Direct Way To Solve Signomial Geometric Programming Problems: Another Illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem from Hou, Shen, and Chen [40, pp. 10-11, Example 9]:

Minimize

(3 + X(1) * X(3)) * (X(1) * X(2) * X(3) * X(4) + 2 * X(1) * X(3) + 2) ^ (2 / 3)

subject to

-3 * (2 * X(1) * X(2) + 3 * X(1) * X(2) * X(4)) * (2 * X(1) * X(3) + 4 * X(1) * X(4) - X(2)) - (X(1) * X(3) + 3 * X(1) * X(2) * X(4)) * (4 * X(3) * X(4) + 4 * X(1) * X(3) * X(4) + X(1) * X(3) - 4 * X(1) * X(2) * X(4)) ^ (1 / 3) + 3 * (X(4) + 3 * X(1) * X(3) * X(4)) * (3 * X(1) * X(2) * X(3) + 3 * X(1) * X(4) + 2 * X(3) * X(4) - 3 * X(1) * X(2) * X(4)) ^ (1 / 4) <= -309.219315,

-2 * (3 * X(3) + 3 * X(1) * X(2) * X(3)) * (X(1) * X(2) * X(3) + 4 * X(2) * X(4) - X(3) * X(4)) ^ 2 + (3 * X(1) * X(2) * X(3)) * (3 * X(3) + 2 * X(1) * X(2) * X(3) + 3 * X(4)) ^ 4 - (X(2) * X(3) * X(4) + X(1) * X(3) * X(4)) * (4 * X(1) - 1) ^ 3 / 4 - 3 * (3 * X(3) * X(4) + 2 * X(1) * X(3) * X(4)) * (X(1) * X(2) * X(3) * X(4) + X(3) * X(4) - 4 * X(1) * X(2) * X(3) - 2 * X(1)) ^ 4 <= -78243.910551,

-3 * (3 * X(1) * X(3) * X(4)) * (2 * X(4) + 2 * X(1) * X(2) - X(2) - X(3)) ^ 2 + 2 * (X(1) * X(2) * X(4) + 3 * X(1) * X(3) * X(4)) * (X(1) * X(2) + 2 * X(2) * X(3) + 4 * X(2) - X(2) * X(3) * X(4) - X(1) * X(3)) ^ 4 <= 9618,

0 <= X(i) <= 5, i=1,...,4.

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)

85 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

111 FOR J44 = 1 TO 5

112 A(J44) = FIX(RND * 6)

113 NEXT J44

128 FOR I = 1 TO 95000

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 * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

186 FOR J44 = 1 TO 4

187 IF X(J44) < 0## THEN 1670

188 IF X(J44) > 5 THEN 1670

189 NEXT J44

315 IF (4 * X(3) * X(4) + 4 * X(1) * X(3) * X(4) + X(1) * X(3) - 4 * X(1) * X(2) * X(4)) < 0## THEN 1670

319 IF (3 * X(1) * X(2) * X(3) + 3 * X(1) * X(4) + 2 * X(3) * X(4) - 3 * X(1) * X(2) * X(4)) < 0## THEN 1670

391 IF -3 * (2 * X(1) * X(2) + 3 * X(1) * X(2) * X(4)) * (2 * X(1) * X(3) + 4 * X(1) * X(4) - X(2)) - (X(1) * X(3) + 3 * X(1) * X(2) * X(4)) * (4 * X(3) * X(4) + 4 * X(1) * X(3) * X(4) + X(1) * X(3) - 4 * X(1) * X(2) * X(4)) ^ (1 / 3) + 3 * (X(4) + 3 * X(1) * X(3) * X(4)) * (3 * X(1) * X(2) * X(3) + 3 * X(1) * X(4) + 2 * X(3) * X(4) - 3 * X(1) * X(2) * X(4)) ^ (1 / 4) > -309.219315 THEN 1670

392 IF -2 * (3 * X(3) + 3 * X(1) * X(2) * X(3)) * (X(1) * X(2) * X(3) + 4 * X(2) * X(4) - X(3) * X(4)) ^ 2 + (3 * X(1) * X(2) * X(3)) * (3 * X(3) + 2 * X(1) * X(2) * X(3) + 3 * X(4)) ^ 4 - (X(2) * X(3) * X(4) + X(1) * X(3) * X(4)) * (4 * X(1) - 1) ^ 3 / 4 - 3 * (3 * X(3) * X(4) + 2 * X(1) * X(3) * X(4)) * (X(1) * X(2) * X(3) * X(4) + X(3) * X(4) - 4 * X(1) * X(2) * X(3) - 2 * X(1)) ^ 4 > -78243.910551 THEN 1670

393 IF -3 * (3 * X(1) * X(3) * X(4)) * (2 * X(4) + 2 * X(1) * X(2) - X(2) - X(3)) ^ 2 + 2 * (X(1) * X(2) * X(4) + 3 * X(1) * X(3) * X(4)) * (X(1) * X(2) + 2 * X(2) * X(3) + 4 * X(2) - X(2) * X(3) * X(4) - X(1) * X(3)) ^ 4 > 9618 THEN 1670

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

1779 IF M < -5.84 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its output of one run through JJJJ= -31287 is summarized below:

4.999359447221122 1.484225924864508D-02 4.312780371342594D-02

4.997601233611117 -5.83953703391377 -31994

.

.

.

4.999624115694577 .014831261808832 4.309994746203202D-02

4.999436033504944 -5.838873471325552 -31287

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 (3.89 GB usable), 64-bit Operating System, and QB64v1000-win [102], the wall-clock time (**not CPU** **time**) for obtaining the output through JJJJ = -31287 was 3 minutes, counting from "Starting program...". One can compare the computational results above with those in Hou, Shen, and Chen [40, p. 10, Table 2, Example 9]. **Acknowledgement**

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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