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Computer program for solving mixed integer signomial programming problems, third edition

Computer program for solving mixed integer signomial programming problems, third edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following signomial programming problem from Chang [18, Example 5 on pp. 1448-1450]:

Minimize
(.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3))

subject to
X(1) - .0193 * X(3) >= 0
X(2) - .00954 * X(3) >= 0
pie * X(3) ^ 2)* X(4) + (4 / 3) * pie * X(3) ^ 3) >= 750 * 1728
1<= X(1) <= 1.375, discrete variable with discreteness .0625
.625<= X(2) <= 1, discrete variable with discreteness .0625
45<= X(3) <= 55, continuous variable
80<= X(4) <= 110, continuous variable.
One notes the following line 196 through 451.

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

118 pie = 3.141592654

119 A(1) = 1 + (.0625 * FIX(RND * 7))
121 A(2) = .625 + (.0625 * FIX(RND * 7))

123 A(3) = 45 + (RND * 10)
124 A(4) = 80 + (RND * 30)
128 FOR i = 1 TO 500000

129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 4)

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

168 NEXT IPP

196 X(1) = 1

198 X(1) = 1 + (.0625 * FIX(RND * 7))
199 X(2) = .625 + (.0625 * FIX(RND * 7))

241 IF X(1) < 1 THEN 1670
242 IF X(1) > 1.375 THEN 1670

244 IF X(2) < .625 THEN 1670
245 IF X(2) > 1 THEN 1670

246 IF X(3) < 45 THEN 1670

247 IF X(3) > 55 THEN 1670
248 IF X(4) < 80 THEN 1670

249 IF X(4) > 110 THEN 1670

251 IF RND < .5 THEN X(1) = X(1) ELSE X(1) = INT(X(1))

333 X(4) = (750 * 1728 - (4 / 3) * pie * X(3) ^ 3) / (pie * X(3) ^ 2)

359 IF X(1) - .0193 * X(3) < 0 THEN 1670
365 IF X(2) - .00954 * X(3) < 0 THEN 1670
441 IF X(1) < 1 THEN 1670
442 IF X(1) > 1.375 THEN 1670

444 IF X(2) < .625 THEN 1670
445 IF X(2) > 1 THEN 1670

446 IF X(3) < 45 THEN 1670

447 IF X(3) > 55 THEN 1670
448 IF X(4) < 80 THEN 1670

449 IF X(4) > 110 THEN 1670

451 IF RND < .5 THEN X(1) = X(1) ELSE X(1) = INT(X(1))

564 PD1 = -(.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3))

569 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
1775 IF M < -9999999 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 [103]. Its complete output of one run through JJJJ= -31999 is shown below. GW-BASIC also can handle this computer program.

1 .625 52.45322925101003 80.00000000000505
-6963.31119194227 -32000

1          .625          52.45322925101045          80.0000000000021
-6963.311191942241          -31999

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999 was 5 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Chang [18, Example 5, pages 1448-1450, Table 2 on p. 1450].
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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Computer program for solving mixed-integer nonlinear programming (MINLP) problems

Computer program for solving mixed-integer nonlinear programming (MINLP) problems

Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following mixed-integer nonlinear programming (MINLP) problem from Molina-Perez et al. [61, p. 19, Test Problem F4].

Minimize - X(1) - X(2)
subject to
X(2) - 3.4 <= 0
X(1) - X(2) <= 0
X(1) epsilon [ -1, 100 ], X(1) is a continuous variable
X(2) { -1, 0, . . ., 100 }, X(2) is an integer variable.

One notes line 226, line 227, and line 228, which are
226 FOR J44 = 1 TO 2
227 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))
228 NEXT J44.

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

91 FOR J44 = 1 TO 2
92 A(J44) = -1 + RND * 101

93 NEXT J44

128 FOR i = 1 TO 5000

129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(RND * 2)

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

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP
211 FOR J44 = 1 TO 2
213 IF X(J44) < -1 THEN 1670
214 IF X(J44) > 100 THEN 1670
215 NEXT J44

226 FOR J44 = 1 TO 2

227 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))
228 NEXT J44

341 IF X(2) - 3.4 > 0 THEN 1670
344 IF X(1) - X(2) > 0 THEN 1670
568 PD1 = X(1) + X(2)

569 p = PD1
1111 IF p <= M THEN 1670
1452 M = p
1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1670 NEXT i
1775 IF M < -999999999 THEN 1999

1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -30080 is shown below. GW-BASIC also can handle this computer program.

3          3          6          -31835
3 3 6 -31702
3 3 6 -31588
3 3 6 -31084
3 3 6 -30173
3 3 6 -30080

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -30080 was 20 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F4].

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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Computer program for solving mixed-integer nonlinear programming (MINLP) problems

Computer program for solving mixed-integer nonlinear programming (MINLP) problems
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following mixed-integer nonlinear programming (MINLP) problem from Molina-Perez et al. [61, p. 19, Test Problem F3].

Minimize     - X(1) - X(2)
subject to
-X(1) + X(2) - 2.005 <= 0
X(1) - X(2) + .5 <= 0
.505 * X(1) + X(2)- 3.505 <= 0
X(1) epsilon [ -1, 100 ], X(1) is a continuous variable
X(2) { -1, 0, . . ., 100 }, X(2) is an integer variable.

Often one or more of the longer lines are binding. Hence one can try the following line 239, which is 239 X(2) = -.505 * X(1) + 3.505.
One also notes line 326, line 327, and line 328, which are
326 FOR J44 = 1 TO 2
327 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))
328 NEXT J44.

 

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
91 FOR J44 = 1 TO 2
92 A(J44) = -1 + RND * 101

93 NEXT J44
128 FOR i = 1 TO 20000

129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(RND * 2)

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
167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP
211 FOR J44 = 1 TO 2
213 IF X(J44) < -1 THEN 1670
214 IF X(J44) > 100 THEN 1670
215 NEXT J44
226 FOR J44 = 1 TO 2
227 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))
228 NEXT J44
239 X(2) = -.505 * X(1) + 3.505

311 FOR J44 = 1 TO 2

313 IF X(J44) < -1 THEN 1670
314 IF X(J44) > 100 THEN 1670
315 NEXT J44
326 FOR J44 = 1 TO 2
327 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))
328 NEXT J44

337 IF -X(1) + X(2) - 2.005 > 0 THEN 1670
338 IF X(1) - X(2) + .5 > 0 THEN 1670

567 PD1 = X(1) + X(2)
569 p = PD1
1111 IF p <= M THEN 1670
1452 M = p
1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1670 NEXT i
1775 IF M < -999999999 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -319841 is shown below. GW-BASIC also can handle this computer program.

1          3          4          -31999
1 3 4 -31950
1 3 4 -31904
1 3 4 -31867
1 3 4 -31862
1 3 4 -31841

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31841 was 5 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F3].

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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Computer program for solving mixed integer nonlinear programming problems, improved edition

Computer program for solving mixed integer nonlinear programming problems, improved edition

Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following mixed integer nonlinear programming problem from Molina-Perez et al. [61, p.19, Test Problem F2].

Minimize X(1) ^ 2 + (X(2) - 1) ^ 2 + (X(3) - 2) ^ 2
subject to
X(1) ^ 2 + X(2) ^ 2 + .5 * X(3) ^ 2 - 1.5 <=0
X(1) epsilon [ -1, 100 ], X(1) is a continuous variable
X(2), X(3) { -1, 0, . . ., 100 }, X(2) and X(3) are integers.

One notes line 416 through 419, which are 416 FOR J44 = 1 TO 3, 415 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44)), and 419 NEXT J44.

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
91 FOR J44 = 1 TO 3
92 A(J44) = -1 + RND * 101
93 NEXT J44
128 FOR i = 1 TO 2000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
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

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

411 FOR J44 = 1 TO 3
412 IF X(J44) < -1 THEN 1670
413 IF X(J44) > 100 THEN 1670
414 NEXT J44
416 FOR J44 = 1 TO 3
415 IF RND < .5 THEN X(J44) = X(J44) ELSE X(J44) = INT(X(J44))

419 NEXT J44
565 PD1 = -X(1) ^ 2 - (X(2) - 1) ^ 2 - (X(3) - 2) ^ 2 - ABS(X(1) ^ 2 + X(2) ^ 2 + .5 * X(3) ^ 2 - 1.5)

569 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
1775 IF M < -999999999 THEN 1999

1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31996 is shown below. GW-BASIC also can handle this computer program.

0          1          1          -1         -32000
0 1 1 -1 -31999
0 1 1 -1 -31998
0 1 1 -1 -31997
0 1 1 -1 -31996

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31996 was 2 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F2].

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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Computer program for solving mixed integer nonlinear programming problems: an illustration

Computer program for solving mixed integer nonlinear programming problems: an illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following mixed integer nonlinear programming problem from Molina-Perez et al. [61, p. 19, Test Problem F2].

Minimize X(1) ^ 2 + (X(2) - 1) ^ 2 + (X(3) - 2) ^ 2

subject to
X(1) ^ 2 + X(2) ^ 2 + .5 * X(3) ^ 2 - 1.5 <=0
X(1) epsilon [ -1, 100 ], X(1) is a continuous variable
X(2), X(3) { -1, 0, . . ., 100 }, X(2) and X(3) are integers.

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
91 FOR J44 = 1 TO 3
92 A(J44) = -1 + RND * 101
93 NEXT J44
128 FOR i = 1 TO 200000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
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

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP
201 X(2) = INT(X(2))
202 X(3) = INT(X(3))
204 REM X(1) = INT(X(1))
211 FOR J44 = 1 TO 3
212 IF X(J44) < -1 THEN 1670
213 IF X(J44) > 100 THEN 1670
214 NEXT J44
401 X(2) = INT(X(2))
402 X(3) = INT(X(3))
411 FOR J44 = 1 TO 3
412 IF X(J44) < -1 THEN 1670
413 IF X(J44) > 100 THEN 1670
414 NEXT J44

565 PD1 = -X(1) ^ 2 - (X(2) - 1) ^ 2 - (X(3) - 2) ^ 2 - ABS(X(1) ^ 2 + X(2) ^ 2 + .5 * X(3) ^ 2 - 1.5)

569 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
1775 IF M < -999999999 THEN 1999

1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its output of one run through JJJJ= -31991 is summarized below. GW-BASIC also can handle this computer program.

.
.
.
-6.369822387695313D-03          1          1          -1.000081021928054
-31998
.
.
.
-6.66874647140503D-03 1 1 -1.000088944359
-31991

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31991 was 7 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F2].

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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Computer program for solving mixed-integer nonlinear programming problems

Computer program for solving mixed-integer nonlinear programming problems

Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the immediately following mixed-integer nonlinear programming problem of Molina-Perez et al. [61, p.19, Test Problem F6]:

Minimize (X(1) - 10) ^ 3 + (X(2) - 20) ^ 3

subject to
-(X(1) - 5) ^ 2 - (X(2) - 4.86) ^ 2 + 100 <= 0
(X(1) - 8) ^ 2 + (X(2) - 5.48) ^ 2 - 60 <= 0
X(1) epsilon [-1, 100], a continuous variable
X(2) epsilon {-1, 0, ..., 100}, an integer variable.

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
125 FOR J44 = 1 TO 2
126 A(J44) = -1 + RND * 101
127 NEXT J44
128 FOR i = 1 TO 10000
129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(RND * 2)
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
203 X(2) = INT(X(2))
211 FOR J44 = 1 TO 2
213 IF X(J44) < -1 THEN 1670

215 IF X(J44) > 100 THEN 1670
219 NEXT J44
377 IF -(X(1) - 5) ^ 2 - (X(2) - 4.86) ^ 2 + 100 > 0 THEN 1670

378 IF (X(1) - 8) ^ 2 + (X(2) - 5.48) ^ 2 - 60 > 0 THEN 1670
565 PD1 = -(X(1) - 10) ^ 3 - (X(2) - 20) ^ 3
569 p = PD1
1111 IF p <= M THEN 1670
1452 M = p
1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1670 NEXT i
1775 IF M < -999999999 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31991 is shown below. GW-BASIC also can handle this computer program.

14.22498780543337 1 6783.581762415023
-31994

14.2249878048725          1          6783.58176244506
-31991

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31991 was 2 seconds, counting from "Starting program...". One can see the computational results of Molina-Perez et al. [61, p. 19, Test Problem F6].
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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Computer program for solving mixed integer signomial programming problems, second edition

Computer program for solving mixed integer signomial programming problems, second edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following signomial programming problem from Chang [18, Example 5 on pp. 1448-1450]:

Minimize
(.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3))

subject to
X(1) - .0193 * X(3) >= 0
X(2) - .00954 * X(3) >= 0
pie * X(3) ^ 2)* X(4) + (4 / 3) * pie * X(3) ^ 3) >= 750 * 1728

1<= X(1) <= 1.375, discrete variable with discreteness .0625
.625<= X(2) <= 1, discrete variable with discreteness .0625
45<= X(3) <= 55, continuous variable
80<= X(4) <= 110, continuous variable.

One notes below pie is 3.141592654 as in line 118, which is 118 pie = 3.141592654. The earlier edition has pie=3.1416. So the outputs here and there are slightly different.

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

118 pie = 3.141592654

119 A(1) = 1 + (.0625 * FIX(RND * 7))
121 A(2) = .625 + (.0625 * FIX(RND * 7))

123 A(3) = 45 + (RND * 10)
124 A(4) = 80 + (RND * 30)
128 FOR i = 1 TO 500000

129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 4)

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

168 NEXT IPP
198 X(1) = 1 + (.0625 * FIX(RND * 7))
199 X(2) = .625 + (.0625 * FIX(RND * 7))

241 IF X(1) < 1 THEN 1670
242 IF X(1) > 1.375 THEN 1670

244 IF X(2) < .625 THEN 1670
245 IF X(2) > 1 THEN 1670

246 IF X(3) < 45 THEN 1670

247 IF X(3) > 55 THEN 1670
248 IF X(4) < 80 THEN 1670

249 IF X(4) > 110 THEN 1670
333 X(4) = (750 * 1728 - (4 / 3) * pie * X(3) ^ 3) / (pie * X(3) ^ 2)
359 IF X(1) - .0193 * X(3) < 0 THEN 1670
365 IF X(2) - .00954 * X(3) < 0 THEN 1670
441 IF X(1) < 1 THEN 1670
442 IF X(1) > 1.375 THEN 1670

444 IF X(2) < .625 THEN 1670
445 IF X(2) > 1 THEN 1670

446 IF X(3) < 45 THEN 1670

447 IF X(3) > 55 THEN 1670
448 IF X(4) < 80 THEN 1670

449 IF X(4) > 110 THEN 1670
564 PD1 = -(.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3))

569 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
1775 IF M < -9999999 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 [103]. Its complete output of one run through JJJJ= -31998 is shown below. GW-BASIC also can handle this computer program.

1 .625 51.81347150212473 84.57852667116167
-7006.780630167471 -32000

1          .625          51.8134715025891          84.57852666778817
-7006.780630135135          -31999

1 .625 51.81347150170441 84.57852667421514
-7006.78063019674 -31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 6 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Chang [18, Example 5, pp. 1448-1450, Table 2 on p. 1450].
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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https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

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Computer program for solving posynomial fractional programming problems

Computer program for solving posynomial fractional programming problems

Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following illustrating example in Chang [19, p. 49]:

Minimize ((X(1) * X(2) ^ .5 * X(3) ^ 1.2) / ((2 * X(2) * X(3)) + (X(2) ^ 2.1 * X(3) ^ 1.3)))
subject to
2 * X(1) + 3 * X(2) + X(3) >= 8
3 * X(1) + X(2) >= 5
X(2) + 2 * X(3) >= 10
2 * X(1) + X(3) <= 4
0<= X(1) <= 5
1<= X(2) <= 5
1<= X(3) <= 5.

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
119 A(1) = 0 + (RND * 5)
121 A(2) = 1 + (RND * 4)
123 A(3) = 1 + (RND * 4)
128 FOR i = 1 TO 20000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)

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

168 NEXT IPP

241 IF X(1) < 0 THEN 1670
242 IF X(1) > 5 THEN 1670

244 IF X(2) < 1 THEN 1670
245 IF X(2) > 5 THEN 1670

246 IF X(3) < 1 THEN 1670

247 IF X(3) > 5 THEN 1670

361 IF 2 * X(1) + 3 * X(2) + X(3) < 8 THEN 1670

363 IF 3 * X(1) + X(2) < 5 THEN 1670
364 IF X(2) + 2 * X(3) < 10 THEN 1670

367 IF 2 * X(1) + X(3) > 4 THEN 1670

564 PD1 = -((X(1) * X(2) ^ .5 * X(3) ^ 1.2) / ((2 * X(2) * X(3)) + (X(2) ^ 2.1 * X(3) ^ 1.3)))

569 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

1711 REM
1775 IF M < -9999922 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31994 is shown below. GW-BASIC also can handle with this computer program.

6.46898105295496D-12 4.99999999999981513 3.97799624330288
-3.502425596265055D-13 -31997

3.852102156925237D-15          4.9999999999999999          3.99890667913426
-2.085096049461978D-16       -31994

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31994 was 2 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Chang [19, pp. 49-51].

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.
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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Computer program for solving mixed integer signomial programming problems

Computer program for solving mixed integer signomial programming problems

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 Chang [18, Example 4, pp. 1445-1448]:

Minimize ((X(1) * X(2) ^ .5 * X(3) ^ 1.2 + 2 * X(1)))
subject to
2 * X(1) + X(2) + X(3) >= 8
X(2) + 2 * X(3) >= 10.5
X(1) + 2 * X(3) <= 10
1 <= X(1), X(2), X(3) <=5.

 

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
119 A(1) = 1 + (RND * 4)

121 A(2) = 1 + (RND * 4)
123 A(3) = 1 + (RND * 4)

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 * 3)

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

168 NEXT IPP

241 IF X(1) < 1 THEN 1670
242 IF X(1) > 5 THEN 1670

244 IF X(2) < 1 THEN 1670
245 IF X(2) > 5 THEN 1670

246 IF X(3) < 1 THEN 1670

247 IF X(3) > 5 THEN 1670

361 IF 2 * X(1) + X(2) + X(3) < 8 THEN 1670

363 REM IF 3 * X(1) + X(2) < 5 THEN 1670
364 IF X(2) + 2 * X(3) < 10.5 THEN 1670

367 IF X(1) + 2 * X(3) > 10 THEN 1670

441 IF X(1) < 1 THEN 1670
442 IF X(1) > 5 THEN 1670

444 IF X(2) < 1 THEN 1670
445 IF X(2) > 5 THEN 1670

446 IF X(3) < 1 THEN 1670

447 IF X(3) > 5 THEN 1670

565 PD1 = -((X(1) * X(2) ^ .5 * X(3) ^ 1.2 + 2 * X(1)))

569 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
1775 IF M < -9.455 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31921 is shown below. GW-BASIC also can handle with this computer program.

1.000000000000501 1.501801238433961 4.499099142364442
-9.44829561746171 -31944

1.000000000000074       1.500426118477611       4.499786787732746
-9.446250309723006          -31921

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31921 was 5 seconds, counting from "Starting program...". One can compare the computational results presented above with the results in Chang [18, Example 4, pp. 1445-1448].

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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Computer program for mixed integer posynomial fractional programming problems

Computer program for mixed integer fractional posynomial programming problems

Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following mathematical programming problem from Chang [18, p. 382, Example 3]:

Minimize (X(1) * X(2) * X(3) + X(1) * X(2))

subject to
2 * X(1) + X(2) + X(3) >= 0
X(1) + 2 * X(2) + X(3) <= 5
-2 <= X(1) <= 2
-3 <= X(2) <= 3
-2 <= X(3) <= 3.
"This problem cannot be solved by LPF approach," Chang [18, p. 382, Example 3]. LPF stands for logarithmic piecewise function.

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
95 PI = 3.1416
116 FOR J44 = 2 TO 5

117 A(J44) = 2 + (RND * 3)
118 NEXT J44

119 A(1) = -2 + (RND * 4)
121 A(2) = -3 + (RND * 6)
123 A(3) = -2 + (RND * 5)

128 FOR i = 1 TO 10000

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)

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

168 NEXT IPP

241 IF X(1) < -2 THEN 1670
242 IF X(1) > 2 THEN 1670

244 IF X(2) < -3 THEN 1670
245 IF X(2) > 3 THEN 1670

246 IF X(3) < -2 THEN 1670

247 IF X(3) > 3 THEN 1670
361 IF X(1) + X(2) + X(3) < 0 THEN 1670

363 IF X(1) + 2 * X(2) + X(3) > 5 THEN 1670

441 IF X(1) < -2 THEN 1670
442 IF X(1) > 2 THEN 1670

444 IF X(2) < -3 THEN 1670
445 IF X(2) > 3 THEN 1670

446 IF X(3) < -2 THEN 1670

447 IF X(3) > 3 THEN 1670

564 PD1 = -(X(1) * X(2) * X(3) + X(1) * X(2))

569 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

1711 REM
1775 IF M < 22 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31998 is shown below. GW-BASIC also can handle this computer program.

1.999999988448793 -2.999999999575211 2.999999999864416
23.99999985717371 -32000

1.999999998683421      -2.999999999929011       2.999999997783706
23.99999997033537          -31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 3 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Chang [18, Example 3, pp. 282-283]. The computer program above can be used as a template to solve mixed integer fractional posynomial programming [MIFPP] problems.

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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