Jsun Yui Wong
The computer program listed below seeks to solve the last instance in Table 5 of Lu, [32, p. 114]:
Minimize X(1) ^ 3 * X(2) * X(3) ^ 3 * X(4) + X(1) ^ 3 * X(2) * X(3) * X(4) ^ 2
subject to
X(1) ^ 3 * X(2) * X(3) ^ 2 + X(3) * X(4)<=-500
- X(1) ^ 3 * X(2) * X(3) + X(3) ^ 2 * X(4)<=500
-6<=X(1) <= 6.75
-6<= X(2) <= 6.75
-1<= X(3) <= 9.2
-9<= X(4) <= 6.3
where X(1) through X(4) are free-sign discrete variables; here the number of discrete values is 512.
X(5) and X(6) below are slack variables.
0 DEFDBL A-Z
2 DEFINT K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32111 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
61 A(1) = -6 + FIX(RND * 512) * 12.75 / 511
63 A(2) = -6 + FIX(RND * 512) * 12.75 / 511
65 A(3) = -1 + FIX(RND * 512) * 10.2 / 511
67 A(4) = -9 + FIX(RND * 512) * 15.3 / 511
128 FOR I = 1 TO 2000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 j = 1 + FIX(RND * 4)
183 r = (1 - RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
268 IF X(1) < -6 THEN 1670
269 IF X(1) > 6.75 THEN 1670
272 IF X(2) < -6 THEN 1670
273 IF X(2) > 6.75 THEN 1670
274 IF X(3) < -1 THEN 1670
275 IF X(3) > 9.2 THEN 1670
284 IF X(4) < -9 THEN 1670
285 IF X(4) > 6.3 THEN 1670
308 X(5) = -500 - X(1) ^ 3 * X(2) * X(3) ^ 2 - X(3) * X(4)
309 X(6) = 500 + X(1) ^ 3 * X(2) * X(3) - X(3) ^ 2 * X(4)
401 FOR J47 = 5 TO 6
402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0
403 NEXT J47
443 POBA = -X(1) ^ 3 * X(2) * X(3) ^ 3 * X(4) - X(1) ^ 3 * X(2) * X(3) * X(4) ^ 2 + 1000000 * (X(5) + X(6))
466 P = POBA
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 72810 THEN 1999
1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ = -31995.0000000008 is shown below:
3.229896900873914 -1.315536597736727 6.060244169953957
6.299724378997351 0 0 72813.92169807518
-31998.09000000031
3.309100297995073 -1.23243415832176 6.043621618088537
6.299938047635811 0 0 72816.111093616
-31997.68000000037
-2.231428778900335 3.97735652428387 6.066947644653092
6.299999972210325 0 0 72813.31551915896
-31997.22000000045
3.441393159311623 -1.092310317177698 6.050514577356587
6.299999660072656 0 0 72816.09250444798
-31995.0000000008
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [49], the wall-clock time for obtaining the output through
JJJJ = -31995.0000000008 was 32 seconds, total, including the time for “Creating .EXE file.” One can compared the computational results above with those on page 114 of Lu [32, Table 5].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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