Errata in: Computer program for solving nonlinear fractional programming problems: another illustration

Jsun Yui Wong

The output should appear as follows:

.7885383 .4086356 -263.8959 -31919**.7887109 .4081472 -263.8958 -31886**

.7890227 .4072663 -263.8959 -31877

.7885319 .4086539 -263.8959 -31834

.7891097 .4070208 -263.896 -31824

.7883261 .4092365 -263.8959 -31740

.789149 .4069097 -263.896 -31714 **.7887858 .4079355 -263.8958 -31603**

That took 25 seconds, not 40 seconds.

Computer program for solving nonlinear fractional programming problems: another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following formulation from Tsai [86, p. 406, Example 2]:

Minimize 200 * 2 ^ .5 * X(1) + 100 * X(2)

subject to

(2 ^ .5 * X(1) + X(2)) / (2 ^ .5 * X(1) ^ 2 + 2 * X(1) * X(2)) - 1 <= 0

1 / (X(1) + 2 ^ .5 * X(2)) - 1 <= 0

X(2) / (2 ^ .5 * X(1) ^ 2 + 2 * X(1) * X(2)) - 1 <= 0

0 < X(1), X(2) <= 1.

0 REM DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

121 A(1) = RND * 1

123 A(2) = RND * 1

126 REM A(3) = 2 + RND * 13

128 FOR i = 1 TO 50000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

140 B = 1 + FIX(RND * 2)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

165 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

213 IF X(1) < .000001 THEN 1670

215 IF X(1) > 1 THEN 1670

222 IF X(2) < .000001 THEN 1670

224 IF X(2) > 1 THEN 1670

375 IF (2 ^ .5 * X(1) + X(2)) / (2 ^ .5 * X(1) ^ 2 + 2 * X(1) * X(2)) - 1 > 0 THEN 1670

378 IF 1 / (X(1) + 2 ^ .5 * X(2)) - 1 > 0 THEN 1670

388 IF X(2) / (2 ^ .5 * X(1) ^ 2 + 2 * X(1) * X(2)) - 1 > 0 THEN 1670

565 PD1 = -200 * 2 ^ .5 * X(1) - 100 * X(2)

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < -263.896 THEN 1999

1904 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31603 is shown below. GW-BASIC also can handle this computer program.

.7885383 .4086356 -263.8959 -31919**.7887109 .4081472 -263.8958 -31886**

.7890227 .4072663 -263.8959 -31877

.7885319 .4086539 -263.8959 -31834

.7891097 .4070208 -263.896 -31824

.7883261 .4092365 -263.8959 -31740

.789149 .4069097 -263.896 -31714**.7887858 .4079355 -263.8958 -31603**

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -31603 was 40 seconds, counting from "Starting program…". One can see other computational results in Tsai [86, p. 408, Table 2].

The computational results presented above were obtained from the following computer system:

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System

**Acknowledgement**

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

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Computer program for solving nonlinear fractional programming problems: another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following formulation from Tsai [86, p. 405, Example 1]:

Minimize (X(3) + 2) * (X(1) * X(2) ^ 2)

subject to

(1 - (X(1) ^ 3 * X(3)) / (71785 * X(2) ^ 4)) <= 0

(4 * X(1) ^ 2 - X(1) * X(2)) / (12566 * (X(1) * X(2) ^ 3 - X(2) ^ 4)) + 1 / (5108 * X(2) ^ 2) - 1 <= 0

1 - (140.45 * X(2)) / (X(2) * X(3)) <= 0

(X(1) + X(2)) / 1.5 - 1 <= 0

.25<= X(1) <= 1.3

.05<= X(2) <= 2.0

2<= X(3) <=15.

0 REM DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

121 A(1) = .25 + RND * 1.05

123 A(2) = .05 + RND * 1.95

126 A(3) = 2 + RND * 13

128 FOR i = 1 TO 90000

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)

140 B = 1 + FIX(RND * 3)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

165 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

213 IF X(1) < .25 THEN 1670

215 IF X(1) > 1.3 THEN 1670

222 IF X(2) < .05 THEN 1670

224 IF X(2) > 2.0 THEN 1670

226 IF X(3) < 2 THEN 1670

268 IF X(3) > 15 THEN 1670

374 IF (1 - (X(1) ^ 3 * X(3)) / (71785 * X(2) ^ 4)) > 0 THEN 1670

376 IF (4 * X(1) ^ 2 - X(1) * X(2)) / (12566 * (X(1) * X(2) ^ 3 - X(2) ^ 4)) + 1 / (5108 * X(2) ^ 2) - 1 > 0 THEN 1670

378 IF 1 - (140.45 * X(2)) / (X(2) * X(3)) > 0 THEN 1670

383 IF (X(1) + X(2)) / 1.5 - 1 > 0 THEN 1670

565 PD1 = -(X(3) + 2) * (X(1) * X(2) ^ 2)

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < -.0126666 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its output of one run through

JJJJ= -29196 is summarized below. GW-BASIC also can handle this computer program.

.3526241 5.151832E-02 11.53306 -1.266577D-02

-31491

.3567026 5.168843E-02 11.42776 -1.266541D-02

-30961

.

.

.**.3567026 5.168843E-02 11.28986 -1.266523D-02****-29196**

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -29196 was 8.5 minutes, counting from "Starting program…". One can see other computational results in Tsai [86, p. 407, Table 1].

The computational results presented above were obtained from the following computer system:

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System

**Acknowledgement**

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

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Computer program for solving nonlinear fractional programming problems, second edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following formulation from Tsai [86, p. 408]:

Minimize (2 + X(5)) / (X(1) * X(2) * (2 * (X(3) + X(4)))) - X(5) ^ .5 * X(3) ^ 1.5 + 2 * X(2) + X(4)

subject to

(8 / (((X(1) * (X(2) + 3 * X(4)) ^ 2)))) + 1 / X(5) ^ 3 <= 2

-2 * X(1) + X(3) - X(4) <= 10

X(1) + X(3) + .5 * X(5) <= 8

.1<= X(1), X(2), X(3), X(4), X(5) <=10.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

125 FOR J44 = 1 TO 5

126 A(J44) = .1 + RND * 9.9

127 NEXT J44

128 FOR i = 1 TO 50000

129 FOR KKQQ = 1 TO 5

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

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

140 B = 1 + FIX(RND * 5)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

165 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

203 REM X(3) = INT(X(3))

205 REM X(5) = INT(X(5))

211 FOR J44 = 1 TO 5

213 IF X(J44) < .1 THEN 1670

215 IF X(J44) > 10 THEN 1670

219 NEXT J44

375 IF (8 / (((X(1) * (X(2) + 3 * X(4)) ^ 2)))) + 1 / X(5) ^ 3 > 2 THEN 1670

377 IF -2 * X(1) + X(3) - X(4) > 10 THEN 1670

379 IF X(1) + X(3) + .5 * X(5) > 8 THEN 1670

565 PD1 = -(2 + X(5)) / (X(1) * X(2) * (2 * (X(3) + X(4)))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4)

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 5

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < 22.7 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its output of one run through

JJJJ= -31892 is summarized below. GW-BASIC also can handle this computer program.

.3500986063829111 .7670734473296763 5.894038072995033

.877583516658183 3.51172664124334 22.88795409341691

-31994

.361524317075925 .8237876654406516 5.74749398141627

.8393303471593814 3.78196340301561 22.83576907831574

-31991

.

.

.**.3251372078803563 .786558404713068 5.924051832218455 ****.9138779562445035 3.501621799571855 22.92127138958****-31892**

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -31892 was 12 seconds, counting from "Starting program…". One can see other computational results in Tsai [86, p. 408].

The computational results presented above were obtained from the following computer system:

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System

**Acknowledgement**

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

References

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Computer program for solving mixed integer signomial programming problems, third edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following signomial programming problem from Chang [18, Example 5 on pp. 1448-1450]:

Minimize

(.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3))

subject to

X(1) - .0193 * X(3) >= 0

X(2) - .00954 * X(3) >= 0

pie * X(3) ^ 2)* X(4) + (4 / 3) * pie * X(3) ^ 3) >= 750 * 1728

1<= X(1) <= 1.375, discrete variable with discreteness .0625

.625<= X(2) <= 1, discrete variable with discreteness .0625

45<= X(3) <= 55, continuous variable

80<= X(4) <= 110, continuous variable.

One notes the following line 196 through 451.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

118 pie = 3.141592654

119 A(1) = 1 + (.0625 * FIX(RND * 7))

121 A(2) = .625 + (.0625 * FIX(RND * 7))

123 A(3) = 45 + (RND * 10)

124 A(4) = 80 + (RND * 30)

128 FOR i = 1 TO 500000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 4)

140 B = 1 + FIX(RND * 4)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

165 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

196 X(1) = 1

198 X(1) = 1 + (.0625 * FIX(RND * 7))

199 X(2) = .625 + (.0625 * FIX(RND * 7))

241 IF X(1) < 1 THEN 1670

242 IF X(1) > 1.375 THEN 1670

244 IF X(2) < .625 THEN 1670

245 IF X(2) > 1 THEN 1670

246 IF X(3) < 45 THEN 1670

247 IF X(3) > 55 THEN 1670

248 IF X(4) < 80 THEN 1670

249 IF X(4) > 110 THEN 1670

251 IF RND < .5 THEN X(1) = X(1) ELSE X(1) = INT(X(1))

333 X(4) = (750 * 1728 - (4 / 3) * pie * X(3) ^ 3) / (pie * X(3) ^ 2)

359 IF X(1) - .0193 * X(3) < 0 THEN 1670

365 IF X(2) - .00954 * X(3) < 0 THEN 1670

441 IF X(1) < 1 THEN 1670

442 IF X(1) > 1.375 THEN 1670

444 IF X(2) < .625 THEN 1670

445 IF X(2) > 1 THEN 1670

446 IF X(3) < 45 THEN 1670

447 IF X(3) > 55 THEN 1670

448 IF X(4) < 80 THEN 1670

449 IF X(4) > 110 THEN 1670

451 IF RND < .5 THEN X(1) = X(1) ELSE X(1) = INT(X(1))

564 PD1 = -(.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3))

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 4

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < -9999999 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31999 is shown below. GW-BASIC also can handle this computer program.

1 .625 52.45322925101003 80.00000000000505

-6963.31119194227 -32000

**1 .625 52.45322925101045 80.0000000000021****-6963.311191942241 -31999**

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -31999 was 5 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Chang [18, Example 5, pages 1448-1450, Table 2 on p. 1450].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

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

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

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

[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values...

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

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).

http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8...

Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[42] R. Israel, A Karush-Kuhn-Tucker Example

https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.

[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming

[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering

and System Safety 152 (2016) 213-227.

[61] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

One can get a Google view of this report.

[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

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

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.

[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.

[8o] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.

[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.

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

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

[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm

[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.

[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.

[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.

[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.

[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.

[91] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. www.mdpi.com/journal/symmetry.

[92] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.

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

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

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

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

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

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

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

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

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

[102] Eric W. Weisstein, "Euler's Sum of Powers Conjecture." https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

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

[104] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourtth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming

[105] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

[106] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.

Computer program for solving mixed-integer nonlinear programming (MINLP) problems

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following mixed-integer nonlinear programming (MINLP) problem from Molina-Perez et al. [61, p. 19, Test Problem F4].

Minimize - X(1) - X(2)

subject to

X(2) - 3.4 <= 0

X(1) - X(2) <= 0

X(1) epsilon [ -1, 100 ], X(1) is a continuous variable

X(2) { -1, 0, . . ., 100 }, X(2) is an integer variable.

One notes line 226, line 227, and line 228, which are

226 FOR J44 = 1 TO 2

227 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))

228 NEXT J44.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

91 FOR J44 = 1 TO 2

92 A(J44) = -1 + RND * 101

93 NEXT J44

128 FOR i = 1 TO 5000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

140 B = 1 + FIX(RND * 2)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

211 FOR J44 = 1 TO 2

213 IF X(J44) < -1 THEN 1670

214 IF X(J44) > 100 THEN 1670

215 NEXT J44

226 FOR J44 = 1 TO 2

227 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))

228 NEXT J44

341 IF X(2) - 3.4 > 0 THEN 1670

344 IF X(1) - X(2) > 0 THEN 1670

568 PD1 = X(1) + X(2)

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < -999999999 THEN 1999

1904 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -30080 is shown below. GW-BASIC also can handle this computer program.

**3 3 6 -31835**

3 3 6 -31702

3 3 6 -31588

3 3 6 -31084

3 3 6 -30173

3 3 6 -30080

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -30080 was 20 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F4].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

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

References

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[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

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

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

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

[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

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

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

[23] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

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

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

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

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values...

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

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).

http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8...

Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[42] R. Israel, A Karush-Kuhn-Tucker Example

https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.

[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming

[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering and System Safety 152 (2016) 213-227.

[60] 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.

[61] D. Molina-Perez, E. Mezura-Montes, E.A. Portilla-Flores, E. Vega-Alvarado, B. Calva-Variez (2024) A differential evolution algorithm for solving mixed-integer nonlinear programming problems, *Swarm and Evolutionary Computation* 84 (2024) 101427, pp. 1-26.

[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

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

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.

[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.

[80] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.

[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.

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

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

[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm

[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.

[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[87] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 37:4, 399-409.

[88] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.

[89] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.

[90] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.

[91] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.

[92] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. www.mdpi.com/journal/symmetry.

[93] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.

[94] 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

[95] 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.

[96] 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.

[97] 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.

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

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

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

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

[102] Eric W. Weisstein, "Euler's Sum of Powers Conjecture." https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

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

[104] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourtth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming

[105] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

[106] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.

Computer program for solving mixed-integer nonlinear programming (MINLP) problems

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following mixed-integer nonlinear programming (MINLP) problem from Molina-Perez et al. [61, p. 19, Test Problem F3].

Minimize - X(1) - X(2)

subject to

-X(1) + X(2) - 2.005 <= 0

X(1) - X(2) + .5 <= 0

.505 * X(1) + X(2)- 3.505 <= 0

X(1) epsilon [ -1, 100 ], X(1) is a continuous variable

X(2) { -1, 0, . . ., 100 }, X(2) is an integer variable.

Often one or more of the longer lines are binding. Hence one can try the following line 239, which is 239 X(2) = -.505 * X(1) + 3.505.

One also notes line 326, line 327, and line 328, which are

326 FOR J44 = 1 TO 2

327 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))

328 NEXT J44.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

91 FOR J44 = 1 TO 2

92 A(J44) = -1 + RND * 101

93 NEXT J44

128 FOR i = 1 TO 20000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

140 B = 1 + FIX(RND * 2)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

211 FOR J44 = 1 TO 2

213 IF X(J44) < -1 THEN 1670

214 IF X(J44) > 100 THEN 1670

215 NEXT J44

226 FOR J44 = 1 TO 2

227 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))

228 NEXT J44

239 X(2) = -.505 * X(1) + 3.505

311 FOR J44 = 1 TO 2

313 IF X(J44) < -1 THEN 1670

314 IF X(J44) > 100 THEN 1670

315 NEXT J44

326 FOR J44 = 1 TO 2

327 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))

328 NEXT J44

337 IF -X(1) + X(2) - 2.005 > 0 THEN 1670

338 IF X(1) - X(2) + .5 > 0 THEN 1670

567 PD1 = X(1) + X(2)

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < -999999999 THEN 1999

1904 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -319841 is shown below. GW-BASIC also can handle this computer program.

**1 3 4 -31999**

1 3 4 -31950

1 3 4 -31904

1 3 4 -31867

1 3 4 -31862

1 3 4 -31841

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -31841 was 5 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F3].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

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

References

[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, *Int. J. of Bio-Inspired Computation*, 2 (2), 100-114, 2010.

[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, *Int. J. of Engg. Science and* *Mgmt.*, Vol. III, Issue 1, January-June 2013.

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

[4] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. *Operations Research*, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[5] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

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

[7] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.

[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.

[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

[11] Hirak Basumatary (1 January 2019). Solve system of equations and inequalities with multiple solutions?

https://www.mathworks.com/matlabcentral/answers/437815-solve-system-of-equations-and-inequalities-with-multiple-solutions?_tid=prof_contriblnk

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton--Approximate nonlinear equations by sequence of linear equations--lecture 6. (Youtube is where I saw this work.)

http://bazziahmad.com/

[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?

https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab

[14] F. Bazikar, M. Saraj (2018), MathLAB Journal, vol. 1 no. 3, 2018.

http://purkh.com/index.php/mathlab

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

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

[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute_values#Numerical_ Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.

[19] Ching-Ter Chang (2006), Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research, volume 173, issue 2, 1 September 2006, pages 370-386.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.

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

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

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

[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

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

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

[23] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

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

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

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

[34] Benjamin Granger, Marta Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values...

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

[37] Mohammad Babul Hasan, Sumi Acharjee (2011), Solving LFP by converting it into a single LP, International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).

http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8...

Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[42] R. Israel, A Karush-Kuhn-Tucker Example

https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.

[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[56] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming

[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering and System Safety 152 (2016) 213-227.

[60] 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.

[61] D. Molina-Perez, E. Mezura-Montes, E.A. Portilla-Flores, E. Vega-Alvarado, B. Calva-Variez (2024) A differential evolution algorithm for solving mixed-integer nonlinear programming problems, *Swarm and Evolutionary Computation* 84 (2024) 101427, pp. 1-26.

[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

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

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.

[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.

[80] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.

[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.

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

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

[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm

[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.

[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[87] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 37:4, 399-409.

[88] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.

[89] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.

[90] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.

[91] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.

[92] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. www.mdpi.com/journal/symmetry.

[93] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.

[94] 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

[95] 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.

[96] 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.

[97] 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.

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

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

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

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

[102] Eric W. Weisstein, "Euler's Sum of Powers Conjecture." https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

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

[104] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourtth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming

[105] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

[106] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.

Computer program for solving mixed integer nonlinear programming problems, improved edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following mixed integer nonlinear programming problem from Molina-Perez et al. [61, p.19, Test Problem F2].

Minimize X(1) ^ 2 + (X(2) - 1) ^ 2 + (X(3) - 2) ^ 2

subject to

X(1) ^ 2 + X(2) ^ 2 + .5 * X(3) ^ 2 - 1.5 <=0

X(1) epsilon [ -1, 100 ], X(1) is a continuous variable

X(2), X(3) { -1, 0, . . ., 100 }, X(2) and X(3) are integers.

One notes line 416 through 419, which are 416 FOR J44 = 1 TO 3, 415 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44)), and 419 NEXT J44.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

91 FOR J44 = 1 TO 3

92 A(J44) = -1 + RND * 101

93 NEXT J44

128 FOR i = 1 TO 2000

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)

140 B = 1 + FIX(RND * 3)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

411 FOR J44 = 1 TO 3

412 IF X(J44) < -1 THEN 1670

413 IF X(J44) > 100 THEN 1670

414 NEXT J44

416 FOR J44 = 1 TO 3

415 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))

419 NEXT J44

565 PD1 = -X(1) ^ 2 - (X(2) - 1) ^ 2 - (X(3) - 2) ^ 2 - ABS(X(1) ^ 2 + X(2) ^ 2 + .5 * X(3) ^ 2 - 1.5)

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < -999999999 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31996 is shown below. GW-BASIC also can handle this computer program.

**0 1 1 -1 -32000**

0 1 1 -1 -31999

0 1 1 -1 -31998

0 1 1 -1 -31997

0 1 1 -1 -31996

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -31996 was 2 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F2].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

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

References

[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, *Int. J. of Bio-Inspired Computation*, 2 (2), 100-114, 2010.

[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.

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

[4] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[5] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

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

[7] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.

[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.

[10] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

[11] Hirak Basumatary (1 January 2019). Solve system of equations and inequalities with multiple solutions?

https://www.mathworks.com/matlabcentral/answers/437815-solve-system-of-equations-and-inequalities-with-multiple-solutions?_tid=prof_contriblnk

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton--Approximate nonlinear equations by sequence of linear equations--lecture 6. (Youtube is where I saw this work.)

http://bazziahmad.com/

[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?

https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab

[14] F. Bazikar, M. Saraj (2018), MathLAB Journal, vol. 1 no. 3, 2018.

http://purkh.com/index.php/mathlab

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

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

[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute_values#Numerical_ Example

[18] Ching-Ter Chang (2005), On the mixed integer signomial programming problems, Applied Mathematics and Computation 170 (2005) pp. 1436-1451.

[20] Ching-Ter Chang (2002), On the posynomial fractional programming problems, European Journal of Operational Research, 143 (2002) pp. 42-52.

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

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

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

[24] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

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

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

[23] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

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

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

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

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

http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8...

Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[42] R. Israel, A Karush-Kuhn-Tucker Example

https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.

[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming

[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering and System Safety 152 (2016) 213-227.

[60] 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.

[61] D. Molina-Perez, E. Mezura-Montes, E.A. Portilla-Flores, E. Vega-Alvarado, B. Calva-Variez (2024) A differential evolution algorithm for solving mixed-integer nonlinear programming problems, *Swarm and Evolutionary Computation* 84 (2024) 101427, pp. 1-26.

[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

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

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.

[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.

[80] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.

[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.

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

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

[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm

[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.

[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[87] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 37:4, 399-409.

[88] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.

[89] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.

[90] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.

[91] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.

[92] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. www.mdpi.com/journal/symmetry.

[93] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.

[94] 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

[95] 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.

[96] 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.

[97] 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.

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

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

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

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

[102] Eric W. Weisstein, "Euler's Sum of Powers Conjecture." https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

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

[104] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourtth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming

[105] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

[106] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.

Computer program for solving mixed integer nonlinear programming problems: an illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following mixed integer nonlinear programming problem from Molina-Perez et al. [61, p. 19, Test Problem F2].

Minimize X(1) ^ 2 + (X(2) - 1) ^ 2 + (X(3) - 2) ^ 2

subject to

X(1) ^ 2 + X(2) ^ 2 + .5 * X(3) ^ 2 - 1.5 <=0

X(1) epsilon [ -1, 100 ], X(1) is a continuous variable

X(2), X(3) { -1, 0, . . ., 100 }, X(2) and X(3) are integers.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

91 FOR J44 = 1 TO 3

92 A(J44) = -1 + RND * 101

93 NEXT J44

128 FOR i = 1 TO 200000

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)

140 B = 1 + FIX(RND * 3)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

201 X(2) = INT(X(2))

202 X(3) = INT(X(3))

204 REM X(1) = INT(X(1))

211 FOR J44 = 1 TO 3

212 IF X(J44) < -1 THEN 1670

213 IF X(J44) > 100 THEN 1670

214 NEXT J44

401 X(2) = INT(X(2))

402 X(3) = INT(X(3))

411 FOR J44 = 1 TO 3

412 IF X(J44) < -1 THEN 1670

413 IF X(J44) > 100 THEN 1670

414 NEXT J44

565 PD1 = -X(1) ^ 2 - (X(2) - 1) ^ 2 - (X(3) - 2) ^ 2 - ABS(X(1) ^ 2 + X(2) ^ 2 + .5 * X(3) ^ 2 - 1.5)

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < -999999999 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its output of one run through JJJJ= -31991 is summarized below. GW-BASIC also can handle this computer program.

.

.

.**-6.369822387695313D-03 1 1 -1.000081021928054****-31998**

.

.

.

-6.66874647140503D-03 1 1 -1.000088944359

-31991

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -31991 was 7 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F2].

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

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

References

[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, *Int. J. of Bio-Inspired Computation*, 2 (2), 100-114, 2010.

[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.

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

[4] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[5] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

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

[7] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.

[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.

[11] Hirak Basumatary (1 January 2019). Solve system of equations and inequalities with multiple solutions?

https://www.mathworks.com/matlabcentral/answers/437815-solve-system-of-equations-and-inequalities-with-multiple-solutions?_tid=prof_contriblnk

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton--Approximate nonlinear equations by sequence of linear equations--lecture 6. (Youtube is where I saw this work.)

http://bazziahmad.com/

[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?

https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab

[14] F. Bazikar, M. Saraj (2018), MathLAB Journal, vol. 1 no. 3, 2018.

http://purkh.com/index.php/mathlab

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

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

[17] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values. https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute_values#Numerical_ Example

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

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

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

[23] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

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

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

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

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

http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8...

Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263

[42] R. Israel, A Karush-Kuhn-Tucker Example

https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[52] Ming-Hua Lin, Jung-Fa Tsai (2012). Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming. European Journal of Operational Research 216 (2012) 17-25.

[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12: 425-443

[57] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.

[58] Costas D. Maranas, Christodoulos A. Floudas, Global Optimization in Generalized Geometric Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming

[59] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering and System Safety 152 (2016) 213-227.

[60] 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.

*Swarm and Evolutionary Computation* 84 (2024) 101427, pp. 1-26.

[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

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

[79] J. R. Sharma, Puneet Gupta (31 October 2013 ). On some efficient techniques for solving systems of nonlinear equations, Advances in Numerical Analysis, Volume 2013, Article ID 252798, pp. 1-11, Hindawi Corporation.

[80] J. R. Sharma, Puneet Gupta (2014 ). An efficient family of Traub-Steffensen-Type Methods for solving systems of nonlinear equations. Advances in Numerical Analysis, Volume 2014, Article ID 152187, 11 pages, Hindawi Corporation.

[81] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.

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

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

[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm

[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.

[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[87] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 37:4, 399-409.

[88] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.

[89] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.

[90] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.

[91] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.

[92] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529. One can get a Google view of this article. www.mdpi.com/journal/symmetry.

[93] Jung-Fa Tsai, Ming-Hua Lin (Summer 2011), An efficient efficient global approach for posynomial geometrical programming problems, INFORMS Journal on Computing, vol, 23, no. 3, summer 2011, pp. 483-492.

[94] 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

[95] 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.

[96] 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.

[97] 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.

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

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

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

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

[102] Eric W. Weisstein, "Euler's Sum of Powers Conjecture." https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

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

[104] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourtth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002. https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming

[105] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

[106] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.

Computer program for solving mixed-integer nonlinear programming problems

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following mixed-integer nonlinear programming problem of Molina-Perez et al. [61, p.19, Test Problem F6]:

Minimize (X(1) - 10) ^ 3 + (X(2) - 20) ^ 3

subject to

-(X(1) - 5) ^ 2 - (X(2) - 4.86) ^ 2 + 100 <= 0

(X(1) - 8) ^ 2 + (X(2) - 5.48) ^ 2 - 60 <= 0

X(1) epsilon [-1, 100], a continuous variable

X(2) epsilon {-1, 0, ..., 100}, an integer variable.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

125 FOR J44 = 1 TO 2

126 A(J44) = -1 + RND * 101

127 NEXT J44

128 FOR i = 1 TO 10000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

140 B = 1 + FIX(RND * 2)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

165 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

203 X(2) = INT(X(2))

211 FOR J44 = 1 TO 2

213 IF X(J44) < -1 THEN 1670

215 IF X(J44) > 100 THEN 1670

219 NEXT J44

377 IF -(X(1) - 5) ^ 2 - (X(2) - 4.86) ^ 2 + 100 > 0 THEN 1670

378 IF (X(1) - 8) ^ 2 + (X(2) - 5.48) ^ 2 - 60 > 0 THEN 1670

565 PD1 = -(X(1) - 10) ^ 3 - (X(2) - 20) ^ 3

569 p = PD1

1111 IF p <= M THEN 1670

1452 M = p

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1775 IF M < -999999999 THEN 1999

1904 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31991 is shown below. GW-BASIC also can handle this computer program.

14.22498780543337 1 6783.581762415023

-31994

**14.2249878048725 1 6783.58176244506 ****-31991**

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -31991 was 2 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F6].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**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 ApMgproach to Find Numerical Solutions of Diophantine Equations, *Int. J. of Engg. Science and* *mt*., 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 Researc*h, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[5] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

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

[7] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.

[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[9] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.

[11] Hirak Basumatary (1 January 2019). Solve system of equations and inequalities with multiple solutions?

https://www.mathworks.com/matlabcentral/answers/437815-solve-system-of-equations-and-inequalities-with-multiple-solutions?_tid=prof_contriblnk

[12 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton--Approximate nonlinear equations by sequence of linear equations--lecture 6. (Youtube is where I saw this work.)

http://bazziahmad.com/

[13] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?

https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab

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