Multiple Solutions in One Run for a System of Nonlinear Equations from Neurophysiology

 

 

 

Jsun Yui Wong

1. The Continuous Case

The computer program listed immediately below seeks to solve simultaneously the following system of six nonlinear equations from Grossan and Abraham  [35, p. 709]:

(X(1) ^ 2 + X(3) ^ 2 ) =1,

(X(2) ^ 2 + X(4) ^ 2 ) =1,

(X(5) * X(3) ^ 3 + X(6) * X(4) ^ 3) = 0,

(X(5) * X(1) ^ 3 + X(6) * X(2) ^ 3) = 0,

(X(5) * X(1) * X(3) ^ 2 + X(6) * X(4) ^ 2 * X(2)) =0,

(X(5) * X(1) ^ 2 * X(3) + X(6) * X(2) ^ 2 * X(4)) =0.

 

In this section X(1) through X(6) are continuous variables.

One notes line 1184, which is  1184 P = -ABS(X(1) ^ 2 + X(3) ^ 2 – 1) – ABS(X(2) ^ 2 + X(4) ^ 2 – 1) – ABS(X(5) * X(3) ^ 3 + X(6) * X(4) ^ 3) – ABS(X(5) * X(1) ^ 3 + X(6) * X(2) ^ 3) – ABS(X(5) * X(1) * X(3) ^ 2 + X(6) * X(4) ^ 2 * X(2)) – ABS(X(5) * X(1) ^ 2 * X(3) + X(6) * X(2) ^ 2 * X(4)).

 

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 6

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 6

 

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

 

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

 

143 j = 1 + FIX(RND * 6)

 

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

 

1178 GOTO 1184

 

1179 FOR J44 = 1 TO 6

1181 X(J44) = INT(X(J44))

1183 NEXT J44

 

1184 P = -ABS(X(1) ^ 2 + X(3) ^ 2 – 1) – ABS(X(2) ^ 2 + X(4) ^ 2 – 1) – ABS(X(5) * X(3) ^ 3 + X(6) * X(4) ^ 3) – ABS(X(5) * X(1) ^ 3 + X(6) * X(2) ^ 3) – ABS(X(5) * X(1) * X(3) ^ 2 + X(6) * X(4) ^ 2 * X(2)) – ABS(X(5) * X(1) ^ 2 * X(3) + X(6) * X(2) ^ 2 * X(4))

 

1221 IF P <= M THEN 1670

 

1420 M = P

 

1444 FOR KLX = 1 TO 6

 

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), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [105].  Selected candidate solutions of a single run through JJJJ= -31939 are shown below:

 

-.994204458861615        .48877353033324      .1075057857963154

.872410703766054          8.891327075313985D-34          2.187408564347586D-33

-2.29850860566927D-17     -32000

.8563172758458552       -.9999998983238686       -.5164501167469455

-4.509459529349398D-04          0           2.098943568360641D-33

-4.185020385794047D-17         -31993

.

.

.

-.5518480097968324              -5.301989893305167D-04         .8339447068500884

-.999999859444506      0         0       -4.600052977421498D-15

-31978

.

.

.

-1.776048262739516D-02          -.1384183674040687               .999842270190825

.9903738463656915            0        0                                            -3.518568249261111D-12

-31976

.

.

.

-.9999995303392074         -.99922335527865361        9.691859288078984D-04

.0391446929429354      0      0      0

-31939

 

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a 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 [105], the wall-clock time (not CPU time) for obtaining the output through  JJJJ  = -31939 was 11 seconds, counting from “Starting program…”.

 

2. The Integer Case

The computer program listed immediately below seeks to solve simultaneously the following system of six nonlinear equations from Grosan and Abraham  [35, p. 709]:

(X(1) ^ 2 + X(3) ^ 2 ) =1,

(X(2) ^ 2 + X(4) ^ 2 ) =1,

(X(5) * X(3) ^ 3 + X(6) * X(4) ^ 3) = 0,

(X(5) * X(1) ^ 3 + X(6) * X(2) ^ 3) = 0,

(X(5) * X(1) * X(3) ^ 2 + X(6) * X(4) ^ 2 * X(2)) =0,

(X(5) * X(1) ^ 2 * X(3) + X(6) * X(2) ^ 2 * X(4)) =0.

 

In this section X(1) through X(6) are integer variables.

One notes line 1184, which is  1184 P = -ABS(X(1) ^ 2 + X(3) ^ 2 – 1) – ABS(X(2) ^ 2 + X(4) ^ 2 – 1) – ABS(X(5) * X(3) ^ 3 + X(6) * X(4) ^ 3) – ABS(X(5) * X(1) ^ 3 + X(6) * X(2) ^ 3) – ABS(X(5) * X(1) * X(3) ^ 2 + X(6) * X(4) ^ 2 * X(2)) – ABS(X(5) * X(1) ^ 2 * X(3) + X(6) * X(2) ^ 2 * X(4)).

 

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 6

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

123 NEXT J44

 

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

 

129 FOR KKQQ = 1 TO 6

 

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

 

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

 

143 j = 1 + FIX(RND * 6)

 

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

 

1179 FOR J44 = 1 TO 6

 

1181 X(J44) = INT(X(J44))

1183 NEXT J44

1184 P = -ABS(X(1) ^ 2 + X(3) ^ 2 – 1) – ABS(X(2) ^ 2 + X(4) ^ 2 – 1) – ABS(X(5) * X(3) ^ 3 + X(6) * X(4) ^ 3) – ABS(X(5) * X(1) ^ 3 + X(6) * X(2) ^ 3) – ABS(X(5) * X(1) * X(3) ^ 2 + X(6) * X(4) ^ 2 * X(2)) – ABS(X(5) * X(1) ^ 2 * X(3) + X(6) * X(2) ^ 2 * X(4))

 

1221 IF P <= M THEN 1670

 

1420 M = P

 

1444 FOR KLX = 1 TO 6

 

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), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [105].  The complete output of a single run through JJJJ= -31988 is shown below:

-1    1    0   0     3

3    0    -32000

0    0   1    1   0

0    0    -31999

0    -1   1    0   0

0    0    -31998

0    0   1    -1   0

0    0    -31997

1    1    0   0     -1

1    0    -31996

-1    1    0   0     2

2    0    -31995

0    0   1    -1   0

0    0    -31994

0    -1   1    0   0

0    0    -31993

1   1   0   0   0

0    0         -31992

-1    0    0    -1   0

0    0    -31991

1    0    0    1   0

0    0    -31990

0    0    1   -1     2

2    0    -31989

1    0    0   -1     0

0    0    -31988

 

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a 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 [105], the wall-clock time (not CPU time) for obtaining the output through  JJJJ  = -31988 was 4 seconds, counting from “Starting program…”.

 

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]  Ward Cheney, David Kincaid (2008), Numerical Mathematics and Computing, Sixth Edition, 2008, Thomsom Brooks/Coles.

[20] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

 

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

 

[22] 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.

 

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

 

[24] 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

 

[25] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000). Mechanical component design for multi objectives using elitist non-dominated sorting GA. Proceedings of the Parallel Probl.Solv. Nat. PPSN VI Conference, Paris, France, pp. 859-868 (2000). (Please see pp. 1-10 in Technical Report No. 200002 via Google search.)

 

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.

 

[27] 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.

 

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

 

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

 

[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] 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.

[35]  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.

[36]  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.

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

[38] 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.

 

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

 

[40] 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.

 

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

 

[42] 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

 

[43] 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).

 

[44] 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

 

[45] 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

 

[46] 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.

 

[47] 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.

 

[48] 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.

 

[49] 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.

 

[50] 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.

 

[51] 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.

 

[52] 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.

 

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

 

[54] 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.

 

[55] 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.

 

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

 

[57] 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.

 

[58] 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.

 

[59] 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

 

[60] 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.

 

[61] 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.

 

[62] 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.

 

[63] 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.

 

[64] 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.

 

[64] 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.)

 

[65]  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.

 

[66] 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.

 

[67] 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.

 

[68] 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.

 

[69] 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.

 

[70] 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.

 

[71] 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/

[72] 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.

 

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

 

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

 

[75]  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.

 

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

 

[77] 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.

 

[78] 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.

 

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

 

[80] 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.

 

[81] 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.

 

[82] 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).

 

[83] 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.

 

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

 

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

 

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

 

[87] 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.

 

[88] 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

 

[89] 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.

 

[90] 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.

 

[91] 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

 

[92] 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.

 

[93] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

 

[94] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.

 

[95] Rahul Varshney, Najmussehar, M. J. Ahsan (2012). An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

 

[96] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.

 

[97] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

 

[98] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.

[99] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

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

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

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

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

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

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

[106] 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.

[107] 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.

 

 

Computer Program for Producing in a Single Run Multiple Solutions of Systems of Nonlinear/Linear Equations

 

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 from Grosan and Abraham [35, p. 8 of 12]:

4 * X(1) ^ 3 + 4 * X(1) * X(2) + 2 * X(2) ^ 2 -42*X(1) – 14   =0,

4 * X(2) ^ 3 + 2 * X(1) ^ 2 + 4 * X(1) * X(2) – 26 * X(2) – 22) =0.

One notes line 1192, which is 1192 P = -ABS(4 * X(1) ^ 3 + 4 * X(1) * X(2) + 2 * X(2) ^ 2 – 42 * X(1) – 14) – ABS(4 * X(2) ^ 3## + 2 * X(1) ^ 2## + 4 * X(1) * X(2) – 26 * X(2) – 22) .

 

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

 

1192 P = -ABS(4 * X(1) ^ 3 + 4 * X(1) * X(2) + 2 * X(2) ^ 2 – 42 * X(1) – 14) – ABS(4 * X(2) ^ 3## + 2 * X(1) ^ 2## + 4 * X(1) * X(2) – 26 * X(2) – 22)

 

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

1896 IF M < -.00001 THEN 1999

1927 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [105].  Selected candidate solutions of a single run through JJJJ= -31906 are shown below:

-.2708445906673476             -.9230385564799815             -3.07046055247895D-16

-31999

3          2          0         -31998

3.584428340330492              -1.848126526964404             -2.86923262926564d-15

-31994

-2.805118086952745              3.131312518250573             -7.133182933216631D-15

31993

8.667750455539627D-02       2.884254701174776             -1.755193212993333D-14

-31981

3.385154177794065               7.385192008917117D-02     -5.79301727813425D-07

-31978

-.1279613467280338               -1.953714980414166          – 4.574661343118646D-09

-31906

 

One notes that seven distinct solutions are 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 [105], the wall-clock time (not CPU time) for obtaining the output through  JJJJ  = -31906 was 12 seconds, counting from “Starting program…”.   One can compare the computational results above with those in Grosan and Abraham [35, p. 8 of 12, Table 3].

 

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]  Ward Cheney, David Kincaid (2008), Numerical Mathematics and Computing, Sixth Edition, 2008, Thomsom Brooks/Coles.

[20] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

 

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

 

[22] 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.

 

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

 

[24] 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

 

[25] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000). Mechanical component design for multi objectives using elitist non-dominated sorting GA. Proceedings of the Parallel Probl.Solv. Nat. PPSN VI Conference, Paris, France, pp. 859-868 (2000). (Please see pp. 1-10 in Technical Report No. 200002 via Google search.)

 

[26] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.

 

[27] 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.

 

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

 

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

 

[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] 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.

 

[35]  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.  (One can google this.)

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

 

[37] 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.

 

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

.

[39] 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.

 

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

 

[41] 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

 

[42] 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).

 

[43] 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

 

[44] 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

 

[45] 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] 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.

 

[47] 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.

 

[48] 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.

 

[49] 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.

 

[50] 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.

 

[51] 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.

 

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

 

[53] 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.

 

[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] 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.

 

[58] 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

 

[59] 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.

 

[60] 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.

 

[61] 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.

 

[62] 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.

 

[63] 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.

 

[64] 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.)

 

[65]  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.

 

[66] 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.

 

[67] 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.

 

[68] 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.

 

[69] 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.

 

[70] 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.

 

[71] 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/

[72] 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.

 

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

 

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

 

[75]  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.

 

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

 

[77] 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.

 

[78] 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.

 

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

 

[80] 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.

 

[81] 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.

 

[82] 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).

 

[83] 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.

 

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

 

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

 

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

 

[87] 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.

 

[88] 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

 

[89] 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.

 

[90] 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.

 

[91] 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

 

[92] 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.

 

[93] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

 

[94] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.

 

[95] Rahul Varshney, Najmussehar, M. J. Ahsan (2012). An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

 

[96] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.

 

[97] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

 

[98] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990)

34:325-334.

[99] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

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

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

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

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

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

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

[106] 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.

[107] 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.

 

 

Computer Program for Producing in a Single Run Multiple Solutions/One Solution of a System of Nonlinear/Linear Equations

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 from Burden, Faires, and Burden [16, p. 665, Exercise Set 10.3, Exercise 5d]:

 

10 * X(1) – 2 * X(2) ^ 2 + X(2) – 2 * X(3) – 5 =0,

8 * X(2) ^ 2 + 4 * X(3) ^ 2 – 9 =0,

8 * X(2) * X(3) + 4 =0.

 

One notes the vicinity of line 1173, which is 1173 X(3) = -4 / (8 * X(2)).

 

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 3

 

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 3

 

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

 

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

 

143 j = 1 + FIX(RND * 3)

 

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

 

1173 X(3) = -4 / (8 * X(2))

 

1175 P = -ABS(10 * X(1) – 2 * X(2) ^ 2## + X(2) – 2 * X(3) – 5) – ABS(8 * X(2) ^ 2## + 4 * X(3) ^ 2## – 9)

 

 

1221 IF P <= M THEN 1670

 

1420 M = P

 

1444 FOR KLX = 1 TO 3

 

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), A(3), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [104].  The complete output of a single run through JJJJ= -31990 is shown below:

 

.9  -1  .5   -2.220446049250313D-16

-32000

.8431980515339465        -.3535533905932737        1.414213562373095

-1.414233313790092D-15  -31999

.2068019484660536      .3535533905932737        -1.414213562373095

-9.610368056911511D-16        -31998

.9  -1  .5   -2.220446049250313D-16

-31997

.9  -1  .5   -2.220446049250313D-16

-31996

..2068019484660537      .3535533905932738        -1.414213562373095

-1.414233313790092D-15  -31995

.5  1  -.5   0   -31994

.9  -1  .5   -2.220446049250313-016

-31993

.5  1  -.5   0 -31992

.2068019484660536      .3535533905932737        -1.414213562373095

-9.610368056911511D-16      -31990

 

One notes that four distinct solutions are 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  = -31990 was 4 seconds, counting from “Starting program…”.   One can compare the computational results above with those in Burden, Faires, and Burden [16, p. 873, Exercise Set 10.3, Exercise 5d].

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.

 

 

 

 

Solving Systems of Nonlinear 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 from Burden, Faires, and Burden [16, p. 677]:

 

(3 * X(1) – COS(X(2) * X(3)) – .5) =0,

(X(1) ^ 2 – 81 * (X(2) + .1) ^ 2 + SIN(X(3)) + 1.06) =0,

(E ^ (-X(1) * X(2)) + 20 * X(3) + ((10 * PI – 3) / 3)) =0.

 

E and PI are shown in line 99 and line 111 below, respectively.

One notes line 1143, which is  1143 P = -ABS(3 * X(1) – COS(X(2) * X(3)) – .5) – ABS(X(1) ^ 2## – 81 * (X(2) + .1) ^ 2## + SIN(X(3)) + 1.06) – ABS(E ^ (-X(1) * X(2)) + 20 * X(3) + ((10 * PI – 3) / 3)) .

 

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 3

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 3

 

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

 

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

 

143 j = 1 + FIX(RND * 3)

 

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

 

1143 P = -ABS(3 * X(1) – COS(X(2) * X(3)) – .5) – ABS(X(1) ^ 2## – 81 * (X(2) + .1) ^ 2## + SIN(X(3)) + 1.06) – ABS(E ^ (-X(1) * X(2)) + 20 * X(3) + ((10 * PI – 3) / 3))

 

1221 IF P <= M THEN 1670

 

1420 M = P

 

1444 FOR KLX = 1 TO 3

 

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), A(3), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [104].  Selected candidate solutions of a single run through JJJJ= -31994 are shown below:

.

.

.

.5         0         -.5235987756666667         -5.92083415715955D-11

-31997

.

.

.

.498144684589087           -.199605895540152         -.528825977640724

-7.452805733665358D-15         -31994

 

One notes the two 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  = -31994  was 3 seconds, counting from “Starting program…”.  One can compare the computational results above with the results presented in Burden, Faires, and Burden [16, p. 678].

 

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.

 

 

 

Solving 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 in Burden, Faires, and Burden [16, p. 680, Exercise Set 10.5, Exercise 1]:

X(1) ^ 2 – X(2) ^ 2 + 2 * X(2) =0,

2 * X(1) + X(2) ^ 2 – 6 =0.

One notes line 1137, which is 1137 P = -ABS(X(1) ^ 2## – X(2) ^ 2## + 2 * X(2)) – ABS(2 * X(1) + X(2) ^ 2## – 6).

 

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

 

1137 P = -ABS(X(1) ^ 2## – X(2) ^ 2## + 2 * X(2)) – ABS(2 * X(1) + X(2) ^ 2## – 6)

 

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, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [104].  Selected candidate solutions of  a single run through JJJJ= -31974 are shown below:

.

.

.

-1.870548165746397         3.121072945557792      -8.89913143176102D-16

-31998

.

.

.

2.109511927133591         -1.33453219526287         -3.446031991271442D-08

-31993

.

.

 

.6252041114794327         2.17935581698839      -3.794643305653211D-08

-31974

 

One notes the three 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  = -31974 took 5 seconds, counting from “Starting program…”.  One can compare the computational results above with those in Burden, Faires, and Burden [16, Exercise Set 10.5, Exercise 1, p. 680, p. 874].

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.

 

 

 

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

 

Jsun Yui Wong

Line 1923 should read 1923 PRINT A(1), A(2), M, JJJJ instead of 1923 PRINT A(1), A(2), M.

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.

 

 

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:

(10 * X(1) – 2 * X(2) ^ 2 + X(2) – 2 * X(3) – 5) =0,

(8 * X(2) ^ 2 + 4 * X(3) ^ 2 – 9) =0,

(8 * X(2) * X(3) + 4) =0.

These three equations are from Burden, Faires, and Burden [16, p. 665, Exercise 5d; p. 680, Exercise 4d].

One notes line 1127, which is 1127 P = -ABS(10 * X(1) – 2 * X(2) ^ 2 + X(2) – 2 * X(3) – 5) – ABS(8 * X(2) ^ 2 + 4 * X(3) ^ 2 – 9) – ABS(8 * X(2) * X(3) + 4).

 

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

 

120 FOR J44 = 1 TO 3

 

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 3

 

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

 

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

143 j = 1 + FIX(RND * 3)

 

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 * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)

 

169 NEXT IPP

 

1127 P = -ABS(10 * X(1) – 2 * X(2) ^ 2 + X(2) – 2 * X(3) – 5) – ABS(8 * X(2) ^ 2 + 4 * X(3) ^ 2 – 9) – ABS(8 * X(2) * X(3) + 4)

 

1221 IF P <= M THEN 1670

 

1420 M = P

 

1444 FOR KLX = 1 TO 3

 

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), A(3), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [104].  Selected candidate solutions of  a single run through JJJJ= -31990 are shown below:

.9   -1   .5       -2.220446049250313D-16

-32000

.

.

.

.8431980515339465       -.3535533905932737   1.414213562373095

-1.48622433804313D-15         -31995

.

.

.

.5000000000000033      1.000000000000002      -.4999999999999972

-6.09512440519211D-14         -31990

One notes the 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  = -31990 took 3 seconds, counting from “Starting program…”.   One can compare the computational results above with those in Burden, Faires, and Burden [16, p. 874, Exercise Set 10.5, Exercise7d].

 

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.