Solving Another System of Nonlinear Equations
Jsun Yui Wong
The computer program listed below seeks to solve the following system of four nonlinear equations:
-1 / (X(1) ^ 2 * X(2) * X(3)) + X(4) * (X(2) + 2 * X(3)) =0,
X(3) - 1 / (X(1) * X(2) ^ 2 * X(3)) + X(4) * X(1) =0,
X(2) - 1 / (X(1) * X(2) * X(3) ^ 2) + 2 * X(4) * X(1) =0,
X(1) *X(2) + X(1) * X(3)= 4,
where X(i) not=0, i=1, 2, 3, 4.
The nonlinear system above is from Broj 1 [15].
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32503), A(32503)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 4
94 A(KK) = RND * 2
95 NEXT KK
128 FOR I = 1 TO 1000000
129 FOR K = 1 TO 4
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 4)
183 R = (1 - RND * 2) * A(B)
187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
191 NEXT IPP
193 FOR J44 = 1 TO 4
195 REM IF X(J44) > 1 THEN 1670
197 IF X(J44) = 0## THEN 1670
199 NEXT J44
444 X(1) = 4 / (X(2) + X(3))
1114 P = -ABS(-1 / (X(1) ^ 2 * X(2) * X(3)) + X(4) * (X(2) + 2 * X(3))) - ABS(X(3) - 1 / (X(1) * X(2) ^ 2 * X(3)) + X(4) * X(1)) - ABS(X(2) - 1 / (X(1) * X(2) * X(3) ^ 2) + 2 * X(4) * X(1))
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 4
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1900 IF M < -.00001 THEN 1999
1912 PRINT A(1), A(2), A(3)
1915 PRINT A(4), M, JJJJ
1939 REM PRINT A(32500), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [121]. The algorithm also works with Microsoft's GW-BASIC. The complete output of one run through JJJJ= -31992 is shown below:
2.93505146152071 .9085587064128428 .4542794081623334
.1547788670805114 -1.457819246802731D-06 -31993
2.935064040914687 .9085548490572718 .4542774245286343
.1547815517379441 -5.05875223751084D-06 -31992
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [121]. The wall-clock time (not CPU time) for obtaining the output shown above was 15 seconds, counting from “Starting program…”.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[0] M. Abdollahi, A. Isazadeh, D. Abdollahi (2013), Imperialist competitive algorithm for solving systems of nonlinear equations, Computers and Mathematics with Applications, vol. 65, number 12, pp. 1894-1908, 2013. (One can directly read this on Google.)
[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] Radwan Alfahel (10 December 2020), Solving 8 equations with 8 unknowns using FSOLVE
[4] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Anonymous (https://math.stackexchange.com/users/217907/anonymous), Finding x in an Olympiad simultaneous equation, URL (version: 2015-09-07): https://math.stackexchange.com/q/1425006
[9] 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.
[10] 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.
[11] 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.
[12] 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.
[13] Madhulima Bhandari (24 February 2015), How to solve nonlinear coupled equations with 6 unknowns by MATLAB?
mathworks.com/mathcentral/answers/180104-how-to-solve-nonlinear-coupled-equations-with-6-unknowns-by-matlab
[14] Victor Blanco, Justo Puerto (2011). Some algebraic methods for solving multiobjective polynomial integer programs, Journal of Symbolic Computation, 46 (2011), pp. 511-533.
[15] Broj 1 (asked September 1, 2021), How to solve the following system of 4 nonlinear equations.
https://math.stackexchange.com
[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] M. Chen (asked July 23 2017), Number of integer solution of system of nonlinear equations
[19] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[20] Ward Cheney, David Kincaid (2008), Numerical Mathematics and Computing, Sixth Edition, 2008, Thomsom Brooks/Coles.
[21] 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.
[22] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[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] Michael B. Cutlip, Mordechai Shacham (1998), POLYMATH verssion 4.1 user-friendly numerical analysis programs. http://www.polymath-software.com.
[25] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[26] 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
[27] 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.)
[28] 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.
[29] Wassila Drici, Mustapha Moulai (2019): An exact method for solving multi-objective integer indefinite quadratic programs, Optimization Methods and Software.
[30] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).
[31] Albus Dumbledore (asked September 4 2020), Finding all positive integers a, b, c, x, y, z satisfying a+b+c=xyz and x+y+z=abc.
https://math.stackexchange.com
[32 ] J. Douglas Faires, Richard L. Burden (1993), Numerical Methods, PWS-Kent Publishing Company, Boston, 1993.
[33] Farid (3 November 2013), How to solve a system of nonlinear equations with three unknowns and three equations?
[34] 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.
[35] 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.
[36] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.
[37] C. A. Floudas(1995), Nonlinear and mixed-integer optimization: fundamentals and applications. Oxford University Press, 1995.
[38] Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More, HYBRD1, Documentation for MINPACK Subroutine HYBRD1, Double Precision Version, Argonne National Laboratory, March 1980. One can directly view this on Google.
[39] Crina Grosan, Ajith Abraham (2008), A New Approach for Solving Nonlinear Equations Systems. IEEE Transactions on Systems, Man and Cybernetics–Part A: Systems and Humans, Volume 38, Number 3, May 2008, pp. 698-714.
[40] Crina Grosan, Ajith Abraham (2008), Multiple Solutions for a System pf Nonlinear Equations. International Journal of Innovative Computing, Information and Control, Volume x, Number x, x 2008.
[41] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf
[42] GUO Chuang-xin, HU Jia-sheng, YE Bin, CAO Yi-jia (2004), Swarm intelligence for mixed-variable design optimization, J. of Zhejiang University SCIENCE 2004 5(7):851-860. One can directly read this on Google.
[43] 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.
[44] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
[45] HAlex (July 5 2018), Solve system of non-linear equations
stackoverflow.com/users/10037155/halex
(One can directly read this on Google.)
[46] 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.
[47] Frederick S. Hillier, Gerald J. Liberman (1990), Introduction to Mathematical Programming, McGraw-Hill Publishing Company, 1990.
[48] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.
[49] 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).
[50] 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
[51] 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.
[52] 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.
[53] 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.
[54] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016). 29, 2016. https://doi:org/10.1063/1.4942987.
[55] 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.
[56] 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.
[57] Angel Kuri (2003). Solution of Simultaneous Non-Linear Equations using Genetic Algorithms. WSEAS Transactions on Systems, January 2003.
(One can directly read this article on Google.)
[58] William La Cruz, Jose Mario Martinez, Marcos Raydan (July 2004). Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004, pp. 1-37.
(One can directly read this article on Google.)
[59] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[60] 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.
[61] Sven Leyffer (December 1993) Deterministic Methods for Mixed Integer Nonlinear Progrmming, PhD Thesis, Department of Mathematics and Computer Science, University of Dundee, Dundee, December 1993. (One can directly read this on Google.)
[62] S. Leyffer, J. Linderoth, J. Luedtke, A. Miller, T. Munson (2009), Applications and algorithms for mixed integer nonlinear programming, J. of Physics: Conference Series 180 (2009) 012014, IOP Publishing. (One can directly read this on Google.)
[63] 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.
[64] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[65] P. Liu, M. J. Lewis (2002), Communication aspects of an asynchronous parallel evolutionary algorithmm, Proceedings of the third international conferenceon communication in in computing (CIC 2002),pp. 190-195, Las Vegas, NV, June 24-27, 2002,. http://grid.cs.binghamton.edu/papers/LiuAPEAComm CIC.pdf.
[66] 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.
[67] 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.
[68] 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
[69] 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. (Table 1 eggholder (dimension=15), p.3, p. 4 of 62.)
[70] 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.
[71] Z. Michalewicz, M. Schoenauer (1996 March), Evolutionary algorithms for constrained parameter optimization problems, Evolutionary Computation, volume 4(1), pp. 1-32, March 1966.
(One can directly read this article on Google Scholar.)
[72] 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.
[73] SK Mishra (2013) Global optimization of some difficult benchmark functions by host-parasite co-evolutionary algorithm, Economic Bulletin, 33 (0:1-18 (2013), electronic copy available at http://ssrn.com/abstract=2343448. Table 4, Eggholder (dimension=15) , p.10, p. 10 of 100.
[74] SK Mishta (2012) Global optimization of some difficult benchmark functions by cuckoo-host co-evolution meta-heuristics. MPRA paper no. 40615, posted 12 August 2012 23:05 UTC. Online at https://mpra.ub.uni-muenchen.de/40615/. Table 1, Eggholder (dimension 15), p. 3, p. 4 of 62.
[75] SK Mishra (June 2007), Minimization of Keane's bump function by repulsive particle swarm and the differential evolution methods, SSRN Electronic Journal, June 2007.
(One can directly read this article on Google.)
[76] MJTHDSN (12 april 2018)
https://www.mathworks.com/matlabcentral/profile/authors/4812205
Please see also Madhulima Bhandari (24 February 2015), How to solve nonlinear coupled equations with 6 unknowns by MATLAB?
mathworks.com/mathcentral/answers/180104-how-to-solve-nonlinear-coupled-equations-with-6-unknowns-by-matlab.
[77] Joaquin Moreno, Miguel Lopez, Raquel Martinez (2018) A new method for solving nonlinear systems of equations that is based on functional iterations, Open Physics 2018; 16: 605-630.
[78] 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.
[79] The Netlib (July 11, 2015) 8.2 Solve System of Nonlinear Equations
http://www.netlib.org/math/docpdf/ch08-02.pdf
[80] 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.
[81] 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.
[82] 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.
[83] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[84] 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.
[85] Piyu (9 July 2017), Solving five nonlinear equations
mathworks.com/matlabcentral/answers/347947-solving-five-nonlinear-equations#accepted_answers_273454
[86] Bindhu PR on 7 April 2020, i need to solve 7 nonlinear equations with 7 unknows
[87] 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.
[88] 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.
[89] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[90] 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.
[91] 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.
[92] 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).
[93] Mordechai Shacham (1986). Numerical solution of constrained nonlinear algebraic equations, International Journal for Numerical Methods in Engineering, Vol. 23, pp. 1455-1481 (1986). (One can read this via Google.)
[94] M. Shacham, R. S. H. Mah (1978). A Newton Type Linearization Method for Solution of Nonlinear Equations, Computers and Chemical Engineering, Vol. 2, pp. 64-66 (1978). (One can read this via Google.)
[95] Yi Shang (1997), Global search methods for solving nonlinear optimization problems, Ph. D. Thesis, University of Illinois at Urbana-Champaign, 1997.
[96] 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.
[97] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[98] K. A. Sidarto, Adhe Kania (2015), Finding all solutions of systems of nonlinear equations using spiral dynamics inspired optimization with clustering,
Journal of Advanced Computational Intelligence and Intelligent Informatics, pp. 697-708, vol. 19, number 5, 2015. (One can directly read this via Google.)
[99] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[100] Snivysteel (June 5, 2014), Math Olympiad Algebra Question
math.stackexchange.com/users/146944/snivysteel
[101] D.G. Sotiropoulos, J. A. Nikas, T. N. Grapsa, Improving the efficiency of a polynomial system solver via a reordering technique, Technical Report No. TR02-03, University of Patras, Department of Mathematics
(One can directly read this article on Google.)
[102] StackExchange.com. Solving nonlinear system with three equations and three variables.
[103] 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
[104] 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.
[105] 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.
[106] 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
[107] 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.
[108] 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.
[109] 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.
[110] 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.
[111] 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.
[112] 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
[113] 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.
[114] 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.
[115] Eric W. Weisstein, “Diophantine Equation–8th Powers.” https://mathworld.wolfram.com/DiophantineEquation8thPowers.html.
[116] Eric W. Weisstein, “Euler’s Sum of 0wers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[117] Eric W. Weisstein, “Diophantine Equation–5th Powers.” https://mathworld.wolfram.com/DiophantineEquation5thPowers.html.
[118] Eric W. Weisstein, “Diophantine Equation–10th Powers.” https://mathworld.wolfram.com/DiophantineEquation10thPowers.html.
[119] Eric W. Weisstein, “Diophantine Equation–9th Powers.” https://mathworld.wolfram.com/DiophantineEquation9thPowers.html.
[120] Rick Wicklin (2018), Solving a system of nonlinear equations with SAS. blogs.sas.com>iml>2018/02/28. (One can directly read this on Google.)
[121] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[122] Jsun Yui Wong (2021, May 3). Solving Another Big Instance (n, m, Dimensions=1200) of the Keane Benchmark Test Problem. https://nonlinearintegerprogrammingsolver.blogspot.com/2021/05/solving-another-big-instance-n-m.html
[123] 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.
[124] Xin-She Yang, Test problems in optimization, arXiv, 3 August 2010, in Engineering Optimization: An Introduction with Metaheuristic Applications (Eds Xin-She Yang), John Wiley and Sons, (2010).