A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration, corrected edition
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following geometric programming problem from Tsai [93, Example 2, p. 840].
Minimize - (-.25 * 3.141592654 ^ 2 * X(1) ^ 2 * X(2) * (X(3) + 2) )
subject to
32 * X(1) ^ -3 * X(2) ^ 2 + 11.68 * X(1) ^ -2 * X(2) - 19.68 * X(1) ^ -1 - 756 * 3.141592654 * (X(2) - X(1)) <= 0
.069565 * X(2) ^ 3 * X(3) + 105 * X(1) ^ 5 * (X(3) + 2) - 1400 * X(1) ^ 4 <= 0
.02087 * X(2) ^ 3 * X(3) - 600 * X(1) ^ 4 <=0
14375 * X(1) ^ 4 - 5.6 * X(2) ^ 3 * X(3) <=0
-3 * X(1) - X(2) <=0
where X(1) epsilon { .207, .225, .244, .263, .283, .307, .331, .362, .394, .4375, .5}
X(2) is a discrete variable with discreteness .01, .6 <= X(2) <= 3
X(3) is an integer variable, 2 <= X(3) <= 38.
Whereas line 92 of the earlier edition is
92 IF RND < 1 / 10 THEN A(J44) = .207 ELSE IF RND < 1 / 9 THEN A(1) = .225 ELSE IF RND < 1 / 8 THEN A(J44) = .244 ELSE IF RND < 1 / 7 THEN A(J44) = .263 ELSE IF RND < 1 / 6 THEN A(J44) = .283 ELSE IF RND < 1 / 5 THEN A(J44) = .307 ELSE IF RND < 1 / 4 THEN A(J44) = .318 ELSE IF RND < 1 / 3 THEN A(1) = .362 ELSE IF RND < 1 / 2 THEN A(1) = .4375 ELSE A(1) = .5,
line 92 here is
92 IF RND < 1 / 11 THEN A(J44) = .207 ELSE IF RND < 1 / 10 THEN A(1) = .225 ELSE IF RND < 1 / 9 THEN A(J44) = .244 ELSE IF RND < 1 / 8 THEN A(J44) = .263 ELSE IF RND < 1 / 7 THEN A(J44) = .283 ELSE IF RND < 1 / 6 THEN A(J44) = .307 ELSE IF RND < 1 / 5 THEN A(J44) = .331 ELSE IF RND < 1 / 4 THEN A(1) = .362 ELSE IF RND < 1 / 3 THEN A(1) = .394 ELSE IF RND < 1 / 2 THEN A(1) = .4375 ELSE A(1) = .5.
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)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
91 FOR J44 = 1 TO 1
92 IF RND < 1 / 11 THEN A(J44) = .207 ELSE IF RND < 1 / 10 THEN A(1) = .225 ELSE IF RND < 1 / 9 THEN A(J44) = .244 ELSE IF RND < 1 / 8 THEN A(J44) = .263 ELSE IF RND < 1 / 7 THEN A(J44) = .283 ELSE IF RND < 1 / 6 THEN A(J44) = .307 ELSE IF RND < 1 / 5 THEN A(J44) = .331 ELSE IF RND < 1 / 4 THEN A(1) = .362 ELSE IF RND < 1 / 3 THEN A(1) = .394 ELSE IF RND < 1 / 2 THEN A(1) = .4375 ELSE A(1) = .5
97 NEXT J44
104 A(2) = .6 + FIX(RND * 241) * .01
113 A(3) = 2 + FIX(RND * 37)
128 FOR i = 1 TO 5000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 3)
156 IF B < 3 THEN 165 ELSE GOTO 167
165 X(B) = A(B)
166 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
211 FOR J44 = 3 TO 3
213 X(3) = INT(X(3))
219 NEXT J44
404 IF 32 * X(1) ^ -3 * X(2) ^ 2 + 11.68 * X(1) ^ -2 * X(2) - 19.68 * X(1) ^ -1 - 756 * 3.141592654 * (X(2) - X(1)) > 0 THEN 1670
406 IF .069565 * X(2) ^ 3 * X(3) + 105 * X(1) ^ 5 * (X(3) + 2) - 1400 * X(1) ^ 4 > 0 THEN 1670
408 IF .02087 * X(2) ^ 3 * X(3) - 600 * X(1) ^ 4 > 0 THEN 1670
409 IF 14375 * X(1) ^ 4 - 5.6 * X(2) ^ 3 * X(3) > 0 THEN 1670
411 IF -3 * X(1) - X(2) > 0 THEN 1670
563 PD1 = -.25 * 3.141592654 ^ 2 * X(1) ^ 2 * X(2) * (X(3) + 2)
569 p = PD1
1111 IF p <= M THEN 1670
1452 M = p
1454 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT i
177 IF M < -2.8 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -20337 is shown below. GW-BASIC also can handle this computer program.
.307 1.67 5 -2.718511 -29870
.283 1.23 9 -2.673686 -26812
.283 1.23 9 -2.673686 -20337
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -20337 was 70 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Tsai [93, Example 2, p. 840].
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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