Direct Finding Multiple Optimal Solutions in One Run of a Generalized Geometric Programming Problem: an Illustration
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following problem from Liu, Wang, and Liu [56, p. 12, Example 4.4]:
Minimize
.3578 * X(3) ^ .1 + .8357 * X(1) * X(5)
subject to
.00002584 * X(3) * X(5) – .00006663 * X(2) * X(5) – .0000734 * X(1) * X(4) <= 5
.00085305 * X(2) * X(5) + .00009395 * X(1) * X(4) – .00033085 * X(3) * X(5) <= 5
1.3294 * X(2) ^ -1 * X(5) ^ -1 – .4200 * X(2) * X(5) ^ -1 – .30575 * X(2) ^ -1 * X(3) ^ -2 * X(5) ^ -1 <= 5
.00024186 * X(2) * X(5) + .00010159 * X(1) * X(2) + .00007379 * X(3) ^ 2 <= 5
2.1327 * X(3) ^ -1 * X(5) ^ -1 – .26680 * X(1) * X(5) ^ -1 – .40584 * X(4) * X(5) ^ -1 <= 5
.000229955 * X(3) * X(5) – .00007992 * X(1) * X(3) + .00012157 * X(3) * X(4) <= 5
1<= X(i) <= 60, i=1, 2, 3,…, 6.
0 DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), 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
89 RANDOMIZE JJJJ
90 M = -3D+30
103 FOR j44 = 1 TO 5
107 A(j44) = 1 + RND * 59
109 NEXT j44
123 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 4)
140 B = 1 + FIX(RND * 5)
144 IF RND < .5 THEN 160 ELSE GOTO 167
160 r = (1 – RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * r
165 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) – FIX(RND * 4) ELSE X(B) = A(B) + FIX(RND * 4)
168 NEXT IPP
172 FOR j44 = 1 TO 5
173 IF X(j44) < 1 THEN 1670
175 IF X(j44) > 60 THEN 1670
176 NEXT j44
302 IF .00002584 * X(3) * X(5) – .00006663 * X(2) * X(5) – .0000734 * X(1) * X(4) > 5 THEN 1670
303 IF .00085305 * X(2) * X(5) + .00009395 * X(1) * X(4) – .00033085 * X(3) * X(5) > 5 THEN 1670
304 IF 1.3294 * X(2) ^ -1 * X(5) ^ -1 – .4200 * X(2) * X(5) ^ -1 – .30575 * X(2) ^ -1 * X(3) ^ -2 * X(5) ^ -1 > 5 THEN 1670
305 IF .00024186 * X(2) * X(5) + .00010159 * X(1) * X(2) + .00007379 * X(3) ^ 2 > 5 THEN 1670
306 IF 2.1327 * X(3) ^ -1 * X(5) ^ -1 – .26680 * X(1) * X(5) ^ -1 – .40584 * X(4) * X(5) ^ -1 > 5 THEN 1670
307 IF .000229955 * X(3) * X(5) – .00007992 * X(1) * X(3) + .00012157 * X(3) * X(4) > 5 THEN 1670
461 PD1 = -.3578 * X(3) ^ .1 – .8357 * X(1) * X(5)
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
1889 IF M < -999999999999 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 [101]. Its complete output of one run through JJJJ= -31999 is shown below:
1 3.489730346287245 1.000000000000006
9.661688138816091 1 -1.1935 -32000
1.00000000000002 3.816821831995267 1.00000000000002
5.502263832853543 1.000000000000022 -1.193500000000035
-31999
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 [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999 was 2 seconds, counting from “Starting program…”. One can compare the computational results above with those in Liu, Wang, and Liu [56, p. 12, Example 4.4].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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