Combinning Mathematical Formulation, “What If”, and Discrete Variables To Help Solve Nonlinear Programming Problems: Another Illustration
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following mathematical formulation in Bunday [15, p. 107, Exercises 6, 10], which is as follows:
Minimize
-X(1) * X(2) * X(3)
subject to
X(1), X(2), X(3) >=0
2* X(1) ^ 2+X(2) ^ 2 + 3 * X(3) ^ 2 <= 51.
One notes lines 92, 98 , and 99, which are 92 A(1) = INT(100 * RND * 4) / 100, 98 A(2) = INT(100 * RND * 4) / 100, and 99 A(3) = INT(100 * RND * 4) / 100.
0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), 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)
79 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
92 A(1) = INT(100 * RND * 4) / 100
98 A(2) = INT(100 * RND * 4) / 100
99 A(3) = INT(100 * RND * 4) / 100
123 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(0 + RND * 3)
144 IF RND < .5 THEN 160 ELSE GOTO 166
160 r = (1 – RND * 2) * A(B)
164 X(B) = A(B) + FIX((RND ^ (RND * 15)) * r)
165 GOTO 168
166 IF RND < .5 THEN X(B) = A(B) – FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
191 IF X(1) < 0 THEN 1670
193 IF X(2) < 0 THEN 1670
195 IF X(3) < 0 THEN 1670
201 IF (-X(2) ^ 2 – 3 * X(3) ^ 2 + 51) / 2 < 0 THEN 1670
202 REM
204 X(1) = ((-X(2) ^ 2 – 3 * X(3) ^ 2 + 51) / 2) ^ .5
205 REM
291 IF X(1) < 0 THEN 1670
293 IF X(2) < 0 THEN 1670
301 IF X(3) < 0 THEN 1670
463 PD1 = X(1) * X(2) * X(3)
466 P = PD1
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
1889 IF M < 28.612 THEN 1999
1899 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [102]. The complete output of one run through -31245 is shown below:
2.918253 4.12 2.38 28.61522 -31715
2.900681 4.11 2.4 28.61232 -31338
2.898862 4.13 2.39 28.61379 -31245
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM, 64-bit Operating System, and QB64v1000-win [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31245 was 77 seconds, counting from “Starting program…”. One can compare the computational results above with those in Bunday [15, p. 126, Exercises 6, 10].
References
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?
[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.)
[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
[14] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, Inc., 1968.
[15] Brian D. Bunday (1984), Basic Optimisation Methods. London: Edward Arnold (Publishers) Ltd., 1984.
[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
[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] H.-L. Li, H.-C. Lu, Global optimization for generalized geometric programs with mixed free-sign variables, Operations Research, 57 (2009), 701-713.
[52] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization (2005) 33:1-13.
[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 ans 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] 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, Han-Lin Li, Nian-Ze Hu (2002). Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 34:6, 613-622.
[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 (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.
[90] Jung-Fa Tsai, Ming-Hua Lin (2013). An improved framewpork for solving NLIPs with signomial terms in the objective or constraints to global optimality, Computers and Chemical Engineering 53 (2013) 44-54.
[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] 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.
[94] 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.
[95] 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.
[96] 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.
[97] 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
[98] 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.
[99] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Journal of Chemical Engineering 28 (1):32-40 January 2011.
[100] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[101] Eric W. Weisstein, “Diophantine Equation–10th Powers.” https://mathworld.wolfram.com/DiophantineEquation10thPowers.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] Helen Wu, (2015), Geometric Programming
https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[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.
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