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Using Absolute Values to Solve Systems of Nonlinear/Linear Equations for One Solution or Multiple Solutions in a Single Run

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [78, 1993, p. 355].

The computer program listed below seeks to solve simultaneously the following system of nonlinear equations:

(SIN(4 * PI * X(1) * X(2)) – 2 * X(2) – X(1)) =0,

((4 * PI – 1) / (4 * PI) * (E ^ (2 * X(1)) – E) + 4 * E * X(2) ^ 2 – 2 * E * X(1)) =0.

These two equations are from Burden, Faires, and Burden [16, p. 656, Exercise 3b/5b].

One notes line 1135, which is 1135 P = -ABS(SIN(4 * PI * X(1) * X(2)) – 2 * X(2) – X(1)) – ABS((4 * PI – 1) / (4 * PI) * (E ^ (2 * X(1)) – E) + 4 * E * X(2) ^ 2 – 2 * E * X(1)).

0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

83 RANDOMIZE JJJJ

87 M = -4E+299

99 E = 2.718281828

111 PI = 3.141592654

120 FOR J44 = 1 TO 2

121 A(J44) = FIX(RND * 6)

122 REM A(J44) = (RND * 5)

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 100000)

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 j = 1 + FIX(RND * 2)

154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162

156 REM

157 R = (1 – RND * 2) * (A(j))

160 X(j) = A(j) + (RND ^ (RND * 15)) * R

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) – FIX(1 + RND * 1.3) ELSE X(j) = A(j) + FIX(1 + RND * 1.3)

169 NEXT IPP

1135 P = -ABS(SIN(4 * PI * X(1) * X(2)) – 2 * X(2) – X(1)) – ABS((4 * PI – 1) / (4 * PI) * (E ^ (2 * X(1)) – E) + 4 * E * X(2) ^ 2 – 2 * E * X(1))

1221 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1557 GOTO 128

1670 NEXT I

1894 IF M < -.00001 THEN 1999

1923 PRINT A(1), A(2), M

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [104]. Selected candidate solutions of a single run through JJJJ= -31916 are shown below:

1.033071472054275 -.2799618358382454 -5.37330596683816D-16

-31981

-.3736982167411166 5.626648942274948D-02 -6.678685382510707D-17

-31979

.4080956624125035 -.4926293940538585 -2.16840434497101D-16

-31974

.1478392372216056 -.4361776221953876 -5.573883368747978D-16

-31916

One notes the four distinct solutions shown above.

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 (R) Core (TM) 2 Duo CPU E8400 @ 3.0 GHz 3.0 GHz, 4.00 GB of RAM (3.9 GB usable), 64-bit Operating System, and QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31916 took 7 seconds, counting from “Starting program…”. One can compare the computational results above with those in Burden, Faires, and Burden [16, p. 872, Exercise Set 10.2, Exercise 5b].

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] Miguel F. Anjos (2012), FLPLIB–Facility Layout Database. Retrieved on September 25 2012 from http://www.gerad.ca/files/Sites/Anjos/indexFR.html

[10] 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.

[11] 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.

[12] H. Bernau (1990). Active constraint strategies in optimization. Geographical data inversion methods and applications, pp. 15-31.

[13] Victor Blanco, Justo Puerto (2011). Some algebraic methods for solving multiobjective polynomial integer programs, Journal of Symbolic Computation, 46 (2011), pp. 511-533.

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

[15] Regina S. Burachik, C. Yalcin Kaya, M. Mustafa Rizvi (March 19, 2019), Algorithms for Generating Pareto Fronts of Multi-objective Integer and Mixed-Integer Programming Problems, arXiv: 1903.07041v1 [math.OC] 17 Mar 2019.

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

[19] 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.

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

[21] 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.

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

[23] Pintu Das, Tapan Kumar Roy (2014). Multi-objective geometric programming and its application in gravel box problem. Journal of global research in computer science volume 5. no.7, July 2014. http://www.jgrcs.info

[24] 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.)

[25] 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.

[26] Anoop K. Dhingra (1992). Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

[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] 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.

[30] 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.

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

[33] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012. Retrieved on September 14 2012 from Google search.

[34] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[35] Neha Gupta, Irfan Ali, Abdul Bari (2013). An optimal chance constraint multivariate stratified sampling design using auxiliary infotmation. Journal of Mathematical Modelling and Algorithms, January 2013.

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

[37] M. Hashish, M. P. duPlessis (1979). Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses. Transactions of ASME: Journal of Engineering for Industry 101 (1979) 311-318.

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

[39] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from http://www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[40] 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).

[41] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout. Retrieved from http://www.optimization-online.org./DB_HTML/2012/04/3432.html

[42] Golam Reza Jahanshahloo, Farhad Hosseinzadeh, Nagi Shoja, Ghasem Tohidi (2003) A method for solvong 0-1 multiple objective liner programming problem using DEA. Journal of the Operations Research Society of Japan (2003), 46 (2): 189-202. http://www.orsj.or.jp/~archive/pdf/e_mag/Vol.46_02_189.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, E. A. Khan, M. J. Ahsan (2003). An optimal multivariate stratified sampling design using dynamic programming. Australian and New Zealand Journal of Statistics, vol. 45, no. 1, 2003, pp. 107-113.

[49] 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.

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

[51] 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.

[52] 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.

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

[54] 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] Rein Luus (1975). Optimization of System Reliability by a New Nonlinear Integer Programming Procedure. IEEE Transactions on Reliability, Vol. R-24, No. 1, April 1975, pp. 14-16.

[57] Milos Madic, Miroslav Radovanovic (2014). Optimization of machining processes using pattern search algorithm. International Journal of Industrial Engineering Computations 5 (2014) 223-234. Homepage: http://www.GrowingScience.com/ijiec

[58] F. Masedu, M Angelozzi (2008). Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming. Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.

[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 (2016). Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic. Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.

[61] 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.

[62] 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.

[63] A. Moradi, A. M. Nafchi, A. Ghanbarzadeh (2015). Multi-objective optimization of truss structures using the bee algorithm. (One can read this via Goodle search.)

[64] Joaquin Moreno, Miguel Lopez, Raquel Martinez (2018) a new method for solving nonlinear systems of equations that is based on functional iterations , Open Physics, Vlume 16, Issue 1, Published online 22 October 2018.

[65] 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.

[66] 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.

[67] 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.

[68] 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.

[69] 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.

[70] Rashmi Ranjan Ota, A. K. Ojha (2015). A comparative study on optimization techniques for solving multi-objective geometric programming problems. Applied mathematical sciences, vol. 9, 2015, no. 22, 1077-1085. http://sites.google.com/site/ijcsis/

[71] 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.

[72] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

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

[74] 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.

[75] Ciara Pike-Burke. Multi-Objective Optimization. https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.

[76] 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.

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

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

[79] 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.

[80] 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.

[81] 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).

[82] 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.

[83] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[84] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.

[85] Isaac Siwale. A Note on Multi-Objective Mathematical Problems. https://www.aptech.com/wp-content/uploads/2012/09/MultObjMath.pdf

[86] 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.

[87] 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

[88] 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.

[89] 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.

[90] 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

[91] 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.

[92] 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.

[93] 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.

[94] 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.

[95] 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.

[96] 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

[97] 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.

[98] 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.

[99] Eric W. Weisstein, “Diophantine Equation–8th Powers.” https://mathworld.wolfram.com/DiophantineEquation8thPowers.html.

[100] Eric W. Weisstein, “Euler’s Sum of Powers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

[101] Eric W. Weisstein, “Diophantine Equation–5th Powers.” https://mathworld.wolfram.com/DiophantineEquation5thPowers.html.

[102] Eric W. Weisstein, “Diophantine Equation–10th Powers.” https://mathworld.wolfram.com/DiophantineEquation10thPowers.html.

[103] Eric W. Weisstein, “Diophantine Equation–9th Powers.” https://mathworld.wolfram.com/DiophantineEquation9thPowers.html.

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

[105] Jsun Yui Wong (2014, March 13). The Domino Method Applied to a System of Four Simultaneous Nonlinear Equations. Retrieved from http://myblogsubstance.typepad.com/substance/2014/03.

[105] Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990). Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

Featured

Solving Another 2-Objective Integer Nonlinear Programming Problem with the Epsilon-Constraint Method

Jsun Yui Wong

The computer program listed below seeks to solve the following 2-objective integer nonlinear programming problem from Sharma, Dahiya, and Verma [54, p. 1924, Example 1]:

Miniimize – (-2 * X(1) ^ 2 – 4 * X(1) * X(2) + X(1) – X(2)) / (2 * X(1) * X(2) + X(2) ^ 2 + X(1))

minimize (X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2))

subject to

X(1) >= 0 and integer

X(2) >= 0 and integer

3 * X(1) + 2 * X(2) >= 6

4 * X(1) + 5 * X(2) <= 20.
0 DEFDBL A-Z

1 DEFINT K

81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ

86 M = -3E+50

88 epsi = RND * 10

92 A(1) = (RND * 14)

93 A(2) = (RND * 14)
128 FOR I = 1 TO 3000

129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 J = 1 + FIX(RND * 2)
154 REM GOTO 162

156 r = (1 – RND * 2) * A(J)

158 X(J) = A(J) + (RND ^ (RND * 15)) * r

161 REM GOTO 169

162 REM IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1

169 NEXT IPP

172 X(1) = INT(X(1))
174 X(2) = INT(X(2))

188 IF X(1) < 0## THEN 1670

189 IF X(2) < 0## THEN 1670
226 IF 3 * X(1) + 2 * X(2) < 6 THEN 1670

227 IF 4 * X(1) + 5 * X(2) > 20 THEN 1670
228 IF X(1) ^ 2 + X(1) * X(2) + X(2) = 0## THEN 1670

229 IF (X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)) > epsi THEN 1670

431 PDU = (-2 * X(1) ^ 2 – 4 * X(1) * X(2) + X(1) – X(2)) / (2 * X(1) * X(2) + X(2) ^ 2 + X(1))

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)
1456 NEXT KLX

1527 gg01star = (-2 * X(1) ^ 2 – 4 * X(1) * X(2) + X(1) – X(2)) / (2 * X(1) * X(2) + X(2) ^ 2 + X(1))

1529 gg02star = (X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2))
1557 GOTO 128
1670 NEXT I

1889 IF M < -99999999999 THEN 1999

1924 PRINT -gg01star, gg02star, epsi

1956 PRINT A(1), A(2), -M, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [60]. The output of a single run through JJJJ= -31980 is summarized below:

.25 4 9.554092884063721
0 4 .25 -31999

3 1 3.04885506629944
2 0 3 -31995

2.142857142857143 1.285714285714286 4.794933199882507
2 1 2.142857142857143 -31994

.3333333333333333 3 3.203887939453125
0 3 .3333333333333333 -31993

2.142857142857143 1.285714285714286 5.443581342697144
2 1 2.142857142857143 -31992

3 1 9.825533032417297
2 0 3 -31990

.25 4 8.194036483764648
0 4 .25 -31989

2.142857142857143 1.285714285714286 8.698806166648865
2 1 2.142857142857143 -31988

3 1 3.1596546649933
2 0 3 -31987

2.142857142857143 1.285714285714286 4.93649301528931
2 1 2.142857142857143 -31985

.25 4 8.634269833564758
0 4 .25 -31983

3 1 9.953094720840454
2 0 3 -31982

.25 4 4.089516997337341
0 4 .25 -31981

.3333333333333333 3 3.526153564453125
0 3 .3333333333333333 -31980

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 [60], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31980 was 1 second, not including the time for “Creating .EXE file” (8 seconds, total, including the time for “Creating .EXE file”). The (-1 3) shown above is a dominated point. One can compare the computational results above with those in Sharma, Dahiya, and Verma [54, pp. 1924-1926, Example 1].
Acknowledgment

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

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] 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.

[3] 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.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] 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.

[6] Miguel F. Anjos (2012), FLPLIB–Facility Layout Database. Retrieved on September 25 2012 from http://www.gerad.ca/files/Sites/Anjos/indexFR.html

[7] 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.

[8] Ritu Arora, S. R. Arora (2014). 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.

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

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

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

[12] 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.

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

[14] 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.

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

[16] Pintu Das, tapan kumar Roy (2014). Multi-objective geometric programming and its application in gravel box problem. Journal of global research in computer science volume 5. no.7, july 2014. http://www.jgrcs.info

[17] 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.

[18] Anoop K. Dhingra (1992). Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

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

[20] 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.

[21] 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.

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

[23] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012. Retrieved on September 14 2012 from Google search.

[24] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[25] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[26] M. Hashish, M. P. duPlessis (1979). Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses. Transactions of ASME: Journal of Engineering for Industry 101 (1979) 311-318.

[27] David M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[28] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[29] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from http://www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[30] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout. Retrieved from http://www.optimization-online.org./DB_HTML/2012/04/3432.html

[31] 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.

[32] 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.

[33] 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.

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

[35] 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.

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

[37] 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.

[38] Rein Luus (1975). Optimization of System Reliability by a New Nonlinear Integer Programming Procedure. IEEE Transactions on Reliability, Vol. R-24, No. 1, April 1975, pp. 14-16.

[39] Milos Madic, Miroslav Radovanovic (2014). Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234. Homepage: http://www.GrowingScience.com/ijiec

[40] F. Masedu, M Angelozzi (2008). Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming. Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.

[41] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html

[42] 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.

[43] Mohamed Arezki Mellal, Edward J. Williams (2016). Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic. Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.

[44] 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.

[45] 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.

[46] 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.

[47] 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.

[48] Rashmi Ranjan Ota, A. K. Ojha (2015). A comparative study on optimization techniques for solving multi-objective geometric programming problems. Applied mathematical sciences, vol. 9, 2015, no. 22, 1077-1085. http://sites.google.com/site/ijcsis/

[49] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

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

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

[52] 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.

[53] 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.

[54] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[55] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.

[56] 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.

[57] 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.

[58] 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

[59] 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.

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

[61] Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990). Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

June 26, 2024 by 123writereadmath

Computer program for solving signomial mixed-integer nonlinear programming problems with free variables

Computer program for solving signomial mixed-integer nonlinear programming problems with free variables
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following signomial programming problem from Tsai and Lin [89, Example 2 on p. 47]:

Minimize X(1) ^ .5 * X(2) + 3 * LOG(X(1))
subject to
-X(1) + X(2) <= 5
X(1) ^ .5 – X(2) <= 6
X(1) epsilon { .1, .5, .7, 1.2 }
-6 <= X(2) <= 4.

119 A(1) = .1 + (RND * 1.1)

121 A(2) = -6 + (RND * 10)

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)

        140 B = 1 + FIX(RND * 2)

        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

    181 IF RND < .25 THEN X(1) = .1 ELSE IF RND < .3333 THEN X(1) = .5 ELSE IF RND < .5 THEN X(1) = .7 ELSE X(1) = 1.2

    376 IF -X(1) + X(2) > 5 THEN 1670

    379 IF X(1) ^ .5 - X(2) > 6 THEN 1670

    441 IF X(1) < .1 THEN 1670
    442 IF X(1) > 1.2 THEN 1670

    444 IF X(2) < -6 THEN 1670
    445 IF X(2) > 4 THEN 1670

    565 REM   
    566 PD1 = -X(1) ^ .5 * X(2) - 3 * LOG(X(1))
    569 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
1775 IF M < -9999999 THEN 1999
1904 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= -31995 is shown below. GW-BASIC also can handle this computer program.

.1 -5.683767 8.70512 -32000
.1 -5.683772 8.705122 -31999
.1 -5.683772 8.705122 -31998
.1 -5.68377 8.705121 -31997
.1 -5.683772 8.705122 -31996
.1 -5.683772 8.705122 -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 [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 2 seconds, counting from “Starting program…”. One can compare the computational results presented above with the computational results in Tsai and Lin [89, Example 2 and Table 1 on page 47].
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

[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] 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] 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.

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

June 25, 2024 by 123writereadmath

Computer program for solving signomial mixed-integer nonlinear programming problems with free variables

Minimize
(X(1) ^ 2 * X(2) ^ -2 * X(3) – 2 * X(2) ^ .7 * X(3) ^ .2 + X(4) * X(5) ^ -2 – 2 * X(1) – 4 * X(3))
subject to
X(1) + 6 * X(2) – X(3) – 5 * X(4) <= 2
X(3) ^ 1.5 * X(4) + .5 * X(2) + 3 * X(1) <= -10
-X(1) – .5 * X(4) + X(5) <= 6
-7<= X(1) <=5
1 <= X(2) <= 10
1 <= X(3) <= 5
2 <= X(4) <= 8
2 <= X(5) <= 9
X(1), X(2), X(4), X(5) epsilon R, X(3) epsilon Z.

89 RANDOMIZE JJJJ
90 M = -3D+30
91 A(1) = -7 + RND * 12

92 FOR J44 = 2 TO 5

    94 A(J44) = 1 + RND * 4

96 NEXT J44

128 FOR i = 1 TO 100000

    129 FOR KKQQ = 1 TO 5
        130 X(KKQQ) = A(KKQQ)
    131 NEXT KKQQ
    139 FOR IPP = 1 TO FIX(1 + RND * 5)

        140 B = 1 + FIX(RND * 5)

        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

    171 FOR J44 = 3 TO 3

        172 X(J44) = INT(X(J44))

    173 NEXT J44

    366 IF X(1) + 6 * X(2) - X(3) - 5 * X(4) > 2 THEN 1670

    372 IF X(3) ^ 1.5 * X(4) + .5 * X(2) + 3 * X(1) > -10 THEN 1670

    376 IF -X(1) - .5 * X(4) + X(5) > 6 THEN 1670


    441 IF X(1) < -7 THEN 1670
    442 IF X(1) > 5 THEN 1670

    444 IF X(2) < 1 THEN 1670
    445 IF X(2) > 10 THEN 1670

    446 IF X(3) < 1 THEN 1670

    447 IF X(3) > 5 THEN 1670
    448 IF X(4) < 2 THEN 1670

    449 IF X(4) > 8 THEN 1670

    450 IF X(5) < 2 THEN 1670

    451 IF X(5) > 9 THEN 1670


    564 PD1 = -(X(1) ^ 2 * X(2) ^ -2 * X(3) - 2 * X(2) ^ .7 * X(3) ^ .2 + X(4) * X(5) ^ -2 - 2 * X(1) - 4 * X(3))

    569 p = PD1
    1111 IF p <= M THEN 1670
    1452 M = p
    1454 FOR KLX = 1 TO 5

        1455 A(KLX) = X(KLX)

    1456 NEXT KLX

1670 NEXT i
1775 IF M < -2.908 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31782 is shown below. GW-BASIC also can handle this computer program.

-5.400118338029741 4.63549561117594 1
3.882571065816902 2.541167194268502 -2.906685370935002
-31847
-5.356924592172367 4.554201442494374 1
3.79365681255878 2.539903814078217 -2.905622554601807
-31837
-5.358627122879355 4.557415346016703 1
3.797172990644173 2.539959372442728 -2.905619641192591
-31782

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 [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31782 was 33 seconds, counting from “Starting program…”. One can compare the computational results presented above with the computational results in Tsai and Lin [89, Example 1, pages 45-47; Table 1 on p. 47].
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