Computer Program for Solving Mixed-Integer/Mixed-Discrete Nonlinear Programming Problems Involving Signomial Parts: an Illustration

Computer Program for Solving Mixed-Integer/Mixed-Discrete Nonlinear Programming Problems Involving Signomial Parts: an Illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem from Porn, Bjork, and Westerlund [72, p. 116 (9 of 13), Example 4]:

Minimize

  -1*( -6 * X(1) - 16 * X(2) + 9 * X(5) - 10 * X(6) + 15 * X(9) )

subject to

     X(1) + X(2) - X(3) - X(4) = 0
     X(3) - X(5) + X(7) =0
     X(4) +X(8) - X(9) = 0
     - X(6) + X(7) + X(8) = 0
     -2.5 * X(5) + 2 * X(7) + X(3) * X(10) <=  0 
     2 * X(8) - 1.5 * X(9) + X(4) * X(10) <=  0 

     3 * X(1) + X(2) -X(3)*X(10) - X(4)*X(10)  =  0

X(i) >=0, i=1,…, 9
X(10)>=1
X(i) = (300, 300, 100, 200, 100, 300, 100, 200, 200, 3).

For this problem, using “what if” analysis via exploring what if X(1) through X(9) are integer variables can be beneficial. One notes line 170, which is
170 X(J44) = INT(X(J44)).

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(20), H(99), L(99), U(99), X(20), 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)
79 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3D+30

104 FOR J44 = 1 TO 9
    107 A(J44) = FIX(RND * 300)



108 NEXT J44

109 A(10) = 1 + RND * 2


123 FOR I = 1 TO 60000

    129 FOR KKQQ = 1 TO 10
        130 X(KKQQ) = A(KKQQ)
    131 NEXT KKQQ
    139 FOR IPP = 1 TO FIX(1 + RND * 5)

        140 B = 1 + FIX(RND * 10)

        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 * 5) ELSE X(B) = A(B) + FIX(RND * 5)

    168 NEXT IPP

    169 FOR J44 = 1 TO 9
        170 X(J44) = INT(X(J44))


    171 NEXT J44




    172 X(3) = X(5) - X(7)
    173 X(4) = -X(8) + X(9)
    174 X(6) = X(7) + X(8)
    175 X(1) = -X(2) + X(3) + X(4)

    178 X(10) = (-3 * X(1) - X(2)) / (-X(3) - X(4))



    198 FOR J44 = 1 TO 9

        199 IF X(J44) < 0## THEN 1670

    200 NEXT J44
    201 IF X(10) < 1## THEN 1670



    207 IF X(1) > 300## THEN 1670
    208 IF X(2) > 300## THEN 1670
    209 IF X(3) > 100## THEN 1670
    217 IF X(4) > 200## THEN 1670
    218 IF X(5) > 100## THEN 1670
    219 IF X(6) > 300## THEN 1670
    227 IF X(7) > 100## THEN 1670
    228 IF X(8) > 200## THEN 1670
    229 IF X(9) > 200## THEN 1670
    231 IF X(10) > 3## THEN 1670

    251 IF -2.5 * X(5) + 2 * X(7) + X(3) * X(10) > 0 THEN 1670

    255 IF 2 * X(8) - 1.5 * X(9) + X(4) * X(10) > 0 THEN 1670




    449 PD1 = -6 * X(1) - 16 * X(2) + 9 * X(5) - 10 * X(6) + 15 * X(9)

    466 P = PD1

    1111 IF P <= M THEN 1670
    1452 M = P
    1454 FOR KLX = 1 TO 10

        1455 A(KLX) = X(KLX)
    1456 NEXT KLX
1670 NEXT I
1889 IF M < -77777777 THEN 1999

1899 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [101]. Its output of one run through JJJJ= -31885 is summarized below:

6 106 0 112 0
88 0 88 200 1.107143
388 -31999
.
.
.
0 100 0 100 0
100 0 100 200 1
400 -31942
.
.
.
0 100 0 100 0
100 0 100 200 1
400 -31885

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM (3.89 GB usable), 64-bit Operating System, and QB64v1000-win [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31885 was 18 seconds, counting from “Starting program…”. One can compare the computational results above with those in Porn, Bjork, and Westerlund [72, p. 117 (10 of 13), Example 4].

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

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