Solving a nonlinear system of 1000 nonlinear equations involving 1000 general integer variables

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program listed below seeks to solve simultaneously the following system of 1000 nonlinear equations involving 1000 general integer variables; this problem is based on Problem 5 in Sharma and Gupta [80, p. 7, Problem 5]:

X(i) ^2 * X(i+1) -1 = 0, (i = 1, 2,..., 999)

X(1000) ^ 2 * X(1) -1 = 0.

One notes the following line 1905 and line 1988, which are 1905 PRINT A(1), A(2), A(3), A(4), A(5) and 1988 PRINT A(996), A(997), A(998), A(999), A(1000), M, JJJJ.

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

89 RANDOMIZE JJJJ

90 M = -3D+30

116 REM

118 A(1) = INT(0 + RND * 10)

128 FOR i = 1 TO 50000

129 FOR KKQQ = 1 TO 1000

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + 1000)

140 B = 1 + FIX(RND * 1000)

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

177 FOR J44 = 1 TO 1000

178 X(J44) = INT(X(J44))

179 NEXT J44

231 FOR J44 = 2 TO 1000

234 X(J44) = 1 / (X(J44 - 1)) ^ 2

239 NEXT J44

333 REM

334 FOR J44 = 1 TO 1000

335 IF X(J44) < 0 THEN 1670

337 NEXT J44

461 REM

463 PD1 = -ABS(X(1000) ^ 2 * X(1) - 1)

466 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 1000

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1905 PRINT A(1), A(2), A(3), A(4), A(5)

1988 PRINT A(996), A(997), A(998), A(999), A(1000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31994 is shown below:

**1 1 1 1 1****1 1 1 1 1****0 -32000****0 1.#INF 0 1.#INF 0****1.#INF 0 1.#INF 0 1.#INF****-1.#IND -31999****0 1.#INF 0 1.#INF 0****1.#INF 0 1.#INF 0 1.#INF****-1.#IND -31998****0 1.#INF 0 1.#INF 0****1.#INF 0 1.#INF 0 1.#INF****-1.#IND -31997****1 1 1 1 1****1 1 1 1 1****0 -31996****1 1 1 1 1****1 1 1 1 1****0 -31995****1 1 1 1 1****1 1 1 1 1****0 -31994**

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 = -31994 was 6 minutes, counting from "Starting program...". One can see the computational results above and the discussion in Sharma and Gupta [80, Problem 5].

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] 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] 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] M. G. M. Khan, T. Maiti, M. J. Ahsan (2010). An optimal multivariate stratified sampling design using auxiliary information: an integer solution using goal programming approach. Journal of Official Statistics, vol. 26, no. 4, 2010, pp. 695-708.

[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] 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] G. Stephanopoulos, A. W. Westerberg, The Use of Hestenes’ Method of Multipliers to Resolve Dual Gaps in Engineering System Optimization. Journal of Optimization Theory and Applications, Vol.15, No. 3, pp. 285-309, 1975.

[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, Fourtth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002.

[104] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

[105] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.