Jsun Yui Wong
The computer program listed below seeks to solve Lin and Lu's Example 2 [27, p. 710], which is shown immediately below:
Minimize X(0) ^ 3 * X(1) * X(2) ^ 3 * X(3) + X(0) ^ 3 * X(1) * X(2) * X(3) ^ 2
subject to
X(0) ^ 3 * X(1) * X(2) ^ 2 + X(2) * X(3)<=-500
- X(0) ^ 3 * X(1) * X(2) + X(2) ^ 2 * X(3)<=500
-5<=X(0) <=5
-5<=X(1) <=5
X(2) epsilon {-1, 0, 1, 4, 5, 6, 7.5, 8, 9, 10}
X(3) epsilon { -27, -18, -9, -7, -4, -1, 1, 3, 4, 5}.
X(4) and X(5) below are slack variables added.
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 32000 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
40 A(0) = -5 + RND * 10
41 A(1) = -5 + RND * 10
43 IF RND < .5 THEN A(2) = 0 ELSE A(2) = 9
46 IF RND < .5 THEN A(3) = -9 ELSE A(3) = 3
128 FOR I = 1 TO 3000
129 FOR KKQQ = 0 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 j = FIX(RND * 4)
200 IF j = 0 THEN GOTO 201 ELSE IF j = 1 THEN GOTO 204 ELSE IF j = 2 THEN GOTO 208 ELSE GOTO 210
201 r = (1 - RND * 2) * A(0)
202 X(0) = A(0) + (RND ^ (RND * 10)) * r
203 GOTO 222
204 r = (1 - RND * 2) * A(1)
205 X(1) = A(1) + (RND ^ (RND * 10)) * r
207 GOTO 222
208 IF RND < .1 THEN X(j) = -1 ELSE IF RND < 1 / 9 THEN X(j) = 0 ELSE IF RND < 1 / 8 THEN X(j) = 1 ELSE IF RND < 1 / 7 THEN X(j) = 4 ELSE IF RND < 1 / 6 THEN X(j) = 5 ELSE IF RND < 1 / 5 THEN X(j) = 6 ELSE IF RND < 1 / 4 THEN X(j) = 7.5 ELSE IF RND < 1 / 3 THEN X(j) = 8 ELSE IF RND < 1 / 2 THEN X(j) = 9 ELSE X(j) = 10
209 GOTO 222
210 IF RND < .1 THEN X(j) = -27 ELSE IF RND < 1 / 9 THEN X(j) = -18 ELSE IF RND < 1 / 8 THEN X(j) = -9 ELSE IF RND < 1 / 7 THEN X(j) = -7 ELSE IF RND < 1 / 6 THEN X(j) = -4 ELSE IF RND < 1 / 5 THEN X(j) = -1 ELSE IF RND < 1 / 4 THEN X(j) = 1 ELSE IF RND < 1 / 3 THEN X(j) = 3 ELSE IF RND < 1 / 2 THEN X(j) = 4 ELSE X(j) = 5
222 NEXT IPP
258 IF X(0) < -5## THEN 1670
259 IF X(0) > 5## THEN 1670
268 IF X(1) < -5## THEN 1670
269 IF X(1) > 5## THEN 1670
308 X(4) = -500 - X(0) ^ 3 * X(1) * X(2) ^ 2 - X(2) * X(3)
309 X(5) = 500 + X(0) ^ 3 * X(1) * X(2) - X(2) ^ 2 * X(3)
401 FOR J47 = 4 TO 5
402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0
403 NEXT J47
443 POBA = -X(0) ^ 3 * X(1) * X(2) ^ 3 * X(3) - X(0) ^ 3 * X(1) * X(2) * X(3) ^ 2 + 1000000 * (X(4) + X(5))
466 P = POBA
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 0 TO 5
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 360000 THEN 1999
1907 PRINT A(0), A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [50]. The complete output through JJJJ = -31999.76000000004 is shown below:
-4.879113209656626 4.537200067061007 1
-27 0 0 369953.9999999876
-31999.89000000002
4.997259945889812 -4.222938844609257 1
-27 0 0 369953.9999998124
-31999.76000000004
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 [50], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =-31999.76000000004 was 3 seconds, not including the time for “Creating .EXE file” (11 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on page 710 of Li and Lu [27].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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