A computer program to solve geometric programming problems: an illustration using Rijckaert and Martens' Problem 6 [76], Second Edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve the following geometric programming problem in Rijckaert and Martens [76, p. 229], Lundell, westerlund, and Westerlund [58, p. 403], and Tsai and Lin [91, p. 490, Example 5]:

Minimize

- ( -2 * X(1) ^ .9 * X(2) ^ -1.5 * X(3) ^ -3 - 5 * X(4) ^ -.3 * X(5) ^ 2.6 - 4.7 * X(6) ^ -1.8 * X(7) ^ -.5 * X(8) )

subject to

7.2 * X(1) ^ -3.8 * X(2) ^ 2.2 * X(3) ^ 4.3 + .5 * X(4) ^ -.7 * X(5) ^ -1.6 + .2 * X(6) ^ 4.3 * X(7) ^ -1.9 * X(8) ^ 8.5 <= 1

10 * X(1) ^ 2.3 * X(2) ^ 1.7 * X(3) ^ 4.5 <= 1

.6 * X(4) ^ -2.1 * X(5) ^.4 <= 1

6.2 * X(6) ^ 4.5 * X(7) ^ -2.7 * X(8) ^ -.6 <= 1

3.1 * X(1) ^ 1.6 * X(2) ^ .4 *X(3) ^ -3.8 <= 1

3.7 * X(4) ^ 5.4 * X(5) ^ 1.3 <= 1

.3 * X(6) ^ -1.1 * X(7) ^ 7.3 * X(8) ^ -5.6 <= 1

.1 <= X(j) <= 10, j=1, ..., 8.

(The superscripts above can be clearer (less confusing) below:)

0 REM DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), G(128), J44(222), 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), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

87 RANDOMIZE JJJJ

88 M = -3D+30

107 FOR J44 = 1 TO 8

108 A(J44) = .1 + RND * 1

109 NEXT J44

128 FOR i = 1 TO 1000000

129 FOR KKQQ = 1 TO 8

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 8)

141 B = 1 + FIX(RND * 8)

142 IF RND < 1.5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r

164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

171 X(5) = (1 / (.6 * X(4) ^ -2.1)) ^ 2.5

175 X(2) = (1 / (3.1 * X(1) ^ 1.6 * X(3) ^ -3.8)) ^ 2.5

176 X(6) = (1 / (6.2 * X(7) ^ -2.7 * X(8) ^ -.6)) ^ .222222222222

177 IF 3.7 * X(4) ^ 5.4 * X(5) ^ 1.3 > 1 THEN 1670

179 IF 10 * X(1) ^ 2.3 * X(2) ^ 1.7 * X(3) ^ 4.5 > 1 THEN 1670

205 IF .3 * X(6) ^ -1.1 * X(7) ^ 7.3 * X(8) ^ -5.6 > 1 THEN 1670

207 IF 7.2 * X(1) ^ -3.8 * X(2) ^ 2.2 * X(3) ^ 4.3 + .5 * X(4) ^ -.7 * X(5) ^ -1.6 + .2 * X(6) ^ 4.3 * X(7) ^ -1.9 * X(8) ^ 8.5 > 1 THEN 1670

211 REM ALLLHS1 = 7.2 * X(1) ^ -3.8 * X(2) ^ 2.2 * X(3) ^ 4.3 + .5 * X(4) ^ -.7 * X(5) ^ -1.6 + .2 * X(6) ^ 4.3 * X(7) ^ -1.9 * X(8) ^ 8.5 - 1

431 FOR J44 = 1 TO 8

433 IF X(J44) < .1 THEN 1670

434 IF X(J44) > 10 THEN 1670

435 NEXT J44

967 PD1 = -2 * X(1) ^ .9 * X(2) ^ -1.5 * X(3) ^ -3 - 5 * X(4) ^ -.3 * X(5) ^ 2.6 - 4.7 * X(6) ^ -1.8 * X(7) ^ -.5 * X(8) - 0 * 1000000 * ABS(ALLLHS1)

969 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 8

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1777 IF M < -29.229999999 THEN 1999

1778 REM IF M < -29.54 THEN 1999

1888 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -29810 is shown below:

.9687111 .1990227 1.121226 .7844015 1.002186

.7001135 1.092237 .9704894 -29.22987 -29893

**.9691276 .1988578 1.12133 .7844079 1.002229****.7020876 1.096323 .9746857 -29.22972 -29810**

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [103], the wall-clock time **(not CPU time)** for obtaining the output through JJJJ = -29810 was 3 hours and 22 minutes, counting from "Starting program...". One can compare the computational results above with those in Rijckaert and Martens [76, p. 229], Lundell, Westerlund, and Westerlund [58, p. 403], and Tsai and Lin [91, p. 491].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

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

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