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

 

Computer Program for Solving Mixed-Integer/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 3]:          

Minimize

    

-1*  ( -2 * X(1) - 3 * X(2) - 1.5 * X(4) - 2 * X(5) + .5 * X(6)    )         

  

subject to 

                  X(1)^2 + X(4)  = 1.25

                 X(2) ^ 1.5 + 1.5 * X(5) = 3

         X(1) + X(4) <= 1.6 

        1.333 * X(2) + X(5) <= 3 

        -X(4) - X(5) + X(6) <= 0

        X(1), X(2), X(3) >= 0    

        (X(4), X(5), X(6)) element  { 0, 1 }.     

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 6

        107 A(J44) = RND

    108 NEXT J44

    123 FOR I = 1 TO 60000

        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)

            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

        177 FOR J44 = 4 TO 6

            178 X(J44) = INT(X(J44))

        179 NEXT J44

        185 IF (1.25 - X(4)) < 0## THEN 1670

        188 X(1) = (1.25 - X(4)) ^ .5

        198 FOR J44 = 1 TO 6

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

        200 NEXT J44

        207 IF X(4) > 1## THEN 1670

        208 IF X(5) > 1## THEN 1670

        209 IF X(6) > 1## THEN 1670

        319 IF X(1) + X(4) > 1.6 THEN 1670

        320 IF 1.333 * X(2) + X(5) > 3 THEN 1670

        321 IF -X(4) - X(5) + X(6) > 0 THEN 1670

        446 PD1 = -2 * X(1) - 3 * X(2) - 1.5 * X(4) - 2 * X(5) + .5 * X(6) - 100000 * ABS(X(2) ^ 1.5 + 1.5 * X(5) - 3)

        466 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

    1889 IF M < -7.7 THEN 1999

    1899 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= -31997 is shown below:

1.118034      1.310371      2.083565       0      1

1         -7.669487         -32000

1.118034   1.310371   .7253669    0   1

1      -7.669487      -31999

1.118034   1.310371   .5079655    0   1

1      -7.669487      -31997

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 = -31997 was 3 seconds, counting from "Starting program...".  One can compare the computational results above with those in Porn, Bjork, and Westerlund [72, p. 116 (9 of 13), Example 3].          

Acknowledgement

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

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Computer Program for Solving a Structural Design Problem Involving Mixed-Discrete Variables    

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem from  in Tsai, Lin, and Wen [88, p. 8/11, Example 2]:

Minimize        

    - 1* ( -.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     

    

          .0193 * X(3) - X(1) <= 0,

        .00954 * X(3) - X(2) <= 0,

        1296000 - 3.1416 * X(3) ^ 2 * X(4) - (4 / 3) * 3.1416 * X(3) ^ 3 <= 0,

         X(4) <= 240  

            

       x(1), x(2) element { .0625, .125, .1875,..., 6.1875 }, x(1) and x(2) are discrete variables.

   10 <= x(3) <= 200, 10 <= x(4)<= 200, x(3) and x(4) are integer variables.

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)

79 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ

    90 M = -3D+30

    111 A(3) = 10 + FIX(RND * 191)

    113 A(4) = 10 + FIX(RND * 191)

    117 A(5) = 1 + FIX(RND * 99)

    120 A(6) = 1 + FIX(RND * 99)

    123 FOR I = 1 TO 30000

        129 FOR KKQQ = 3 TO 6

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

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

            140 B = 3 + 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 * 4) ELSE X(B) = A(B) + FIX(RND * 4)

        168 NEXT IPP

        182 FOR J44 = 3 TO 6

            184 X(J44) = INT(X(J44))

        187 NEXT J44

        214 IF .0193 * X(3) > .0625 * X(5) THEN 1670

        215 IF .00954 * X(3) > .0625 * X(6) THEN 1670

        218 IF 1296000 - 3.1416 * X(3) ^ 2 * X(4) - (4 / 3) * 3.1416 * X(3) ^ 3 > 0 THEN 1670

        219 IF X(4) > 240 THEN 1670

        301 IF X(3) < 10 THEN 1670

        302 IF X(3) > 200 THEN 1670

        303 IF X(4) < 10 THEN 1670

        304 IF X(4) > 200 THEN 1670

        305 IF X(5) < 1 THEN 1670

        306 IF X(5) > 99 THEN 1670

        307 IF X(6) < 1 THEN 1670

        308 IF X(6) > 99 THEN 1670

        444 X(1) = .0625 * X(5): X(2) = .0625 * X(6)

        459 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)

        466 P = PD1

        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 3 TO 6

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

    1670 NEXT I

    1889 IF M < -6111 THEN 1999

    1899 PRINT 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= -31934 is shown below:

42      178      13      7      -6074.999

-31976

42   178   13   7   -6074.999

-31951

42   178   13   7   -6074.999

-31934

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 = -31934 was 5 seconds, counting from "Starting program...".  One can compare the computational results above with those in Tsai, Lin, and Wen [88, p. 10/11, Table 3].  

Acknowledgement

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

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