A computer program to solve bilinear problems: an illustration using a classical bilinear problem in Porn, Bjork, and Westerlund [71]

A computer program to solve bilinear problems: an illustration using a classical bilinear problem in Porn, Bjork, and Westerlund [71]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve the following classical bilinear problem in Porn et al. [71, p. 111]:
Minimize
X(1) * X(2) - X(1) - X(2) )
subject to
-6 * X(1) + 8 * X(2) <= 3
3 * X(1) - X(2) <= 3
0 <= X(1), X(2) <= 1.5
X(1), X(2) epsilon R.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), 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), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
108 A(1) = 0 + RND * 1.5

111 A(2) = 0 + RND * 1.5

128 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)

141 B = 1 + FIX(RND * 2)

142 IF RND < 1.5 THEN 160 ELSE GOTO 167

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

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

172 REM
195 REM
197 REM
222 REM
226 X(2) = (3 * X(1) - 3)

233 IF -6 * X(1) + 8 * X(2) > 3 THEN 1670

433 IF X(1) < 0 THEN 1670

434 IF X(1) > 1.5 THEN 1670

437 IF X(2) < 0 THEN 1670

439 IF X(2) > 1.5 THEN 1670

972 PD1 = -X(1) * X(2) + X(1) + X(2)

979 P = PD1

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

1778 IF M < -999999999 THEN 1999
1888 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31993 is shown below:

1.166651       .4999523       1.083333       -32000

1.166767 .5003003 1.083333 -31998

1.166744 .5002327 1.083333 -31996

1.166796 .5003872 1.083333 -31995

1.166441 .499324 1.083333 -31994

1.166796 .5003886 1.083333 -31993

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 [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31993 was 3 seconds, counting from "Starting program...". One can compare the computational results above with those in Porn, Bjork, and Westerlund [71, p. 112].

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 (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[6] 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.
[7] 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.
[8] 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.
[9] 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.
[10] B. V. Babu, Rakesh Angira (2006). Modified differential evolution (MDE) for optimization of non-linear chemical processes. Computers and Chemical Engineering 30 (2006) 989-1002.
[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.
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] 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] Andreas Lundell, Tapio Westerlund, Convex underestimation strategies for signomial functions, Optimization Methods and Software 24 (4) 505-522, August 2009.

[58] A. Lundell, J. Westerlund , T. Westerlund (2009). Some transformation techniques with applications in global optimization. J. Global Optim. 43 (2-3), 391-405.

[59] 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

[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] 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] Ray Porn, Kaj-Mikael Bjork, Tapio Westerlund (2008). Global solution of optimization problems with signomial parts, Discrete Optimization 5 (2008) 108-120.

[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] Singiresu S. Rao, Engineering Optimization, Fifth Edition, Wiley, New York, 2020.

[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] 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).

[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 Sharma, 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] 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.

[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 (2011). An efficient global approach for posynomial geometric programming problems. INFORMS Journal on Computing, Vol. 23, No. 3, Summer 2011, pp. 483-492.

[92] 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.

[93] 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

[94] 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.

[95] 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.

[96] 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.

[97] 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.

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

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

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

[106] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.
One can easily view this dissertation on Google.