Solving Another Big Instance (n, m, Dimensions=1200) of the Keane Benchmark Test Problem

 

 

 

Solving Another Big Instance (n, m, Dimensions=1200) of the Keane Benchmark Test Problem    

 

Jsun Yui Wong

 

The computer program listed below seeks to solve the following big instance from Michalewicz and Schoenauer [72, p. 7/32, MIT Press version, here with n=1200] and from Mishra [74, here with dimensions=1200]:

    

maximize               ABS((SUM1 - 2 * PRO1) / ((SUM2) ^ .5))

 

where SUM1, PRO1, and SUM2 are defined below (PRO stands for product)

 

subject  to      

 

 X(1) * X(2) * X(3) * X(4) * X(5) * X(6) * X(7) * X(8) * X(9) * X(10)*...*X(1200)          >=.75

 

 X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) +...+X(1200)   <= 7.5 * 1200   

 

0<=  X(i) <=10, i=1, 2,..., 1200. 

 

One notes line 181, which is 181 IF PRO22 < .75 THEN 1670.

One also notes line 172, which is 172 X(j) = A(j - 1) - RND ^ (RND * 12), which is a possible implementation of a conjecture of Mishra [74, p. 5, p. 7 of 13].  Mishra's conjecture is also the basis of line 22, which is 22 A(J44) = A(J44 - 1) - RND * .001 * A(J44 - 1), the starting values.

 

0 DEFDBL A-Z

 

2 REM    DEFINT K

 

3 DIM B(99), N(99), A(100255), H(99), L(99), U(99), X(100250), D(111), P(511), PS(33), J(30003), J44(30003), KKQQ(30003), KLX(30003), W(10111)

 

12 FOR JJJJ = -32000 TO 32000 STEP .01

 

    13 RANDOMIZE JJJJ

 

    16 M = -1D+37

 

    17 A(0) = 4 + RND * 3

 

    18 FOR J44 = 1 TO 1200

 

        22 A(J44) = A(J44 - 1) - RND * .001 * A(J44 - 1)

 

    25 NEXT J44

 

    29 A(1) = 6.5

 

    128 FOR I = 1 TO 10000

 

        129 FOR KKQQ = 1 TO 1200

 

            130 X(KKQQ) = A(KKQQ)

 

        131 NEXT KKQQ

 

        135 FOR IPP = 1 TO (1 + FIX(RND ^ 5 * 5))

 

 

            148 j = 1 + FIX(RND * 1210)

 

 

            153 REM    GOTO 162

 

            154 IF RND < .333 THEN GOTO 156 ELSE IF RND < .5 THEN GOTO 162 ELSE GOTO 172

 

 

 

            156 REM

 

            157 R = (1 - RND * 2) * (A(j))

 

            160 X(j) = A(j) + (RND ^ (RND * 30)) * R

 

            161 GOTO 174

 

            162 REM  

 

            164 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * .3) ELSE X(j) = A(j) + FIX(1 + RND * .3)

 

            166 GOTO 174

 

            172 X(j) = A(j - 1) - RND ^ (RND * 12)

 

        174 NEXT IPP

 

        175 PRO22 = 1

 

        176 FOR J44 = 1 TO 1200

 

            177 PRO22 = PRO22 * X(J44)

 

        178 NEXT J44

 

        180 REM   X(1) = .75 / PRO22

        181 IF PRO22 < .75 THEN 1670

        182 SUM21 = 0

        184 FOR J44 = 1 TO 1200

            186 SUM21 = SUM21 + X(J44)

        188 NEXT J44

 

        191 REM

 

        193 IF SUM21 > 7.5 * 1200 THEN 1670

 

 

 

        195 FOR J44 = 1 TO 1200

 

            197 IF X(J44) < 0 THEN 1670

 

            198 IF X(J44) > 10 THEN 1670

 

        204 NEXT J44

 

        243 SUM1 = 0

 

        244 FOR J44 = 1 TO 1200

 

 

            249 SUM1 = SUM1 + (COS(X(J44))) ^ 4

 

        259 NEXT J44

 

        263 SUM2 = 0

 

        264 FOR J44 = 1 TO 1200

 

            267 SUM2 = SUM2 + J44 * X((J44)) ^ 2

 

        269 NEXT J44

 

        363 SUM3 = 0

 

        364 FOR J44 = 1 TO 1200

 

            367 SUM3 = SUM3 + X((J44))

 

        369 NEXT J44

 

        543 PRO1 = 1

 

        544 FOR J44 = 1 TO 1200

 

 

            549 PRO1 = PRO1 * (COS(X(J44))) ^ 2

 

        559 NEXT J44

 

        663 PRO2 = 1

 

        664 FOR J44 = 1 TO 1200

 

            667 PRO2 = PRO2 * X(J44)

 

        669 NEXT J44

 

        928 P = ABS((SUM1 - 2 * PRO1) / ((SUM2) ^ .5))

 

        1111 IF P <= M THEN 1670

 

        1452 M = P

 

        1454 FOR KLX = 1 TO 1200

 

            1459 A(KLX) = X(KLX)

 

        1460 NEXT KLX

        1522 REM     PRINT A(1), M, JJJJ

        1524 REM  PRINT A(1), A(2), A(3), A(4), A(5), A(6)

        1529 REM   PRINT A(1197), A(1198), A(1199), A(1200), M, JJJJ

 

        1557 GOTO 128

 

    1670 NEXT I

 

    1671 REM  IF M < .852 THEN GOTO 1999

 

    1900 REM PRINT A(1), A(2), A(3)

 

    1901 GOTO 1914

 

    1903 PRINT A(6), A(7), A(8), A(9), A(10)

    1907 PRINT A(11), A(12), A(13), A(14), A(15)

    1911 PRINT A(16), A(17), A(18), A(19), A(20)

    1913 REM   PRINT A(21), A(22), A(23), A(24), A(25)

    1914 PRINT A(1), M, JJJJ

1999 NEXT JJJJ

 

 

This BASIC computer program was run with qb64v1000-win [120].  The output of one run through JJJJ=-31999.94000000001 is summarized below:

  

6.282312902744658                 .8586113416788073         -32000    

6.282331266411133                 .7902501566926868         -31999.99

6.282340625831971                 .764690084816953           -31999.98

9.423532043980726                 .7613485497691257         -31999.97000000001

........................                       .8455172668259275         -31999.96000000001

........................                       .750052944850952           -31999.95000000001

........................                       .8773641776234751        -31999.94000000001  

 

Only one (A(1)) of the 1200 values of A(1) through A(1200) is shown above, in accordance with line 1914.  The best M value shown above is .8773641776234751.

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 [120].  The wall-clock time (not CPU time) for obtaining the output shown above was 13 hours, counting from “Starting program…”. 

Acknowledgement

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

References

[1]  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.)

[2] 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.

[3] 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.

[4]  A.R. Al-Roomi (2015).  Unconstrained single-objective benchmark functions repository.  https://www.al-roomi.org/benchmarks/unconstrained.

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

[6] 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.

[7] 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.

 

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

 

[9]  George Anescu (2016), Gradual and cumulative improvements to the classical differential evolution scheme through experiments, Analele  Universitatii de Vest, Timisoara, Seria Mathematica–Informatica, Vol. LIV, 2, (2016), 13-35.   (One can directly read this on Google.)

 

[10] 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.

 

[11] 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.

 

[12] 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.

 

[13] 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.

 

[14]   H. Bernau (1990).  Active constraint strategies in optimization.  Geographical Data Inversion Methods and Applications, pp. 15-31.

 

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

 

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

 

[17] Richard L. Burden, Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016, Cengage Learning.

 

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

 

[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]  A. J. Conejo, L. Baringo (2018), Power System Operations, Springer International Publishing, 2018.

 

[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] Michael B. Cutlip, Mordechai Shacham (1998), POLYMATH verssion 4.1 user-friendly numerical analysis programs.  http://www.polymath-software.com.

 

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

 

[27] 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

 

[28] 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.)

 

[29] 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.

 

[30] 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.

 

[31] Wassila Drici, Mustapha Moulai (2019): An exact method for solving multi-objective integer indefinite quadratic programs, Optimization Methods and Software.

 

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

 

[33] 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.

 

[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] 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.

 

[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] 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.

 

[46]  www – optima.amp.i.kyoto – u.ac.jp/member/student/hedar/Hedar_fil…

 

[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, E. A. Khan, M. J. Ahsan (2003). An optimal multivariate stratified sampling design using dynami programming. Australian and New Zealand Journal of Statistics, vol. 45, no. 1, 2003, pp. 107-113.

 

[57] 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.

 

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

 

[59] 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.

 

[60] 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.)

 

[61] 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.)

 

[62] 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.

 

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

 

[64]   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.

 

[65] 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.

 

[66] 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.

 

[67] 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

 

[68] 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.

 

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

 

[71] 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.

 

[72]  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.)

 

[73] 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.

[74]  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.

[75]  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.  

 

[76]  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.)

 

[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] 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.

 

[80] 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.

 

[81] 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.

 

[82] 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/

 

[83] 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.

 

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

 

[85] 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.

 

 

[86] 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.

 

[87] 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.

 

[88] W. Rheinboldt, Some Nonlinear Testproblems. One can google this 10-page paper.

 

[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] 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.

 

[101] Sonja Surjanovic, Derek Bingham (2013), Virtual library of simulation experiments: test functions and datasets. Simon Fraser University, Burnaby, BC, Canada, 2013. (One can directly read this on Google.)

 

[102] 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

 

[103] 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.

 

[104] 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.

 

[105] 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

 

[106] 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.

 

[107] 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.

 

[108] 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.

 

[109] 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.

 

[110] 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.

 

[111] 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

 

[112] 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.

 

[113] 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.

 

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

 

[115] Eric W. Weisstein, “Euler’s Sum of 0wers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

 

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

 

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

 

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

 

[119] 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.)

 

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

 

[121] 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.

 

[122] 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.