A computer program to solve mixed-integer nonlinear programming (MINLP) problems: an illustration using Example 3 in Lundell and Westerlund [58]
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve the following noncovex mixed-integer nonlinear programming (MINLP) example in Lundell and Westerlund [58, p. 129, Example 3]:
Minimize
(X(1) - 2) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 4.5) ^ 2
subject to
X(1) ^ 2 * X(2) - X(2) ^ -1 + X(2) ^ 2 * X(3) + 5 * SIN(X(2)) * COS(X(3)) <= 12
1 <= X(1), X(2) <= 3
1 <= X(3) <= 8
X(1), X(2) EPSILON R, X(3) EPSILON Z.
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
87 RANDOMIZE JJJJ
88 M = -3D+30
108 A(1) = 1 + RND * 2
111 A(2) = 1 + RND * 2
115 A(3) = 1 + FIX(RND * 8)
128 FOR i = 1 TO 50000
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)
142 IF RND < 1.5 THEN 160 ELSE GOTO 167
160 r = (1 - RND * 2) * A(B)
162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168
167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1
168 NEXT IPP
172 X(3) = INT(X(3))
195 IF ((X(2) ^ -1 - X(2) ^ 2 * X(3) - 5 * SIN(X(2)) * COS(X(3)) + 12) / X(2)) < 0 THEN 1670
197 X(1) = ((X(2) ^ -1 - X(2) ^ 2 * X(3) - 5 * SIN(X(2)) * COS(X(3)) + 12) / X(2)) ^ .5
433 IF X(1) < 1 THEN 1670
434 IF X(1) > 3 THEN 1670
437 IF X(2) < 1 THEN 1670
439 IF X(2) > 3 THEN 1670
444 IF X(3) < 1 THEN 1670
446 IF X(3) > 8 THEN 1670
972 PD1 = -(X(1) - 2) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 4.5) ^ 2
979 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
1778 IF M < -10 THEN 1999
1888 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= -31990 is shown below:
1.860231 1.606857 4 -.4240965 -31998
1.860309 1.60683 4 -.4240965 -31997
1.860341 1.606818 4 -.4240965 -31995
1.860216 1.606863 4 -.4240965 -31994
1.86032 1.606826 4 -.4240965 -31991
1.860316 1.606827 4 -.4240965 -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 [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31990 was 4 seconds, counting from "Starting program...". One can compare the computational results above with those in Lundell and Westerlund [58, p. 130, Example 3].
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.
References
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