Direct Finding Multiple Optimal Solutions in One Run of a Signomial Discrete Programming Problem with Free Variables

 

 

 

 

Direct Finding Multiple Optimal Solutions in One Run of a Signomial Discrete Programming Problem with Free Variables

 

Jsun Yui Wong

 

Similar to the computer programs of the preceding paper, the computer program listed below aims to solve directly the following signomial discrete programming problem in Lin and Tsai [52, pp. 433-434, Example 1]:

 

Minimize

- ( -X(1) ^ 3 * X(3) ^ .5 * X(4) ^ .5 * X(5) ^ .5 + LOG(X(4) * X(5) * X(6) + 1) - X(2) - 5 * X(3) )

 

subject to

 

         X(4) ^ 2 + X(5) - X(1) ^ 3 + 3 * X(3) <= 11

         X(6) ^ 2 - X(1) ^ 2 + 2 * X(2) ^ .5 * X(3) ^ -2 <=<3

         X(4) + X(5) + X(6) + X(1) + X(2) + X(3) <= 6

         -2< X(1) <= 4

          0<=  X(2) <=3  

          4<= X(3) <= 7 

X(1) through X(3) are integer variables

X(4) through X(6) are 0-1 variables.

 

0 REM   DEFDBL A-Z

1 REM    DEFINT I, J, K

2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), 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)

9 FOR JJJJ = -32000 TO 32000

 

    89 RANDOMIZE JJJJ

    90 M = -3D+30

 

    104 IF RND < 1 / 10 THEN A(2) = -1 ELSE IF RND < 1 / 9 THEN A(2) = 0 ELSE IF RND < 1 / 8 THEN A(2) = 1 ELSE IF RND < 1 / 7 THEN A(2) = 4 ELSE IF RND < 1 / 6 THEN A(2) = 5 ELSE IF RND < 1 / 5 THEN A(2) = 6 ELSE IF RND < 1 / 4 THEN A(2) = 7.5 ELSE IF RND < 1 / 3 THEN A(2) = 8 ELSE IF RND < 1 / 2 THEN A(2) = 9 ELSE A(2) = 10

 

    105 IF RND < 1 / 10 THEN A(3) = -27 ELSE IF RND < 1 / 9 THEN A(3) = -18 ELSE IF RND < 1 / 8 THEN A(3) = -9 ELSE IF RND < 1 / 7 THEN A(3) = -7 ELSE IF RND < 1 / 6 THEN A(3) = -4 ELSE IF RND < 1 / 5 THEN A(3) = -1 ELSE IF RND < 1 / 4 THEN A(3) = 1 ELSE IF RND < 1 / 3 THEN A(3) = 3 ELSE IF RND < 1 / 2 THEN A(3) = 4 ELSE A(3) = 5

 

    111 A(1) = -2 + RND * 6

 

    112 A(2) = 0 + RND * 3

 

    113 A(3) = 4 + RND * 3

 

    114 FOR J44 = 4 TO 6

 

        115 A(J44) = INT(RND)

 

    116 NEXT J44

 

    128 FOR I = 1 TO 50000

 

        129 FOR KKQQ = 1 TO 6

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

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

 

            140 B = 1 + FIX(RND * 6)

 

            142 IF B < 4 THEN 160 ELSE GOTO 179

 

            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 188

 

            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 4) ELSE X(B) = A(B) + FIX(RND * 4)

            169 GOTO 188

 

            179 IF A(B) = 0 THEN X(B) = 1 ELSE X(B) = 0

 

        188 NEXT IPP

        191 FOR J44 = 1 TO 6

            192 X(J44) = INT(X(J44))

        194 NEXT J44

 

        227 IF X(1) < -2 THEN 1670

        228 IF X(1) > 4 THEN 1670

 

        229 IF X(2) < 0 THEN 1670

        230 IF X(2) > 3 THEN 1670

 

        231 IF X(3) < 4 THEN 1670

        233 IF X(3) > 7 THEN 1670

        234 FOR J44 = 4 TO 6

 

            235 IF X(J44) < 0 THEN 1670

 

            237 IF X(J44) > 1 THEN 1670

 

        238 NEXT J44

        264 IF X(4) + X(5) + X(6) + X(1) + X(2) + X(3) > 6 THEN 1670

        265 IF X(4) ^ 2 + X(5) - X(1) ^ 3 + 3 * X(3) > 11 THEN 1670

        267 IF X(4) ^ 2 - X(1) ^ 2 + X(6) + X(1) + 2 * X(2) ^ .5 * X(3) ^ -2 > 3 THEN 1670

 

        478 PD1 = -X(1) ^ 3 * X(3) ^ .5 * X(4) ^ .5 * X(5) ^ .5 + LOG(X(4) * X(5) * X(6) + 1) - X(2) - 5 * X(3)

 

        479 P = PD1

 

        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 6

 

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

 

    1670 NEXT I

 

    1677 IF M < -9999999 THEN 1999

 

    1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

 

1999 NEXT JJJJ

 

 

 

This computer program was run with qb64v1000-win [101].  Its complete output of one run through JJJJ= -31980 is shown below:

 

1   0   4   0   0

1   -20   -32000

 

1   0   4   0   0

1   -20   -31999

 

1   0   4   0   0

0   -20   -31996

 

2   0   4   0   0

0   -20   -31994

 

1   0   4   0   0

1   -20   -31988

 

1   0   4   0   0

0   -20   -31984

 

1   0   4   0   0

0   -20   -31983

 

1   0   4   0   0

0   -20   -31982

 

1   0   4   0   0

1   -20   -31980

 

Three distinct solutions are shown above.

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM, 64-bit Operating System, and QB64v1000-win [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31980 was 4 seconds, counting from "Starting program...".  One can compare the computational results above with those in Lin and Tsai [52, p. 436 and p. 437, Table 2]. 

 

Acknowledgement

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

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