"What If..." and Discrete Variables To Help Solve Nonlinear Programming Problems: Another Illustration
Second Edition
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following mathematical formulation in Wang, Zhang, and Gao [99, p. 1514, Example 3], which is as follows:
Minimize
X(1)
subject to
.274 * X(4) * X(5) ^ 4 + 2520.66 * X(2) * X(5) ^ 5 + X(1) * X(4) ^ 2 - X(1) * X(2) * X(3) * X(4) + 1 <= 1
X(2) * X(3) ^ -1 * X(4) <= 1
X(2) * X(5) ^ 4 <= 1
X(4) * X(5) ^ 3 <= 1
10 ^ -12 <= X(1) <= 2
20<= X(2) <= 35
120 <= X(3) <= 160
1 <= X(4) <= 10
10 ^ 6 <= X(5) <= 1.
0 DEFDBL A-Z
1 REM DEFINT I, J
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)
79 FOR JJJJ = -32000 TO 32000 STEP .01
89 RANDOMIZE JJJJ
90 M = -3D+30
95 A(1) = (100 * (1D-12 + RND * (2 - 1D-12))) / 100
97 A(2) = INT(100 * (20 + RND * 15)) / 100
98 A(3) = INT(100 * (120 + RND * 40)) / 100
99 A(4) = INT(100 * (1 + RND * 9)) / 100
100 A(5) = (100 * (1D-06 + RND * (1 - 1D-06))) / 100
124 FOR I = 1 TO 4000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 5)
144 IF RND < .5 THEN 160 ELSE GOTO 166
160 r = (1 - RND * 2) * A(B)
164 X(B) = A(B) + FIX((RND ^ (RND * 15)) * r)
165 GOTO 168
166 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
191 IF X(1) < 1D-12 THEN 1670
192 IF X(1) > 2 THEN 1670
193 IF X(2) < 20 THEN 1670
194 IF X(2) > 35 THEN 1670
195 IF X(3) < 120 THEN 1670
196 IF X(3) > 160 THEN 1670
197 IF X(4) < 1 THEN 1670
198 IF X(4) > 10 THEN 1670
199 IF X(5) < 1D-06 THEN 1670
200 IF X(5) > 1 THEN 1670
205 IF X(2) * X(5) ^ 4 > 1 THEN 1670
206 IF X(4) * X(5) ^ 3 > 1 THEN 1670
208 IF X(2) * X(3) ^ -1 * X(4) > 1 THEN 1670
277 IF .274 * X(4) * X(5) ^ 4 + 2520.66 * X(2) * X(5) ^ 5 + X(1) * X(4) ^ 2 - X(1) * X(2) * X(3) * X(4) + 1 > 1 THEN 1670
463 PD1 = -X(1)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 5
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1891 IF M < -1D-05 THEN 1999
1899 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [102]. The complete output of one run through JJJJ=
-26174.04000093257 is shown below:
3.457070396970928D-06 26.81 121.14 3.87
3.604132965558767D-02 -3.457070396970928D-06 -31841.3100000254
1.311302685103044D-06 22.19 131 5.220000000000001
2.179428408056498D-02 -1.311302685103044D-06 -31183.17000013075
3.337860607466325D-06 34.47 136.08 3.03
7.472077123641968D-03 -3.337860607466325D-06 -26174.04000093257
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 [102], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -26174.040000993257 was 45 minutes, counting from "Starting program...". One can compare the computational results above with those in Wang, Zhang, and Gao [99, p. 1515, Table 1, Example 3].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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