A computer program for solving signomial discrete programming programs with discrete variables

 

 A computer program for solving signomial discrete programming programs with discrete variables

Jsun Yui Wong

The computer program listed below attempts to solve the following nonlinear program from Li and Lu [52] and Tsai, Lin, and Peng [90, pp. 179-180, Example 3]:   

Minimize 

  -1*  (-X(1) ^ 2 * X(2) ^ .816 + X(2) ^ .5 + X(3) ^ 1.2)

subject to  

         X(1) ^ .8 + X(2) ^ .9 + X(3) ^ .5 >= 16 

         X(1) ^ -1.5 + X(2) ^ 1.7 + X(3) ^ 1.2 <= 31 

        

where X(1) epsilon {1.1, 3.2, 5.3, 7.4, 9.5, 12.6, 14.7, 16.8}

X(1), X(2) epsilon { the 128 possible values of X(2) are shown in line 91 through line 97 below; same 128 possible values for X(2) }.

        

0 REM DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), G(128), J44(222), 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), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ

    90 M = -3D+30

    91 G(1) = 1.1: G(2) = 1.2: G(3) = 1.3: G(4) = 1.4: G(5) = 1.5: G(6) = 2: G(7) = 2.6: G(8) = 2.7: G(9) = 2.8: G(10) = 2.9: G(11) = 3.1: G(12) = 3.2: G(13) = 3.3: G(14) = 3.4: G(15) = 3.5: G(16) = 4: G(17) = 4.6: G(18) = 4.7: G(19) = 4.8: G(20) = 4.9

    92 G(21) = 5.1: G(22) = 5.2: G(23) = 5.3: G(24) = 5.4: G(25) = 5.5: G(26) = 6: G(27) = 6.6: G(28) = 6.7: G(29) = 6.8: G(30) = 6.9: G(31) = 7.1: G(32) = 7.2: G(33) = 7.3: G(34) = 7.4: G(35) = 7.5: G(36) = 8: G(37) = 8.6: G(38) = 8.7: G(39) = 8.8: G(40) = 8.9

    93 G(41) = 9.1: G(42) = 9.2: G(43) = 9.3: G(44) = 9.4: G(45) = 9.5: G(46) = 10: G(47) = 10.6: G(48) = 10.7: G(49) = 10.8: G(50) = 10.9: G(51) = 11.1: G(52) = 11.2: G(53) = 11.3: G(54) = 11.4: G(55) = 11.5: G(56) = 12: G(57) = 12.6: G(58) = 12.7: G(59) = 12.8: G(60) = 12.9

    94 G(61) = 13.1: G(62) = 13.2: G(63) = 13.3: G(64) = 13.4: G(65) = 13.5: G(66) = 14: G(67) = 14.6: G(68) = 14.7: G(69) = 14.8: G(70) = 14.9: G(71) = 15.1: G(72) = 15.2: G(73) = 15.3: G(74) = 15.4: G(75) = 15.5: G(76) = 16: G(77) = 16.6: G(78) = 16.7: G(79) = 16.8: G(80) = 16.9

    95 G(81) = 17.1: G(82) = 17.2: G(83) = 17.3: G(84) = 17.4: G(85) = 17.5: G(86) = 18: G(87) = 18.6: G(88) = 18.7: G(89) = 18.8: G(90) = 18.9: G(91) = 19.1: G(92) = 19.2: G(93) = 19.3: G(94) = 19.4: G(95) = 19.5: G(96) = 20: G(97) = 20.6: G(98) = 20.7: G(99) = 20.8: G(100) = 20.9

    96 G(101) = 21.1: G(102) = 21.2: G(103) = 21.3: G(104) = 2.4: G(105) = 21.5: G(106) = 22: G(107) = 22.6: G(108) = 22.7: G(109) = 22.8: G(110) = 22.9: G(111) = 23.1: G(112) = 23.2: G(113) = 23.3: G(114) = 23.4: G(115) = 23.5: G(116) = 24: G(117) = 24.6: G(118) = 24.7: G(119) = 24.8: G(120) = 24.9

    97 G(121) = 25.1: G(122) = 25.2: G(123) = 25.3: G(124) = 25.4: G(125) = 25.5: G(126) = 26: G(127) = 26.6: G(128) = 26.7

    121 IF RND < 1 / 8 THEN A(1) = 1.1 ELSE IF RND < 1 / 7 THEN A(1) = 3.2 ELSE IF RND < 1 / 6 THEN A(1) = 5.3 ELSE IF RND < 1 / 5 THEN A(1) = 7.4 ELSE IF RND < 1 / 4 THEN A(1) = 9.5 ELSE IF RND < 1 / 3 THEN A(1) = 12.6 ELSE IF RND < 1 / 2 THEN A(1) = 14.7 ELSE A(1) = 16.8

    122 A(2) = 1.1 + RND * 25.6

    123 A(3) = 1.1 + RND * 25.6

    128 FOR i = 1 TO 20000

        129 FOR KKQQ = 1 TO 3

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

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

            141 B = 1 + FIX(RND * 3)

            144 REM  IF RND < .9999 THEN 160 ELSE GOTO 167

            147 IF B = 1 THEN 167 ELSE IF B = 2 THEN 215 ELSE GOTO 259

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

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

            165 REM GOTO 168

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

            167 IF RND < 1 / 8 THEN X(1) = 1.1 ELSE IF RND < 1 / 7 THEN X(1) = 3.2 ELSE IF RND < 1 / 6 THEN X(1) = 5.3 ELSE IF RND < 1 / 5 THEN X(1) = 7.4 ELSE IF RND < 1 / 4 THEN X(1) = 9.5 ELSE IF RND < 1 / 3 THEN X(1) = 12.6 ELSE IF RND < 1 / 2 THEN X(1) = 14.7 ELSE X(1) = 16.8

            168 GOTO 280

            215 X(2) = G(1 + FIX(RND * 128))

            219 GOTO 280

            259 X(3) = G(1 + FIX(RND * 128))

        280 NEXT IPP

        281 IF X(1) ^ .8 + X(2) ^ .9 + X(3) ^ .5 < 16 THEN 1670

        284 IF X(1) ^ -1.5 + X(2) ^ 1.7 + X(3) ^ 1.2 > 31 THEN 1670

        324 IF X(1) < 1.1 THEN 1670

        325 IF X(1) > 16.8 THEN 1670

        343 IF X(2) < 1.1 THEN 1670

        345 IF X(2) > 26.7 THEN 1670

        348 IF X(3) < 1.1 THEN 1670

        349 IF X(3) > 26.7 THEN 1670

        441 PD1 = -X(1) ^ 2 * X(2) ^ .816 + X(2) ^ .5 + X(3) ^ 1.2

        469 P = PD1

        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 3

            1455 A(KLX) = X(KLX)

        1459 NEXT KLX

    1670 NEXT i

    1777 IF M < -999999 THEN 1999

    1888 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102].  Its complete output of one run through JJJJ=  -31990 is shown below:

14.7      5.2      9.2      -813.0268      -32000

16.8      3.2      13.5      -704.6483      -31999

14.7      5.2      9.2      -813.0268      -31998

16.8      3.1      14      -685.0154      -31997

16.8      3.1      14      -685.0154      -31996

16.8      3.1      14      -685.0154      -31995

16.8      3.1      14      -685.0154      -31994

16.8      3.1      14      -685.0154      -31993

16.8      3.1      14      -685.0154      -31992

16.8      3.1      14      -685.0154      -31991

16.8      3.1      14      -685.0154      -31990

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.  By using the following computer system and QB64v1000-win [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31990 was 2 seconds, counting from "Starting program...".   One can compare the computational results above with those in Tsai and Lin [90, p. 180].      

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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[94] 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.

[95] 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.

[96] 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.

[97] 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.

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[99] 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.

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

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

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

[103] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002.

[104] Helen Wu, (2015), Geometric Programming

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.  One can easily view this dissertation on Google.