999 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output in one run through
JJJJ=-31998 is shown below. GW-BASIC also can handle this computer program.

29.99998       2.847438E-07       3.714286       -32000
29.99999       2.158129E-07       3.714286       -31999
30                   3.202235E-07       3.714286       -31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was one second, counting from “Starting program…”. Please see the computational results in Raouf and Hezam [73, Problem f1, page 5 of 10; free access].
The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
[3] Math Admiral (July 12, 2022), How-to-solve-non-linear-system-of-four-equations-with-four-variables
https://math.stackexchange.com/questions/4491203/how-to-solve-non-linear-system-of-four-equations-with-four-variables
[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.
[11] Dana H. Ballard, C. O. Jelinek, R. Schinzinger (November 1972). An algorithm for the solution of constrained generalised polynomial programming problems, The Computer Journal, 17(3): 261-266.

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton–Approximate nonlinear equations by sequence of linear equations–lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[ 14] Dennis L. Bricker, Jae Chul Choi (1995), Obtaining a feasible geometric programming primal solution, given a near-optimal dual solution, Engineering Optimization, 1995,
[15] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.
[16] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute values#Numerical Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.
[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.
[21] Ta-Cheng Chen (2006). As based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[22] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[23] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.
[27] Ampon Dhamacharoen (2014), An efficient hybrid method for solving nonlinear equations, vol.263, June 2014, 59-68., J. of computational and applied mathematics.
https://www.sciencedirect.com/science/article/pii/S0377042713006766

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values…
[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8…

[38] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[39] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[40] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems,
Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[42] M. Jaberipour, E. Khorram (2010), Solving the sum-of-ratios problemsby a harmony search algorithm, J. of Computational and Applied Mathematics 234 (2010).

[43] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.

[44] H. Jiao, Z. Wang, Y. Chen (2013), Global Optimization Algorithm for Generalized Polynomial Sum of Ratios Problem, Appl Math Model 37 (1) : 187-197.

[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.
[48] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[49] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[50] Learncommunolizer. Germany Math Olympiad/A Nice Algebra Problem/How to solve for "a", "b" and "c" in this Problem ? Please see Youtube.
https://www.youtube.com/watch?v=kuWb5-GXfOk
[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[54] C. S. Liu, S. N. Atluri (2008), A Novel time integration method for solving a large system of nonlinear algebraic equations, CMES Computer Modeling in Engineering and Sciences 31 (2) 71-83.

[55] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming
[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Jaleh Shirin Nejad, Mansour Saraj , Sara Shokrolahi Yancheshmeh1, Fatemeh Kiany Harchegani (2024), Global optimization of mixed integer signomial fractional programming problems, Messure and Control (a journal), Sagepub.com/journals.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[69] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[70] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.

[71] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[72] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[73] O. A. Raouf, I. M. Hezam (2014), Solving fractional programming problems based on swarm intelligence, J. of Industrial Engineering International, volumn 10, article numer 56, (2014)

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.
[8o] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.
[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[82] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[83] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.
[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. http://www.mdpi.com/journal/symmetry.
[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.
[93] Jung-Fa Tsai (2010), Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 42(9), 833-843.
[94] Jung-Fa Tsai (2005), Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 2005, 399-409.
[95] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.
[96] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[97] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.
[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[100] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.

[101] Chuanwei Wang, Yiju Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. of Global Optimization, 03 July 2008, Vol. 44, 283-296 (2009).
[102] T. Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011), Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[103] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[104] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[105] Wayne L. Winston (2004), Operations Research–Applications and Algorithms, Fourtth Edition, Brooks/Cole–Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

Computer program to solve nonlinear/linear fractional programming problems: another illustration

Computer program to solve nonlinear/linear fractional programming problems: another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following fractional programming problem based on Shen et al. [83, Example 2, p. 452] and on Raouf and Hezam [73, Problem f6, page 4 of 10]:

Maximize (-X(1) ^ 2 + 3 * X(1) - X(2) ^ 2 + 3 * X(2) + 3.5) / (X(1) + 1) + X(2) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)

subject to
2 * X(1) + X(2) <= 6
3 * X(1) + X(2) <= 8
-X(1) + X(2) >= -1
1 <= X(1) <= 2.25
1 <= X(2) <= 4.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 2
121 A(J44) = 0 + (RND * 5)
126 NEXT J44
127 FOR I = 1 TO 5000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(RND * 2)
156 IF RND < .5 THEN 160 ELSE GOTO 167
160 r = (1 - RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * r
166 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
186 FOR J44 = 1 TO 2
188 REM X(J44) = INT(X(J44))

189 NEXT J44

422 FOR J44 = 1 TO 2
435 REM
436 REM

455 NEXT J44

457 IF X(1) < 1 THEN 1670
458 IF X(1) > 2.25 THEN 1670
459 IF X(2) < 1 THEN 1670
460 IF X(2) > 4 THEN 1670

468 IF 2 * X(1) + X(2) > 6 THEN 1670

469 IF 3 * X(1) + X(2) > 8 THEN 1670
470 IF -X(1) + X(2) < -1 THEN 1670

626 P = (-X(1) ^ 2 + 3 * X(1) - X(2) ^ 2 + 3 * X(2) + 3.5) / (X(1) + 1) + X(2) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M < -999999 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output in one run through
JJJJ=-31998 is shown below. GW-BASIC also can handle this computer program.

1                  1.743462       4.060819       -32000
1                  1.743692       4.060819       -31999
1.000005  1.744078       4.060812       -31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was one second, counting from “Starting program…”. Please see the computational results in Shen et al. [83, Example 2, p. 452] and in Raouf and Hezam [73, Problem f6, page 5 of 10].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
[3] Math Admiral (July 12, 2022), How-to-solve-non-linear-system-of-four-equations-with-four-variables
https://math.stackexchange.com/questions/4491203/how-to-solve-non-linear-system-of-four-equations-with-four-variables
[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.
[11] Dana H. Ballard, C. O. Jelinek, R. Schinzinger (November 1972). An algorithm for the solution of constrained generalised polynomial programming problems, The Computer Journal, 17(3): 261-266.

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton–Approximate nonlinear equations by sequence of linear equations–lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[ 14] Dennis L. Bricker, Jae Chul Choi (1995), Obtaining a feasible geometric programming primal solution, given a near-optimal dual solution, Engineering Optimization, 1995,
[15] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.
[16] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute values#Numerical Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.
[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.
[21] Ta-Cheng Chen (2006). As based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[22] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[23] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.
[27] Ampon Dhamacharoen (2014), An efficient hybrid method for solving nonlinear equations, vol.263, June 2014, 59-68., J. of computational and applied mathematics.
https://www.sciencedirect.com/science/article/pii/S0377042713006766

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values…
[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8…

[38] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[39] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[40] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems,
Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[42] M. Jaberipour, E. Khorram (2010), Solving the sum-of-ratios problemsby a harmony search algorithm, J. of Computational and Applied Mathematics 234 (2010).

[43] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.

[44] H. Jiao, Z. Wang, Y. Chen (2013), Global Optimization Algorithm for Generalized Polynomial Sum of Ratios Problem, Appl Math Model 37 (1) : 187-197.

[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.
[48] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[49] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[50] Learncommunolizer. Germany Math Olympiad/A Nice Algebra Problem/How to solve for "a", "b" and "c" in this Problem ? Please see Youtube.
https://www.youtube.com/watch?v=kuWb5-GXfOk
[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[54] C. S. Liu, S. N. Atluri (2008), A Novel time integration method for solving a large system of nonlinear algebraic equations, CMES Computer Modeling in Engineering and Sciences 31 (2) 71-83.

[55] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming
[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Jaleh Shirin Nejad, Mansour Saraj , Sara Shokrolahi Yancheshmeh1, Fatemeh Kiany Harchegani (2024), Global optimization of mixed integer signomial fractional programming problems, Messure and Control (a journal), Sagepub.com/journals.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[69] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[70] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.

[71] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[72] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[73] O. A. Raouf, I. M. Hezam (2014), Solving fractional programming problems based on swarm intelligence, J. of Industrial Engineering International, volumn 10, article numer 56, (2014)

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.
[8o] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.
[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[82] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[83] P. Shen, Y. Ma, Y. Chen (2011), Global optimization for the generalized polynomial sum of ratios problem, J. of Global Optimization 2011 50:439-455.

[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.
[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. http://www.mdpi.com/journal/symmetry.
[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.
[93] Jung-Fa Tsai (2010), Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 42(9), 833-843.
[94] Jung-Fa Tsai (2005), Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 2005, 399-409.
[95] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.
[96] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[97] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.
[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[100] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.

[101] Chuanwei Wang, Yiju Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. of Global Optimization, 03 July 2008, Vol. 44, 283-296 (2009).
[102] T. Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011), Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[103] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[104] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[105] Wayne L. Winston (2004), Operations Research–Applications and Algorithms, Fourtth Edition, Brooks/Cole–Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

Computer program to solve nonlinear/linear fractional programming problems: an illustration

Computer program to solve nonlinear/linear fractional programming problems: an illustration
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following nonlinear fractional programming problem in Raouf and Hezam [73, Problem f6, page 4 of 10]:

Maximize
((-X(1) ^ 2 + 3 * X(1) - X(2) ^ 2 + 3 * X(2) + 3.5) / (X(1) + 1)) + (X(2)) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)
subject to
2 * X(1) + X(2) <= 6
3 * X(1) + X(2) <= 8
-X(1) + X(2) >= -1
1<= X(1) <=2.25
1<= X(2) <= 4 .

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 2
121 A(J44) = 1 + (RND * 3)
126 NEXT J44
127 FOR I = 1 TO 10000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(RND * 2)
156 IF RND < .5 THEN 160 ELSE GOTO 167
160 r = (1 - RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * r
166 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
186 FOR J44 = 1 TO 2
188 REM X(J44) = INT(X(J44))

189 NEXT J44

422 FOR J44 = 1 TO 2
435 REM
436 REM

455 NEXT J44

457 IF X(1) < 0 THEN 1670
458 REM

459 IF X(2) < 0 THEN 1670
460 IF X(1) < 1 THEN 1670
461 IF X(1) > 2.25 THEN 1670
462 IF X(2) < 1 THEN 1670
463 IF X(2) > 4 THEN 1670
464 IF 2 * X(1) + X(2) > 6 THEN 167

466 IF 3 * X(1) + X(2) > 8 THEN 1670

467 IF -X(1) + X(2) < -1 THEN 1670
624 P = ((-X(1) ^ 2 + 3 * X(1) - X(2) ^ 2 + 3 * X(2) + 3.5) / (X(1) + 1)) + (X(2)) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I

1777 IF M < -999999 THEN 1999

1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output through
JJJJ=-31999 in one run is shown below. GW-BASIC also can handle this computer program.

1       1.744018       4.060819       -32000
1       1.743725       4.060819       -31999

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999 was 2 seconds, counting from “Starting program…”. Please see the computational results in Raouf and Hezam [73, Table 1, Problem f6, page 5 of 10].
All computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
[3] Math Admiral (July 12, 2022), How-to-solve-non-linear-system-of-four-equations-with-four-variables
https://math.stackexchange.com/questions/4491203/how-to-solve-non-linear-system-of-four-equations-with-four-variables
[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.
[11] Dana H. Ballard, C. O. Jelinek, R. Schinzinger (November 1972). An algorithm for the solution of constrained generalised polynomial programming problems, The Computer Journal, 17(3): 261-266.

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton–Approximate nonlinear equations by sequence of linear equations–lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[ 14] Dennis L. Bricker, Jae Chul Choi (1995), Obtaining a feasible geometric programming primal solution, given a near-optimal dual solution, Engineering Optimization, 1995,
[15] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.
[16] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute values#Numerical Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.
[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.
[21] Ta-Cheng Chen (2006). As based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[22] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[23] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.
[27] Ampon Dhamacharoen (2014), An efficient hybrid method for solving nonlinear equations, vol.263, June 2014, 59-68., J. of computational and applied mathematics.
https://www.sciencedirect.com/science/article/pii/S0377042713006766

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values…
[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8…

[38] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[39] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[40] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems,
Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[42] M. Jaberipour, E. Khorram (2010), Solving the sum-of-ratios problemsby a harmony search algorithm, J. of Computational and Applied Mathematics 234 (2010).

[43] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.
[44] Jorge Donis del Alamo (March 14 2017) Easy and reliable method for solving four nonlinear equations
https://math.stackexchange.com/questions/2186612/easy-and-reliable-method-for-solving-four-nonlinear-equations
[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.
[48] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[49] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[50] Learncommunolizer. Germany Math Olympiad/A Nice Algebra Problem/How to solve for "a", "b" and "c" in this Problem ? Please see Youtube.
https://www.youtube.com/watch?v=kuWb5-GXfOk
[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[54] C. S. Liu, S. N. Atluri (2008), A Novel time integration method for solving a large system of nonlinear algebraic equations, CMES Computer Modeling in Engineering and Sciences 31 (2) 71-83.

[55] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming
[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Jaleh Shirin Nejad, Mansour Saraj , Sara Shokrolahi Yancheshmeh1, Fatemeh Kiany Harchegani (2024), Global optimization of mixed integer signomial fractional programming problems, Messure and Control (a journal), Sagepub.com/journals.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[69] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[70] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.

[71] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[72] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[73] O. A. Raouf, I. M. Hezam (2014), Solving fractional programming problems based on swarm intelligence, J. of Industrial Engineering International, volumn 10, article numer 56, (2014)

[74] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[75] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[76] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[77] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[78] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.
[79] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.
[80] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[81] Vikas Shar-55. of global optimization,50 ():239m for generalizedpolynomial sum of ratios problem, Ja, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[82] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.

[ 83 ] P. Shen, Y. Ma, Y. Chen (2011) Global optimization for generalized polynomial sum of ratios problem, J. of Global Optimization 50(3): 439-455.
[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.
[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. http://www.mdpi.com/journal/symmetry.
[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.
[93] Jung-Fa Tsai (2010), Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 42(9), 833-843.
[94] Jung-Fa Tsai (2005), Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 2005, 399-409.
[95] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.
[96] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[97] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.
[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[100] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.

[101] Chuanwei Wang, Yiju Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. of Global Optimization, 03 July 2008, Vol. 44, 283-296 (2009).
[102] T. Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011), Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[103] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[104] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[105] Wayne L. Winston (2004), Operations Research–Applications and Algorithms, Fourtth Edition, Brooks/Cole–Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

Computer program for solving integer/continuous nonlinear/linear fractional programming problems: an illustration

Computer program for solving integer/continuous nonlinear/linear fractional programming problems: an illustration

Jsun Yui Wong

The computer program listed below aims to solve directly the following nonlinear programming problem from Reference 83:

Maximize (2 * X(1) + X(2) - X(3)) / (5 * X(1) - 7 * X(2) + 3)
subject to
0 <= X(1) + X(2) + X(3) <= 1
1<= X(1) - X(2) + X(3) <= 2
X(1) - X(2) - X(3) = 3.

One notes line 619 below, which is 619 P = (2 * X(1) + X(2) - X(3)) / (5 * X(1) - 7 * X(2) + 3).

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 3
121 A(J44) = FIX(-5 + RND * 10)
126 NEXT J44
127 FOR I = 1 TO 5000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 3)
156 IF RND < .5 THEN 160 ELSE GOTO 167
160 r = (1 - RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * r
166 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
186 FOR J44 = 1 TO 3
188 REM X(J44) = INT(X(J44))
189 NEXT J44
422 FOR J44 = 1 TO 3
435 REM
436 REM

455 NEXT J44

463 IF X(1) + X(2) + X(3) < 0 THEN 1670
464 IF X(1) + X(2) + X(3) > 1 THEN 1670

465 IF X(1) - X(2) + X(3) < 1 THEN 1670
466 IF X(1) - X(2) + X(3) > 2 THEN 1670

468 IF X(1) - X(2) - X(3) < 3 THEN 1670
469 IF X(1) - X(2) - X(3) > 3 THEN 1670

619 P = (2 * X(1) + X(2) - X(3)) / (5 * X(1) - 7 * X(2) + 3)

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M < -999999 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output in one run through JJJJ=-31914 is shown below. GW-BASIC also can handle this computer program.

2    0    -1    .3846154    -31991
2 0 -1 .3846154 -31959
2 0 -1 .3846154 -31949
2 0 -1 .3846154 -31936
2 0 -1 .3846154 -31934
2 0 -1 .3846154 -31931
2 0 -1 .3846154 -31914

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31914 was 3 seconds, counting from “Starting program…”. Please see the results in Reference 83. 
All computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
[3] Math Admiral (July 12, 2022), How-to-solve-non-linear-system-of-four-equations-with-four-variables
https://math.stackexchange.com/questions/4491203/how-to-solve-non-linear-system-of-four-equations-with-four-variables
[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.
[11] Dana H. Ballard, C. O. Jelinek, R. Schinzinger (November 1972). An algorithm for the solution of constrained generalised polynomial programming problems, The Computer Journal, 17(3): 261-266.

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton–Approximate nonlinear equations by sequence of linear equations–lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[ 14] Dennis L. Bricker, Jae Chul Choi (1995), Obtaining a feasible geometric programming primal solution, given a near-optimal dual solution, Engineering Optimization, 1995,
[15] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.
[16] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute values#Numerical Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.
[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.
[21] Ta-Cheng Chen (2006). As based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[22] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[23] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.
[27] Ampon Dhamacharoen (2014), An efficient hybrid method for solving nonlinear equations, vol.263, June 2014, 59-68., J. of computational and applied mathematics.
https://www.sciencedirect.com/science/article/pii/S0377042713006766

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values…
[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8…

[38] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[39] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[40] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems,
Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[42] R. Israel, A Karush-Kuhn-Tucker Example
https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

[43] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.
[44] Jorge Donis del Alamo (March 14 2017) Easy and reliable method for solving four nonlinear equations
https://math.stackexchange.com/questions/2186612/easy-and-reliable-method-for-solving-four-nonlinear-equations
[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.
[48] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[49] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[50] Learncommunolizer. Germany Math Olympiad/A Nice Algebra Problem/How to solve for "a", "b" and "c" in this Problem ? Please see Youtube.
https://www.youtube.com/watch?v=kuWb5-GXfOk
[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[54] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.

[55] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming

[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Jaleh Shirin Nejad, Mansour Saraj , Sara Shokrolahi Yancheshmeh1, Fatemeh Kiany Harchegani (2024), Global optimization of mixed integer signomial fractional programming problems, Messure and Control (a journal), Sagepub.com/journals.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[69] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[70] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.
[71] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statitiscal Computation and Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370.
[72] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[73] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[74] O. A. Raouf, I. M. Hezam (11 April 2014), Solving fractional programming problems based on swarm intelligence, Journal of Industrial Engineering International, vol. 10, article number 56 (2014).

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.
[80] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.
[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[82] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[83] Stackoverflow.com (2011), Mathematica: help solving a system of non-linear equations with inequality constraints.
https://stackoverflow.com/questions/5918928/mathematica-help-solving-a-system-of-non-linear-equations-with-inequality-const

[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.
[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. http://www.mdpi.com/journal/symmetry.
[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.
[93] Jung-Fa Tsai (2010), Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 42(9), 833-843.
[94] Jung-Fa Tsai (2005), Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 2005, 399-409.
[95] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.
[96] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[97] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.
[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[100] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.

[101] Chuanwei Wang, Yiju Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. of Global Optimization, 03 July 2008, Vol. 44, 283-296 (2009).
[102] T. Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011), Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[103] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[104] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[105] Wayne L. Winston (2004), Operations Research–Applications and Algorithms, Fourtth Edition, Brooks/Cole–Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

Erratum in: Here is an industry application of nonlinear fractional programming solving a gear train design problem

Erratum in: Here is an industry application of nonlinear fractional programming solving a gear train design problem

Jsun Yui Wong

Line 1777 of the preceding paper should read
1777 IF M < -3D-11 THEN 1999
instead of
1777 IF M < -999999 THEN 1999.

Here is an industry application of nonlinear fractional programming solving a gear train design problem

Here is an industry application of nonlinear fractional programming solving a gear train design problem

Jsun Yui Wong

The computer program listed below aims to solve directly the following nonlinear programming problem designing a gear train from Raouf and Hezam [74, page 6 of 10]:

Minimize ((1 / 6.931) - (X(1) * X(2)) / (X(3) * X(4))) ^ 2
12 <= X(i) <= 60, i = 1, 2, 3, 4; 4 integer variables representing number of teeth of each of the 4 gears.

The key line of the following computer program is 618 P = -((1 / 6.931) - (X(1) * X(2)) / (X(3) * X(4))) ^ 2.

0 DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30

117 FOR J44 = 1 TO 4

121 A(J44) = 12 + RND * 48

126 NEXT J44

127 FOR I = 1 TO 500000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 4)

140 B = 1 + FIX(RND * 4)

156 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

166 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

186 FOR J44 = 1 TO 4

188 X(J44) = INT(X(J44))

189 NEXT J44

422 FOR J44 = 1 TO 4

435 IF X(J44) < 12 THEN 1670

436 IF X(J44) > 60 THEN 1670

455 NEXT J44
618 P = -((1 / 6.931) - (X(1) * X(2)) / (X(3) * X(4))) ^ 2
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M < -999999 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output in one run through JJJJ=-31992 is shown below. GW-BASIC also can handle this computer program.

30 13 51 53 -2.30781573331217D-11
-31998

16       19       43       49       -2.700857148881926D-12
-31992

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31992 was  8 seconds, counting from “Starting program…”.  Please see the computational results in Raouf and Hezam [74, page 7 of 10, Table 2].
All computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
[3] Math Admiral (July 12, 2022), How-to-solve-non-linear-system-of-four-equations-with-four-variables
https://math.stackexchange.com/questions/4491203/how-to-solve-non-linear-system-of-four-equations-with-four-variables
[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.
[11] Dana H. Ballard, C. O. Jelinek, R. Schinzinger (November 1972). An algorithm for the solution of constrained generalised polynomial programming problems, The Computer Journal, 17(3): 261-266.

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton–Approximate nonlinear equations by sequence of linear equations–lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[ 14] Dennis L. Bricker, Jae Chul Choi (1995), Obtaining a feasible geometric programming primal solution, given a near-optimal dual solution, Engineering Optimization, 1995,
[15] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.
[16] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute values#Numerical Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.
[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.
[21] Ta-Cheng Chen (2006). As based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[22] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[23] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.
[27] Ampon Dhamacharoen (2014), An efficient hybrid method for solving nonlinear equations, vol.263, June 2014, 59-68., J. of computational and applied mathematics.
https://www.sciencedirect.com/science/article/pii/S0377042713006766

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values…
[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8…

[38] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[39] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[40] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems,
Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[42] R. Israel, A Karush-Kuhn-Tucker Example
https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

[43] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.
[44] Jorge Donis del Alamo (March 14 2017) Easy and reliable method for solving four nonlinear equations
https://math.stackexchange.com/questions/2186612/easy-and-reliable-method-for-solving-four-nonlinear-equations
[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.
[48] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[49] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[50] Learncommunolizer. Germany Math Olympiad/A Nice Algebra Problem/How to solve for "a", "b" and "c" in this Problem ? Please see Youtube.
https://www.youtube.com/watch?v=kuWb5-GXfOk
[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[54] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.

[55] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming
[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Jaleh Shirin Nejad, Mansour Saraj , Sara Shokrolahi Yancheshmeh1, Fatemeh Kiany Harchegani (2024), Global optimization of mixed integer signomial fractional programming problems, Messure and Control (a journal), Sagepub.com/journals.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[69] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[70] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.
[71] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statitiscal Computation and Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370.
[72] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[73] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[74] O. A. Raouf, I. M. Hezam (11 April 2014), Solving fractional programming problems based on swarm intelligence, Journal of Industrial Engineering International, vol. 10, article number 56 (2014).

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.
[8o] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.
[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[82] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[83] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.
[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. http://www.mdpi.com/journal/symmetry.
[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.
[93] Jung-Fa Tsai (2010), Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 42(9), 833-843.
[94] Jung-Fa Tsai (2005), Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 2005, 399-409.
[95] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.
[96] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[97] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.
[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[100] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.

[101] Chuanwei Wang, Yiju Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. of Global Optimization, 03 July 2008, Vol. 44, 283-296 (2009).
[102] T. Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011), Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[103] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[104] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[105] Wayne L. Winston (2004), Operations Research–Applications and Algorithms, Fourtth Edition, Brooks/Cole–Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

Direct solution of a system of nonlinear equations in the literature

Direct solution of a system of nonlinear equations in the literature

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following unusual system of nonlinear equations in Freundenstein and Roth [34, p. 555]:

(-13 + X(1) - 2 * X(2) + 5 * X(2) ^ 2 - 1 * X(2) ^ 3) = 0
(-29 + X(1) - 14 * X(2) + 1 * X(2) ^ 2 + X(2) ^ 3) = 0.

 

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 2

121 A(J44) = -50 + RND * 100

122 NEXT J44

127 FOR I = 1 TO 5000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

140 B = 1 + FIX(RND * 2)

156 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

166 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

211 FOR J44 = 1 TO 2

213 REM X(J44) = INT(X(J44))

219 NEXT J44

422 REM FOR J44 = 1 TO 2

435 REM
437 REM

455 REM NEXT J44

551 LHS1 = (-13 + X(1) - 2 * X(2) + 5 * X(2) ^ 2 - 1 * X(2) ^ 3)

553 LHS2 = (-29 + X(1) - 14 * X(2) + 1 * X(2) ^ 2 + X(2) ^ 3)

611 P = -ABS(LHS1) - ABS(LHS2)

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1670 NEXT I
1777 IF M < -.001 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output in one run through JJJJ=-31986 is shown below. GW-BASIC also can handle this computer program.

4.999913 4.000002 -1.230241E-04 -31993
5.000061       3.999998       -8.630747E-05       -31991
5       4       0       -31987
4.999921 4.000002 -1.230241E-04 -31986

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31986 was 2 seconds, counting from “Starting program…”. Please see the computations in Freudenstein and Roth [34, p. 555].

All computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
[3] Math Admiral (July 12, 2022), How-to-solve-non-linear-system-of-four-equations-with-four-variables
https://math.stackexchange.com/questions/4491203/how-to-solve-non-linear-system-of-four-equations-with-four-variables
[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.
[11] Dana H. Ballard, C. O. Jelinek, R. Schinzinger (November 1972). An algorithm for the solution of constrained generalised polynomial programming problems, The Computer Journal, 17(3): 261-266.

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton–Approximate nonlinear equations by sequence of linear equations–lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[ 14] Dennis L. Bricker, Jae Chul Choi (1995), Obtaining a feasible geometric programming primal solution, given a near-optimal dual solution, Engineering Optimization, 1995,
[15] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.
[16] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute values#Numerical Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.
[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.
[21] Ta-Cheng Chen (2006). As based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[22] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[23] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.
[27] Ampon Dhamacharoen (2014), An efficient hybrid method for solving nonlinear equations, vol.263, June 2014, 59-68., J. of computational and applied mathematics.
https://www.sciencedirect.com/science/article/pii/S0377042713006766

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[34] Ferdinand Freudenstein, Bernhard Roth (1 October 1963 ), Numerical Solution of Systems of Nonlinear Equations, Journal of the ACM (JACM), Volume 10, Issue 4, Pages 550-556.
https://dl.acm.org

[35] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values…
[36] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[37] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[38] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8…

[39] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[40] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[41] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems,
Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[42] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[43] R. Israel, A Karush-Kuhn-Tucker Example
https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

[44] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.

[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.
[48] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[49] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[50] Learncommunolizer. Germany Math Olympiad/A Nice Algebra Problem/How to solve for "a", "b" and "c" in this Problem ? Please see Youtube.
https://www.youtube.com/watch?v=kuWb5-GXfOk
[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[54] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.

[55] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming
[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Jaleh Shirin Nejad, Mansour Saraj , Sara Shokrolahi Yancheshmeh1, Fatemeh Kiany Harchegani (2024), Global optimization of mixed integer signomial fractional programming problems, Messure and Control (a journal), Sagepub.com/journals.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[69] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[70] Pkrzysiek Pavel (2024), How do you deal with system of 3 nonlinear equations
https://www.reddit.com/r/askmath/comments/18vwd5s/how_do_you_deal_with_system_of_3_nonlinear/

[71] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.
[72] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statitiscal Computation and Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370.
[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[74] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.
[8o] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.
[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[82] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[83] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.
[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. http://www.mdpi.com/journal/symmetry.
[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.
[93] Jung-Fa Tsai (2010), Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 42(9), 833-843.
[94] Jung-Fa Tsai (2005), Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 2005, 399-409.
[95] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.
[96] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[97] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.
[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[100] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.

[101] Chuanwei Wang, Yiju Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. of Global Optimization, 03 July 2008, Vol. 44, 283-296 (2009).
[102] T. Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011), Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[103] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[104] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[105] Wayne L. Winston (2004), Operations Research–Applications and Algorithms, Fourtth Edition, Brooks/Cole–Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

An easiest method to solve systems of nonlinear equations plus nonlinear constraints: an illustration

An easiest method to solve systems of nonlinear equations plus nonlinear constraints: an illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following system of two nonlinear equations from Mathworks [58] plus two nonlinear constraints:

 

(X(1) + 1) * (10 - X(1)) * (1 + X(2) ^ 2) / (1 + X(2) ^ 2 + X(2)) = 0
(X(2) + 2) * (20 - X(2)) * (1 + X(1) ^ 2) / (1 + X(1) ^ 2 + X(1)) =0
X(1) ^ 2 + X(2) ^ 2 >= 100
X(1) ^ 2 + X(2) ^ 2 >= 200.

 

The four key lines in the following computer program are 435 IF X(1) ^ 2 + X(2) ^ 2 >= 100 THEN 437 ELSE GOTO 1670, 437 IF X(1) ^ 2 + X(2) ^ 2 >= 200 THEN 551 ELSE GOTO 1670, 551 LHS1 = (X(1) + 1) * (10 - X(1)) * (1 + X(2) ^ 2) / (1 + X(2) ^ 2 + X(2)), and 554 LHS2 = (X(2) + 2) * (20 - X(2)) * (1 + X(1) ^ 2) / (1 + X(1) ^ 2 + X(1)).

 

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30

117 FOR J44 = 1 TO 2

121 A(J44) = -50 + RND * 100

122 NEXT J44

127 FOR I = 1 TO 20000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)

140 B = 1 + FIX(RND * 2)

156 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

166 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

211 FOR J44 = 1 TO 2

213 X(J44) = INT(X(J44))

219 NEXT J44

422 REM FOR J44 = 1 TO 2

435 IF X(1) ^ 2 + X(2) ^ 2 >= 100 THEN 437 ELSE GOTO 1670
437 IF X(1) ^ 2 + X(2) ^ 2 >= 200 THEN 551 ELSE GOTO 1670

551 LHS1 = (X(1) + 1) * (10 - X(1)) * (1 + X(2) ^ 2) / (1 + X(2) ^ 2 + X(2))

554 LHS2 = (X(2) + 2) * (20 - X(2)) * (1 + X(1) ^ 2) / (1 + X(1) ^ 2 + X(1))

611 P = -ABS(LHS1) - ABS(LHS2)

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT I

1777 IF M < -.1 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output of one run through JJJJ= -31982 is shown below. GW-BASIC also can handle this computer program.

-1    20    0    -31999
-1 20 0 -31998
-1 20 0 -31995
10 20 0 -31994
-1 20 0 -31991
-1 20 0 -31987
-1 20 0 -31984
10 20 0 -31983
10 20 0 -31982

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31982 was 3 seconds, counting from “Starting program…”. Please see Mathworks [58].
All computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
[3] Math Admiral (July 12, 2022), How-to-solve-non-linear-system-of-four-equations-with-four-variables
https://math.stackexchange.com/questions/4491203/how-to-solve-non-linear-system-of-four-equations-with-four-variables
[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

[11] Dana H. Ballard, C. O. Jelinek, R. Schinzinger (November 1972). An algorithm for the solution of constrained generalised polynomial programming problems, The Computer Journal, 17(3): 261-266.
[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton–Approximate nonlinear equations by sequence of linear equations–lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[ 14] Dennis L. Bricker, Jae Chul Choi (1995), Obtaining a feasible geometric programming primal solution, given a near-optimal dual solution, Engineering Optimization, 1995,
[15] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.
[16] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute values#Numerical Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.
[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.
[21] Ta-Cheng Chen (2006). As based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[22] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[23] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.
[27] Ampon Dhamacharoen (2014), An efficient hybrid method for solving nonlinear equations, vol.263, June 2014, 59-68., J. of computational and applied mathematics.
https://www.sciencedirect.com/science/article/pii/S0377042713006766

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values…
[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8…

[38] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[39] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[40] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems,
Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[42] R. Israel, A Karush-Kuhn-Tucker Example
https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

[43] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.
[44] Jorge Donis del Alamo (March 14 2017) Easy and reliable method for solving four nonlinear equations
https://math.stackexchange.com/questions/2186612/easy-and-reliable-method-for-solving-four-nonlinear-equations
[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.
[48] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[49] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[50] Learncommunolizer. Germany Math Olympiad/A Nice Algebra Problem/How to solve for "a", "b" and "c" in this Problem ? Please see Youtube.
https://www.youtube.com/watch?v=kuWb5-GXfOk
[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443
[54] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.
[55] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.
[56] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[57] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming

[58] MathWorks, Nonlinear System of Equations with Constraints, Problem-Based.
https://www.mathworks.com/help/optim/ug/systems-of-equations-with-constraints-problem-based.html

[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering and System Safety 152 (2016) 213-227.
[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Jaleh Shirin Nejad, Mansour Saraj , Sara Shokrolahi Yancheshmeh1, Fatemeh Kiany Harchegani (2024), Global optimization of mixed integer signomial fractional programming problems, Messure and Control (a journal), Sagepub.com/journals.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[69] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[70] Pkrzysiek Pavel (2024), How do you deal with system of 3 nonlinear equations
https://www.reddit.com/r/askmath/comments/18vwd5s/how_do_you_deal_with_system_of_3_nonlinear/

[71] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.
[72] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statitiscal Computation and Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370.
[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[74] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.
[8o] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.
[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[82] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[83] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.
[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. http://www.mdpi.com/journal/symmetry.
[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.
[93] Jung-Fa Tsai (2010), Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 42(9), 833-843.
[94] Jung-Fa Tsai (2005), Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 2005, 399-409.
[95] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.
[96] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[97] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.
[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[100] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.
[101] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[102] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

[103] Whis (Nov.29, 2023), About solving the following system of inequalities.
https://math.stackexchange.com/questions/4816417/about-solving-the-following-system-of-inequalities?rq=1

[104] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[105] Wayne L. Winston (2004), Operations Research–Applications and Algorithms, Fourtth Edition, Brooks/Cole–Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

 

Computer program easily solving systems of nonlinear equations: another illustration

Computer program easily solving systems of nonlinear equations: another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following nonlinear system of equations in Mathworks [58]:

EXP(-EXP(-(X(1) + X(2))) = X(2) * (1 + X(1) ^ 2)
X(1) * COS(X(2) + X(2) * SIN(X(1) = 1/2.

The main line of the following computer program is 613 PD1 = -ABS(EXP(-EXP(-(X(1) + X(2)))) - X(2) * (1 + X(1) ^ 2)) - ABS(X(1) * COS(X(2)) + X(2) * SIN(X(1)) - .5), which involves the two given equations.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 5
121 A(J44) = -5 + RND * 10
126 NEXT J44
127 FOR I = 1 TO 5000
129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

140 B = 1 + FIX(RND * 2)

156 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

166 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

211 FOR J44 = 1 TO 2

213 REM X(J44) = INT(X(J44))

214 NEXT J44

613 PD1 = -ABS(EXP(-EXP(-(X(1) + X(2)))) - X(2) * (1 + X(1) ^ 2)) - ABS(X(1) * COS(X(2)) + X(2) * SIN(X(1)) - .5)

1111 IF PD1 <= M THEN 1670

1452 M = PD1

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT I
1779 IF M < -.001 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output through
JJJJ=-31995 is shown below. GW-BASIC also can handle this computer program.

.3532466       .6060817       -2.076433E-08       -31999
.3532466 .6060817 -2.076433E-08 -31997
.3532466 .6060817 -2.076433E-08 -31996
.3532466 .6060817 -2.076433E-08 -31995

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 1 second, counting from “Starting program…”. Please see the computational results in Mathworks [58].
All computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.

[3] Math Admiral (July 12, 2022), How-to-solve-non-linear-system-of-four-equations-with-four-variables
https://math.stackexchange.com/questions/4491203/how-to-solve-non-linear-system-of-four-equations-with-four-variables

[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.

[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

[11] Dana H. Ballard, C. O. Jelinek, R. Schinzinger (November 1972). An algorithm for the solution of constrained generalised polynomial programming problems, The Computer Journal, 17(3): 261-266.

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton–Approximate nonlinear equations by sequence of linear equations–lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
[ 14] Dennis L. Bricker, Jae Chul Choi (1995), Obtaining a feasible geometric programming primal solution, given a near-optimal dual solution, Engineering Optimization, 1995,
[15] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.
[16] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute values#Numerical Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.
[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.
[21] Ta-Cheng Chen (2006). As based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[22] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[23] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.
[27] Ampon Dhamacharoen (2014), An efficient hybrid method for solving nonlinear equations, vol.263, June 2014, 59-68., J. of computational and applied mathematics.
https://www.sciencedirect.com/science/article/pii/S0377042713006766

[28] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[29] Joseph G. Ecker, Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley & Sons, New York (1988).

[30] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[31] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[32] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[33] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values…
[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8…

[38] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[39] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[40] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems,
Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[42] R. Israel, A Karush-Kuhn-Tucker Example
https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

[43] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.
[44] Jorge Donis del Alamo (March 14 2017) Easy and reliable method for solving four nonlinear equations
https://math.stackexchange.com/questions/2186612/easy-and-reliable-method-for-solving-four-nonlinear-equations
[45] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[46] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[47] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.
[48] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[49] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[50] Learncommunolizer. Germany Math Olympiad/A Nice Algebra Problem/How to solve for "a", "b" and "c" in this Problem ? Please see Youtube.
https://www.youtube.com/watch?v=kuWb5-GXfOk
[51] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[54] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.
[55] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[56] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[57] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming

[58] Mathworks, Solve Nonlinear System of Equations, Problem-Based.
https://www.mathworks.com/help/optim/ug/nonlinear-equations-problem-based.html

[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[62] Riley Murray, Venkat Chandrasekaran, Adam Wierman. Signomial and polynomial optimization via relative entropy and partial dualization. [math.OC] 21 July 2019. eprint: arXve:1907.00814v2.

[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[64] Jaleh Shirin Nejad, Mansour Saraj , Sara Shokrolahi Yancheshmeh1, Fatemeh Kiany Harchegani (2024), Global optimization of mixed integer signomial fractional programming problems, Messure and Control (a journal), Sagepub.com/journals.

[65] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[66] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.

[67] Max M. J. Opgenoord, Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including equality constraints in signomial programming. ACDL Technical Report TR-2017-1. August 31 2017. pp.1-23.
One can get a Google view of this report.

[68] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.

[69] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[70] Pkrzysiek Pavel (2024), How do you deal with system of 3 nonlinear equations
https://www.reddit.com/r/askmath/comments/18vwd5s/how_do_you_deal_with_system_of_3_nonlinear/

[71] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.
[72] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statitiscal Computation and Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370.
[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[74] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.
[8o] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.
[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[82] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[83] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm
[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.
[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.
[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.
[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.
[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. http://www.mdpi.com/journal/symmetry.
[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.
[93] Jung-Fa Tsai (2010), Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 42(9), 833-843.
[94] Jung-Fa Tsai (2005), Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 2005, 399-409.
[95] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.
[96] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[97] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.
[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[100] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.
[101] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.
[102] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

[103] Whis (Nov.29, 2023), About solving the following system of inequalities.
https://math.stackexchange.com/questions/4816417/about-solving-the-following-system-of-inequalities?rq=1

[104] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[105] Wayne L. Winston (2004), Operations Research–Applications and Algorithms, Fourtth Edition, Brooks/Cole–Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[106] G. Xu (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.