"What If..." and
Discrete Variables To Help Solve Nonlinear Programming Problems
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 2], which is as follows:
Minimize
X(1)
subject to
X(1) ^ -1 * X(3) ^ -1 * X(4) ^ -1 * X(6) + 5
* X(1) ^ -1 * X(2) ^ .5 * X(5) * X(6) <= 1
X(3) ^ .3333333 * X(4) - X(5) ^ .5
<= -1
-X(6) - 2 * X(1) * X(2) * X(3) * X(4)
^ 4 * X(5) ^ -1 * X(6) <= -1
30<= X(1) <= 40
.01<= X(2) <= 1
.0001<= X(3) <= 1
15<= X(4) <= 20
15<= X(5) <= 20
.1 <= X(6) <= 1.
0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
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
80 RANDOMIZE JJJJ
90 M = -3D+30
95 A(1) = INT(100 * (30 + RND * 10)) / 100
97 A(2) = INT(100 * (.01 + RND * .99)) /
100
98 A(3) = INT(100 * (.0001 + RND * .9999))
/ 100
99 A(4) = INT(100 * (15 + RND * 5)) / 100
100 A(5) = INT(100 * (15 + RND * 5)) / 100
101 A(6) = INT(100 * (.1 + RND * .9)) / 100
123 FOR I = 1 TO 10000
129 FOR KKQQ = 1 TO 6
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 6)
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) < 30 THEN 1670
192 IF X(1) > 40 THEN 1670
193 IF X(2) < .01 THEN 1670
194 IF X(2) > 1 THEN 1670
195 IF X(3) < .0001 THEN 1670
196 IF X(3) > 1 THEN 1670
197 IF X(4) < 15 THEN 1670
198 IF X(4) > 20 THEN 1670
199 IF X(5) < 15 THEN 1670
200 IF X(5) > 20 THEN 1670
202 IF X(6) < .1 THEN 1670
203 IF X(6) > 1 THEN 1670
204 IF X(3) ^ .3333333 * X(4) - X(5) ^
.5 > -1 THEN 1670
226 IF -X(6) - 2 * X(1) * X(2) * X(3) *
X(4) ^ 4 * X(5) ^ -1 * X(6) > -1 THEN 1670
277 IF X(1) ^ -1 * X(3) ^ -1 * X(4) ^
-1 * X(6) + 5 * X(1) ^ -1 * X(2) ^ .5 * X(5) * X(6) > 1 THEN 1670
463 PD1 = -X(1)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1891 IF M < -30.01 THEN 1999
1899 PRINT A(1), A(2), A(3), A(4), A(5),
A(6), M, JJJJ
1999 NEXT JJJJ
This computer program was
run with qb64v1000-win [102]. The
complete output of one run through 32000 is shown below:
30 .31
.01 15.61 19.95
.43 -30
-19130
30 .44
.01 15.64 19.9
.1 -30
22037
30.01 .18
.01 15.8 19.49
.62 -30.01 23855
30.01 .32
.01 15.3 19.75
.19 -30.01 31850
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 = 32000 was 12 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 2].
Acknowledgement
I would like to acknowledge
the encouragement of Roberta Clark and Tom Clark.
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