Using Selected Lower Bounds and Upper Bounds as Active Constraints to Improve Given Solutions

 

Jsun Yui Wong

Looking at the preceding paper's computer program and its candidate solutions, one may want to try the upper bound X(1) = 4 and the lowed bounds X(5) = 1, X(6) = 1, X(7) = 1, and  X(8) = 1 as active constraints. Thus, that computer program becomes the following computer program, where one notes line 1601, which is 1061 X(1) = 4: X(5) = 1: X(6) = 1: X(7) = 1: X(8) = 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

    120 FOR J44 = 1 TO 8

        121 A(J44) = FIX(RND * 6)

 

        122 REM   A(J44) = (RND)

    123 NEXT J44

    128 FOR I = 0 TO FIX(RND * 130000)  

 

        129 FOR KKQQ = 1 TO 8

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

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

 

            143 J = 1 + FIX(RND * 8.5)

 

            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 * 30)) * R

            161 GOTO 169

            162 REM     IF RND < .16666 THEN X(J) = A(J) - FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(J) = A(J) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(J) = A(J) - FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(J) = A(J) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(J) = A(J) - FIX(1 + RND * 200.3) ELSE X(J) = A(J) + FIX(1 + RND * 200.3)

 

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

 

        169 NEXT IPP

 

 

        1061 X(1) = 4: X(5) = 1: X(6) = 1: X(7) = 1: X(8) = 1 

 

 

        1066 X(2) = -2 - X(8) - X(3) * X(4) * X(5) * X(6) + X(1) * X(4) + X(6) * X(7)

        1068 IF X(1) < -1 THEN 1670

        1069 IF X(2) < -2.5 THEN 1670

        1070 IF X(3) < -.5 THEN 1670

        1071 IF X(4) < -.5 THEN 1670

 

        1072 IF X(5) < 1 THEN 1670

        1073 IF X(6) < 1 THEN 1670

        1074 IF X(7) < 1 THEN 1670

        1075 IF X(8) < 1 THEN 1670

 

        1077 IF X(1) > 4 THEN 1670

        1078 IF X(2) > 4 THEN 1670

        1079 IF X(3) > 6 THEN 1670

        1080 IF X(4) > 6 THEN 1670

        1081 IF X(5) > 6 THEN 1670

        1082 IF X(6) > 5 THEN 1670

        1083 IF X(7) > 3 THEN 1670

 

        1087 IF X(1) * X(2) * X(3) * X(4) - X(1) * X(2) - X(4) * X(5) + X(5) + X(6) < 230 THEN 1670

 

        1349 P = -4 * (X(1) + X(2) + X(3)) - 3 * (X(4) + X(5)) - 3.5 * (X(6) + X(7)) - 2.5 * X(8)

 

        1351 IF P <= M THEN 1670

        1420 M = P

        1444 FOR KLX = 1 TO 8

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

        1557 GOTO 128

    1670 NEXT I

    1688 IF M < -72.18 THEN 1999

    1924 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [118].  Selected candidate solutions of a single run through JJJJ= -31958 are shown below:

.

.

.

4       3.639673933557837         3.0057388980006932

5.672511992069006       1          1       1

1         -72.09938763046608      -31974

4       3.511777620470184          3.047845372448122

5.788742144373893       1          1       1

1         -72.10471840479491      -31970 

4       3.62O673663541915         3.012028543192354

5.689105312519407       1          1       1

1         -72.0981247644953        -31967

4       3.605649299110592          3.01696467471956

5.70238846453602       1          1       1

1         -72.09762128892866      -31958

 

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 [118], the wall-clock time (not CPU time) for obtaining the output through  JJJJ  = -31958 was 7 seconds, counting from "Starting program...". 

 

Acknowledgement

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

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[110] 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.

 

[111] 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.

 

[112]  Eric W. Weisstein, "Diophantine Equation--8th Powers."  https://mathworld.wolfram.com/DiophantineEquation8thPowers.html.

 

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

 

[114]  Eric W. Weisstein, "Diophantine Equation--5th Powers."  https://mathworld.wolfram.com/DiophantineEquation5thPowers.html.

 

[115]  Eric W. Weisstein, "Diophantine Equation--10th Powers."  https://mathworld.wolfram.com/DiophantineEquation10thPowers.html.

 

[116]  Eric W. Weisstein, "Diophantine Equation--9th Powers."  https://mathworld.wolfram.com/DiophantineEquation9thPowers.html.

 

[117]  Rick Wicklin (2018), Solving a system of nonlinear equations with SAS.

blogs.sas.com>iml>2018/02/28.  One can directly read this on Google.

 

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

 

[119] 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.

 

[120] 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.