Jsun Yui Wong

The computer program listed below is about extending and solving Lu's Example 2 for r =1384 discrete values [32, p. 111 and Table 5], which is shown immediately below:

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 1384 for each discrete variable, [32, p. 114, Table 5].

X(5) and X(6) below are slack variables added. One notes line 268, which is 268 IF X(1) < -6## THEN 1670.

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

31 unitsize(1) = 12.75 / 1383

33 unitsize(2) = 12.75 / 1383

35 unitsize(3) = 10.2 / 1383

37 unitsize(4) = 15.3 / 1383

61 A(1) = -6 + FIX(RND * 1384) * unitsize(1)

63 A(2) = -6 + FIX(RND * 1384) * unitsize(2)

65 A(3) = -1 + FIX(RND * 1384) * unitsize(3)

67 A(4) = -9 + FIX(RND * 1384) * unitsize(4)

128 FOR I = 1 TO 6000

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 REM r = (1 - RND * 2) * A(j)

187 REM X(j) = A(j) + (RND ^ (RND * 10)) * r

197 IF RND < .5 THEN X(j) = A(j) - FIX(RND * 11) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 11) * unitsize(j)

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 < 72815.55 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 = -31950.61000000791 is shown below:

5.339479392624724 -.293383947939263 6.043383947939264

6.299999999999996 0 0 72815.81934346557

-31958.45000000665

5.339479392624726 -.293383947939263 6.043383947939264

6.299999999999995 0 0 72815.81934346558

-31950.61000000791

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 (not CPU time) for obtaining the output through

JJJJ = -31950.61000000791 was 220 seconds, including the time for “Creating .EXE file.” One can compare 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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