Four Similar Computer Programs for Solving Geometric Programming Problems, Including Those with Free Variables
Jsun Yui Wong
The four computer programs listed below seek to solve all the geometric programming examples in Li and Tsai [51].
Example 1 in Li and Tsai, 2005 [51]
Similar to the computer program of the preceding paper, the first computer program listed below seeks to solve the formulation on page 9 of Li and Tsai [51] :
Minimize
X(1) ^ 2.1 * X(2) * X(3) ^ 3 + X(1)
subject to
-X(1) - X(2) ^ 2 <= -5
X(2) - X(1) + X(3) <= 13
0 <= X(1) <=3
-2 <= X(2) <= 3
-2 <= X(3) <= 3.
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)
9 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 A(1) = 0 + (RND * 3)
112 A(2) = -2 + (RND * 5)
113 A(3) = -2 + (RND * 5)
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(RND * 3)
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
196 IF X(1) < 0 THEN 1670
197 IF X(1) > 3 THEN 1670
198 IF X(2) < -2 THEN 1670
199 IF X(2) > 3 THEN 1670
200 IF X(3) < -2 THEN 1670
201 IF X(3) > 3 THEN 1670
219 IF -X(1) - X(2) ^ 2 > -5 THEN 1670
221 IF X(2) - X(1) + X(3) > 13 THEN 1670
459 PD1 = -X(1) ^ 2.1 * X(2) * X(3) ^ 3 - X(1)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M < 333 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [101]. Its complete output of one run through JJJJ= -31981 is shown below:
3 -2 3 539.4359 -31998
3 -2 3 539.4359 -31997
3 -2 3 539.4359 -31981
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 [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31981 was 3 seconds, counting from "Starting program...". One can compare the computational results above with those in Li and Tsai [51].
Example 2 in Li and Tsai, 2005 [51]
Similar to the computer program above, the computer program listed immediately below seeks to solve the formulation on page 11 of Li and Tsai [51] :
Minimize
400 * X(1) ^ .9 + 1000 + 22 * (X(2) - 14.7) ^ 1.2 + X(4)
subject to
X(2) = EXP(-3950 / (X(3) + 460) + 11.86)
144 * (80 - X(3)) = X(1) * X(4)
0 <= X(1) <= 15.1
14.7 <= X(2) <= 94.2
-459.67 <= X(3) <= 80
0 <= X(4).
0 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)
9 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 A(1) = 0 + (RND * 15.1)
112 A(2) = 14.7 + (RND * 79.5)
113 A(3) = -459.67 + (RND * 539.67)
114 A(4) = (RND * 10)
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 4)
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
196 IF X(1) < 0 THEN 1670
197 IF X(1) > 15.1 THEN 1670
198 IF X(2) < 14.7 THEN 1670
199 IF X(2) > 94.2 THEN 1670
200 IF X(3) < -459.67 THEN 1670
201 IF X(3) > 80 THEN 1670
202 IF X(4) < 0 THEN 1670
218 X(3) = (144 * 80 - X(1) * X(4)) / 144
220 X(2) = EXP(-3950 / (X(3) + 460) + 11.86)
459 PD1 = -400 * X(1) ^ .9 - 1000 - 22 * (X(2) - 14.7) ^ 1.2 - X(4)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M <= -9999999999 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 [101]. Its complete output of one run through JJJJ= -31999 is shown below:
7.103713991400354D-18 94.1778659402041 80
8.136067055234835D-13 -5194.866244203784 -32000
2.404310389455815D-17 94.1778659402041 80
3.586516987154525D-13 -5194.866244203784 -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 (3.89 GB usable), 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 Li and Tsai [51].
Example 3 in Li and Tsai, 2005 [51]
The computer program listed immediately below seeks to solve the second formulation on page 11 of Li and Tsai [51] :
Minimize
X(1) ^ -2 * X(2) ^ -.5 * X(3) ^ -1 + 8 * X(1) ^ -1 * X(4) ^ 2 - 8 * X(4)
subject to
X(1) - X(2) ^ .5 * X(3) ^ .5 <= 3
2 * X(1) + X(2) - X(3) + X(4) <= 6
1 <= X(1) <= 5
3 <= X(2) <= 7
1 <= X(3) <= 10
1 <= X(4) <= 5.
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)
9 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 A(1) = 1 + (RND * 4)
112 A(2) = 3 + (RND * 4)
113 A(3) = 1 + (RND * 9)
114 A(4) = 1 + (RND * 4)
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 4)
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
196 IF X(1) < 1 THEN 1670
197 IF X(1) > 5 THEN 1670
198 IF X(2) < 3 THEN 1670
199 IF X(2) > 7 THEN 1670
200 IF X(3) < 1 THEN 1670
201 IF X(3) > 10 THEN 1670
202 IF X(4) < 1 THEN 1670
203 IF X(4) > 5 THEN 1670
235 IF X(1) - X(2) ^ .5 * X(3) ^ .5 > 3 THEN 1670
237 IF 2 * X(1) + X(2) - X(3) + X(4) > 6 THEN 1670
461 PD1 = -X(1) ^ -2 * X(2) ^ -.5 * X(3) ^ -1 - 8 * X(1) ^ -1 * X(4) ^ 2 + 8 * X(4)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M < -999999999 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 [101]. Its complete output of one run through JJJJ= -31998 is shown below:
5 3.497652 9.999683 2.500422 9.997861
-32000
5 3.498328 9.99999 2.500001 9.997862
-31999
5 3.498937 9.999455 2.500042 9.997862
-31998
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 [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 2 seconds, counting from "Starting program...". One can compare the computational results above with those in Li and Tsai [51].
Example 4 in Li and Tsai, 2005 [51]
The computer program listed immediately below seeks to solve the formulation on page 12 of Li and Tsai [51] :
Minimize
- ( -(X(4) - 1) ^ 2 - (X(5) - 2) ^ 2 - (X(6) - 1) ^ 2 + LOG(X(7) + 1) - (X(1) - 1) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 3) ^ 2)
subject to
X(4) + X(5) + X(6) + X(1) + X(2) + X(3) <= 5
X(6) ^ 2 + X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 <= 5.5
X(4) + X(1) <= 1.2
X(5) + X(2) <= 1.8
X(6) + X(3) <= 2.5
X(7) + X(1) <= 1.2
X(5) ^ 2 + X(2) ^ 2 <= 1.64
X(6) ^ 2 + X(3) ^ 2 <= 4.25
X(5) ^ 2 + X(3) ^ 2 <= 4.64
0 <= X(1) <= 1.2
0 <= X(2) <= 1.8
0 <= X(3) <= 2.5
X(i) element {0, 1}, i= 4, 5, 6, 7.
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)
9 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 A(1) = 0 + (RND * 1.2)
112 A(2) = 0 + (RND * 1.8)
113 A(3) = 0 + (RND * 2.5)
116 FOR J44 = 4 TO 7
117 A(J44) = INT(RND)
118 NEXT J44
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 7
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 5)
140 B = 1 + FIX(RND * 7)
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
196 IF X(1) < 0 THEN 1670
197 IF X(1) > 1.2 THEN 1670
198 IF X(2) < 0 THEN 1670
199 IF X(2) > 1.8 THEN 1670
200 IF X(3) < 0 THEN 1670
201 IF X(3) > 2.5 THEN 1670
214 FOR J44 = 4 TO 7
215 X(J44) = INT(X(J44))
216 NEXT J44
311 IF X(4) + X(1) > 1.2 THEN 1670
312 IF X(5) + X(2) > 1.8 THEN 1670
313 IF X(6) + X(3) > 2.5 THEN 1670
314 IF X(7) + X(1) > 1.2 THEN 1670
315 IF X(5) ^ 2 + X(2) ^ 2 > 1.64 THEN 1670
316 IF X(6) ^ 2 + X(3) ^ 2 > 4.25 THEN 1670
317 IF X(5) ^ 2 + X(3) ^ 2 > 4.64 THEN 1670
318 IF X(6) ^ 2 + X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 > 5.5 THEN 1670
319 IF X(4) + X(5) + X(6) + X(1) + X(2) + X(3) > 5 THEN 1670
458 IF (X(7) + 1) < .0001 THEN 1670
461 PD1 = -(X(4) - 1) ^ 2 - (X(5) - 2) ^ 2 - (X(6) - 1) ^ 2 + LOG(X(7) + 1) - (X(1) - 1) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 3) ^ 2
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 7
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M < -5 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [101]. Its complete output of one run through JJJJ= -31994 is shown below:
.1978585 .7999998 1.907878 1 1
0 1 -4.583015 -31998
.2 .8 1.907878 1 1
0 1 -4.579583 -31997
.1999998 .8 1.907878 1 1
0 1 -4.579583 -31994
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 [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31994 was 3 seconds, counting from "Starting program...". One can compare the computational results above with those in Li and Tsai [51].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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