Substance

A Correction for the Preceding Paper

A Correction for the Preceding Paper

Jsun Yui Wong

The title of the preceding paper should read as follows:

A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems:  yet another illustration.

A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: still another illustration Jsun Yui Wong Similar to the computer program of the preceding paper, the computer program listed below aims to solve di

A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: still another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following geometric programming problem from Ballard et al. [11, Problem 3].

Minimize

-1*(-1 * .0204 * (X(1) ^ 2 * X(4) + X(1) * X(2) * X(4) + X(1) * X(3) * X(4)) - 1 * .0187 * (X(1) * X(2) * X(3) + 1.57 * X(2) ^ 2 * X(3) + X(2) * X(3) * X(4)) - 1 * .0607 * (X(1) ^ 2 * X(4) * X(5) ^ 2 + X(1) * X(2) * X(4) * X(5) ^ 2) - 1 * .0437 * (X(1) * X(2) * X(3) * X(6) ^ 2 + 1.57 * X(2) ^ 2 * X(3) * X(6) ^ 2 + X(2) * X(3) * X(4) * X(6) ^ 2) - 1 * .0607 * X(1) * X(3) * X(4) * X(5) ^ 2 )
subject to
2070 / (X(1) * X(2) * X(3) * X(4) * X(5) * X(6)) <= 1
.00062 * (X(1) ^ 2 * X(4) * X(5) ^ 2 + X(1) * X(2) * X(4) * X(5) ^ 2 + X(1) * X(3) * X(4) * X(5) ^ 2) + .00058 * (X(1) * X(2) * X(3) * X(6) ^ 2 + X(2) * X(3) * X(4) * X(6) ^ 2 + 1.57 * X(2) ^ 2 * X(3) * X(6) ^ 2) <= 1
X(J44) >= 0, J44= 1, 2, 3,..., 6

One notes line 117, line 121, and line 126, which are
117 FOR J44 = 1 TO 7
121 A(J44) = RND * 20
126 NEXT J44.

 

0 REM 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 STEP .1
87 RANDOMIZE JJJJ
88 M = -3D+30
90 GOTO 117
117 FOR J44 = 1 TO 7
121 A(J44) = RND * 20
126 NEXT J44
127 REM A(1) = 1000
128 FOR i = 1 TO 50000
129 FOR KKQQ = 1 TO 6
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 6)
140 B = 1 + FIX(RND * 6)

156 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 REM X(B) = A(B)

166 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
211 FOR J44 = 1 TO 1

213 REM X(J44) = INT(X(J44))

219 NEXT J44

408 IF 2070 / (X(1) * X(2) * X(3) * X(4) * X(5) * X(6)) > 1 THEN 1670

409 IF .00062 * (X(1) ^ 2 * X(4) * X(5) ^ 2 + X(1) * X(2) * X(4) * X(5) ^ 2 + X(1) * X(3) * X(4) * X(5) ^ 2) + .00058 * (X(1) * X(2) * X(3) * X(6) ^ 2 + X(2) * X(3) * X(4) * X(6) ^ 2 + 1.57 * X(2) ^ 2 * X(3) * X(6) ^ 2) > 1 THEN 1670

422 FOR J44 = 1 TO 6

433 IF X(J44) < 0 THEN 1670

455 NEXT J44

567 PD1 = -1 * .0204 * (X(1) ^ 2 * X(4) + X(1) * X(2) * X(4) + X(1) * X(3) * X(4)) - 1 * .0187 * (X(1) * X(2) * X(3) + 1.57 * X(2) ^ 2 * X(3) + X(2) * X(3) * X(4)) - 1 * .0607 * (X(1) ^ 2 * X(4) * X(5) ^ 2 + X(1) * X(2) * X(4) * X(5) ^ 2) - 1 * .0437 * (X(1) * X(2) * X(3) * X(6) ^ 2 + 1.57 * X(2) ^ 2 * X(3) * X(6) ^ 2 + X(2) * X(3) * X(4) * X(6) ^ 2) - 1 * .0607 * X(1) * X(3) * X(4) * X(5) ^ 2

669 p = PD1
1111 IF p <= M THEN 1670
1452 M = p
1454 FOR KLX = 1 TO 6

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1670 NEXT i
1777 IF M < -135.5 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its output of one run through
JJJJ= -31129.31 is summarized below. GW-BASIC also can handle this computer program.

5.428269 4.49112 11.08345 11.77721 .732458
.8880848    -135.3105 -31639.71
.
.
.
5.184072       4.761842       10.4606       12.09968       .7760271
.8537226       -135.248       -31129.31

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 = -31129.31 was 22 minutes, counting from "Starting program...". One can compare the computational results presented above with the computational results in Ballard et al. [11, Problem 3, Table 1, p. 265].
Incidentally, one notes 135.248 / 135.1023 = 1.001078442/1.
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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A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration

A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following geometric programming problem from Ballard et al. [11, Problem 4, pp. 265-266].

Minimize - 1.1 * X(1) ^ .51 * X(2) ^ .47 * X(3) ^ .24 - .9 * X(1) ^ .51 * X(4) ^ .53 * X(5) ^ .19 - 1.4 * X(1) ^ .51 * X(6) ^ .5 * X(7) ^ .21
subject to
(50 * X(2) + 120 * X(4) + 85 * X(6)) / 10000 <= 1
X(1) / 1000 <= 1
(X(3) + X(5) + X(7)) / 500 <= 1
X(J44) >=0, J44=1, 2, 3,..., 7.

One notes line 117, line 121, and line 126, which are
117 FOR J44 = 1 TO 7
121 A(J44) = 0 + RND * 1100
126 NEXT J44
One also notes line 156 and line 167, which are
156 IF RND < .5 THEN 160 ELSE GOTO 167
167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 10) ELSE X(B) = A(B) + FIX(RND * 10).

0 REM 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 STEP .1
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 7
121 A(J44) = 0 + RND * 1100
126 NEXT J44
128 FOR i = 1 TO 10000
129 FOR KKQQ = 1 TO 7
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 7)
140 B = 1 + FIX(RND * 7)

156 IF RND < .5 THEN 160 ELSE GOTO 167

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

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

166 GOTO 168

167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 10) ELSE X(B) = A(B) + FIX(RND * 10)

168 NEXT IPP
211 FOR J44 = 1 TO 1

213 REM X(J44) = INT(X(J44))

219 NEXT J44

408 IF (50 * X(2) + 120 * X(4) + 85 * X(6)) / 10000 > 1 THEN 1670

409 IF X(1) / 1000 > 1 THEN 1670

411 IF (X(3) + X(5) + X(7)) / 500 > 1 THEN 1670

422 FOR J44 = 1 TO 7

433 IF X(J44) < 0 THEN 1670

455 NEXT J44
565 PD1 = 1.1 * X(1) ^ .51 * X(2) ^ .47 * X(3) ^ .24 + .9 * X(1) ^ .51 * X(4) ^ .53 * X(5) ^ .19 + 1.4 * X(1) ^ .51 * X(6) ^ .5 * X(7) ^ .21

569 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
177 IF M < 2410 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 [103]. Its complete output of one run through
JJJJ= -30472.89 is shown below. GW-BASIC also can handle this computer program.

999.9918 106.1718 298.1491 8.401453 29.61746
43.31565 172.1945 2410.47 -31908.46

999.9968     107.1903     299.0378     3.924303     15.09245
49.03944     185.8448     2418.106     -30472.89

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 = -30472.89 was 5 minutes, counting from "Starting program...". One can compare the computational results presented above with the computational results in Ballard et al. [11, Problem 4, Table 1, p. 265].

Incidentally, one notes 2418.106/2420.284=.999100105/1.

 

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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A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration, corrected edition

A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration, corrected edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following geometric programming problem from Tsai [93, Example 2, p. 840].

Minimize - (-.25 * 3.141592654 ^ 2 * X(1) ^ 2 * X(2) * (X(3) + 2) )

subject to
32 * X(1) ^ -3 * X(2) ^ 2 + 11.68 * X(1) ^ -2 * X(2) - 19.68 * X(1) ^ -1 - 756 * 3.141592654 * (X(2) - X(1)) <= 0

.069565 * X(2) ^ 3 * X(3) + 105 * X(1) ^ 5 * (X(3) + 2) - 1400 * X(1) ^ 4 <= 0

.02087 * X(2) ^ 3 * X(3) - 600 * X(1) ^ 4 <=0

14375 * X(1) ^ 4 - 5.6 * X(2) ^ 3 * X(3) <=0
-3 * X(1) - X(2) <=0

where X(1) epsilon { .207, .225, .244, .263, .283, .307, .331, .362, .394, .4375, .5}
X(2) is a discrete variable with discreteness .01, .6 <= X(2) <= 3
X(3) is an integer variable, 2 <= X(3) <= 38.

Whereas line 92 of the earlier edition is
92 IF RND < 1 / 10 THEN A(J44) = .207 ELSE IF RND < 1 / 9 THEN A(1) = .225 ELSE IF RND < 1 / 8 THEN A(J44) = .244 ELSE IF RND < 1 / 7 THEN A(J44) = .263 ELSE IF RND < 1 / 6 THEN A(J44) = .283 ELSE IF RND < 1 / 5 THEN A(J44) = .307 ELSE IF RND < 1 / 4 THEN A(J44) = .318 ELSE IF RND < 1 / 3 THEN A(1) = .362 ELSE IF RND < 1 / 2 THEN A(1) = .4375 ELSE A(1) = .5,
line 92 here is
92 IF RND < 1 / 11 THEN A(J44) = .207 ELSE IF RND < 1 / 10 THEN A(1) = .225 ELSE IF RND < 1 / 9 THEN A(J44) = .244 ELSE IF RND < 1 / 8 THEN A(J44) = .263 ELSE IF RND < 1 / 7 THEN A(J44) = .283 ELSE IF RND < 1 / 6 THEN A(J44) = .307 ELSE IF RND < 1 / 5 THEN A(J44) = .331 ELSE IF RND < 1 / 4 THEN A(1) = .362 ELSE IF RND < 1 / 3 THEN A(1) = .394 ELSE IF RND < 1 / 2 THEN A(1) = .4375 ELSE A(1) = .5.

 

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

87 RANDOMIZE JJJJ
88 M = -3D+30

91 FOR J44 = 1 TO 1
92 IF RND < 1 / 11 THEN A(J44) = .207 ELSE IF RND < 1 / 10 THEN A(1) = .225 ELSE IF RND < 1 / 9 THEN A(J44) = .244 ELSE IF RND < 1 / 8 THEN A(J44) = .263 ELSE IF RND < 1 / 7 THEN A(J44) = .283 ELSE IF RND < 1 / 6 THEN A(J44) = .307 ELSE IF RND < 1 / 5 THEN A(J44) = .331 ELSE IF RND < 1 / 4 THEN A(1) = .362 ELSE IF RND < 1 / 3 THEN A(1) = .394 ELSE IF RND < 1 / 2 THEN A(1) = .4375 ELSE A(1) = .5

97 NEXT J44
104 A(2) = .6 + FIX(RND * 241) * .01
113 A(3) = 2 + FIX(RND * 37)

128 FOR i = 1 TO 5000

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)

156 IF B < 3 THEN 165 ELSE GOTO 167
165 X(B) = A(B)

166 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
211 FOR J44 = 3 TO 3

213 X(3) = INT(X(3))

219 NEXT J44

404 IF 32 * X(1) ^ -3 * X(2) ^ 2 + 11.68 * X(1) ^ -2 * X(2) - 19.68 * X(1) ^ -1 - 756 * 3.141592654 * (X(2) - X(1)) > 0 THEN 1670
406 IF .069565 * X(2) ^ 3 * X(3) + 105 * X(1) ^ 5 * (X(3) + 2) - 1400 * X(1) ^ 4 > 0 THEN 1670

408 IF .02087 * X(2) ^ 3 * X(3) - 600 * X(1) ^ 4 > 0 THEN 1670

409 IF 14375 * X(1) ^ 4 - 5.6 * X(2) ^ 3 * X(3) > 0 THEN 1670

411 IF -3 * X(1) - X(2) > 0 THEN 1670

563 PD1 = -.25 * 3.141592654 ^ 2 * X(1) ^ 2 * X(2) * (X(3) + 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
177 IF M < -2.8 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= -20337 is shown below. GW-BASIC also can handle this computer program.

.307 1.67 5 -2.718511 -29870
.283       1.23       9       -2.673686       -26812
.283 1.23 9 -2.673686 -20337

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 = -20337 was 70 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Tsai [93, Example 2, p. 840].

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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A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration

A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following geometric programming problem from Tsai [93, Example 2, p. 840].

Minimize - (-.25 * 3.141592654 ^ 2 * X(1) ^ 2 * X(2) * (X(3) + 2) )
subject to
32 * X(1) ^ -3 * X(2) ^ 2 + 11.68 * X(1) ^ -2 * X(2) - 19.68 * X(1) ^ -1 - 756 * 3.141592654 * (X(2) - X(1)) <= 0

.069565 * X(2) ^ 3 * X(3) + 105 * X(1) ^ 5 * (X(3) + 2) - 1400 * X(1) ^ 4 <= 0

.02087 * X(2) ^ 3 * X(3) - 600 * X(1) ^ 4 <=0

14375 * X(1) ^ 4 - 5.6 * X(2) ^ 3 * X(3) <=0

-3 * X(1) - X(2) <=0

where X(1) epsilon { .207, .225, .244, .263, .283, .307, .318, .362, .394, .4375, .5}
X(2) is a discrete variable with discreteness .01, .6 <= X(2) <= 3
X(3) is an in teger variable, 2 <= X(3) <= 38.

One notes line 165 and line 167, which are  165 X(B) = A(B) and 167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2).

 

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

87 RANDOMIZE JJJJ
88 M = -3D+30

91 FOR J44 = 1 TO 1
92 IF RND < 1 / 10 THEN A(J44) = .207 ELSE IF RND < 1 / 9 THEN A(1) = .225 ELSE IF RND < 1 / 8 THEN A(J44) = .244 ELSE IF RND < 1 / 7 THEN A(J44) = .263 ELSE IF RND < 1 / 6 THEN A(J44) = .283 ELSE IF RND < 1 / 5 THEN A(J44) = .307 ELSE IF RND < 1 / 4 THEN A(J44) = .318 ELSE IF RND < 1 / 3 THEN A(1) = .362 ELSE IF RND < 1 / 2 THEN A(1) = .4375 ELSE A(1) = .5

97 NEXT J44
104 A(2) = .6 + FIX(RND * 241) * .01
113 A(3) = 2 + FIX(RND * 37)

128 FOR i = 1 TO 5000

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)

156 IF B < 3 THEN 165 ELSE GOTO 167
165 X(B) = A(B)

166 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
211 FOR J44 = 3 TO 3

213 X(3) = INT(X(3))

219 NEXT J44

404 IF 32 * X(1) ^ -3 * X(2) ^ 2 + 11.68 * X(1) ^ -2 * X(2) - 19.68 * X(1) ^ -1 - 756 * 3.141592654 * (X(2) - X(1)) > 0 THEN 1670
406 IF .069565 * X(2) ^ 3 * X(3) + 105 * X(1) ^ 5 * (X(3) + 2) - 1400 * X(1) ^ 4 > 0 THEN 1670

408 IF .02087 * X(2) ^ 3 * X(3) - 600 * X(1) ^ 4 > 0 THEN 1670

409 IF 14375 * X(1) ^ 4 - 5.6 * X(2) ^ 3 * X(3) > 0 THEN 1670

411 IF -3 * X(1) - X(2) > 0 THEN 1670
563 PD1 = -.25 * 3.141592654 ^ 2 * X(1) ^ 2 * X(2) * (X(3) + 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
177 IF M < -2.8 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= -21490 is shown below. GW-BASIC also can handle this computer program.

.307 1.67 5 -2.718511 -31098
.283       1.23       9       -2.673686       -29424
.283 1.23 9 -2.673686 -294241  
.283 1.23 9 -2.673686 -21490

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 = -21490 was 70 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Tsai [93, Example 2, p. 840].

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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A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration

A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems: another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following geometric programming problem from Tsai [93, Example 3, pp. 840-841].

Minimize 2 * X(1) + X(2) + 2 ^ .5 * X(3)

subject to
-1 + (3 ^ .5 * X(2) + 1.932 * X(3)) /( 1.5 * X(1) * X(2) + 2 ^ .5 * X(2) * X(3) + 1.319 * X(1) * X(3)) <= 0
-1 + (.634 * X(1) + 2.828 * X(3)) / (1.5 * X(1) * X(2) + 2 ^ .5 * X(2) * X(3) + 1.319 * X(1) * X(3)) <= 0
-1 + (.5 * X(1) - 2 * X(2)) / (1.5 * X(1) * X(2) + 2 ^ .5 * X(2) * X(3) + 1.319 * X(1) * X(3)) <= 0
-1 - (.5 * X(1) - 2 * X(2)) / (1.5 * X(1) * X(2) + 2 ^ .5 * X(2) * X(3) + 1.319 * X(1) * X(3)) <= 0
where X(i) epsilon { .1, .2, .3, .5, .8, 1, 1.2 }, i=1, 2, 3.

One notes line 165, which is 165 X(B) = A(B).

0 REM 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
87 RANDOMIZE JJJJ
88 M = -3D+30

91 FOR J44 = 1 TO 3

92 IF RND < 1 / 7 THEN A(J44) = .1 ELSE IF RND < 1 / 6 THEN A(J44) = .2 ELSE IF RND < 1 / 5 THEN A(J44) = .3 ELSE IF RND < 1 / 4 THEN A(J44) = .5 ELSE IF RND < 1 / 3 THEN A(J44) = .8 ELSE IF RND < 1 / 2 THEN A(J44) = 1 ELSE A(J44) = 1.2
97 NEXT J44

128 FOR i = 1 TO 5000

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

164 REM X(B) = A(B) + (RND ^ (RND * 15)) * r
165 X(B) = A(B)

166 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 REM X(3) = INT(X(3))

205 REM X(4) = INT(X(4))

403 downstair = 1.5 * X(1) * X(2) + 2 ^ .5 * X(2) * X(3) + 1.319 * X(1) * X(3)
405 IF -1 + (3 ^ .5 * X(2) + 1.932 * X(3)) / downstair > 0 THEN 1670

406 IF -1 + (.634 * X(1) + 2.828 * X(3)) / downstair > 0 THEN 1670

408 IF -1 + (.5 * X(1) - 2 * X(2)) / downstair > 0 THEN 1670

409 IF -1 - (.5 * X(1) - 2 * X(2)) / downstair > 0 THEN 1670
563 PD1 = -2 * X(1) - X(2) - 2 ^ .5 * 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
1777 IF M < -3.25 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= -31754 is shown below. GW-BASIC also can handle this computer program.

1.2       .5       .1       -3.041421      -31988
1.2 .5 .2 -3.182843 -31986
1.2 .5 .2 -3.182843 -31956
1.2 .5 .1 -3.041421 -31766
1.2 .5 .2 -3.182843 -31754

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 = -31954 was 3 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Tsai [93, p. 841].

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.

References

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A most-general-purpose computer program for solving mixed-integer nonlinear programming (MINLP) problems

A most-general-purpose 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 geometric programming problem from Tsai [93, p. 838].
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
-3.1416 * X(3) ^ 2 * X(4) - (4 / 3) * 3.1416 * X(3) ^ 3 + 1296000 <= 0
-240 + X(4) <= 0
1 <= X(1) <= 1.375
.625 <= X(2) <= 1
48 <= X(3) <= 52
90 <= X(4) <= 112
where X(1) and X(2) are discrete variables with discreteness .0625 and X(3) and X(4) are integer variables.

0 REM 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
92 A(1) = 1 + FIX(RND * 7) * .0625
94 A(2) = .625 + FIX(RND * 7) * .0625

96 A(3) = 48 + FIX(RND * 5)

98 A(4) = 90 + FIX(RND * 22)
128 FOR i = 1 TO 10000
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
203 X(3) = INT(X(3))

205 X(4) = INT(X(4))

211 FOR J44 = 1 TO 4
213 IF X(J44) <= 0 THEN 1670

219 NEXT J44

379 IF -X(1) + .0193 * X(3) > 0 THEN 1670
399 IF -X(2) + .00954 * X(3) > 0 THEN 1670
401 IF -3.1416 * X(3) ^ 2 * X(4) - (4 / 3) * 3.1416 * X(3) ^ 3 + 1296000 > 0 THEN 1670

403 IF -240 + X(4) > 0 THEN 1670

501 IF X(1) < 1 THEN 1670
502 IF X(2) < .625 THEN 1670
503 IF X(3) < 48 THEN 1670
504 IF X(4) < 90 THEN 1670
505 IF X(1) > 1.375 THEN 1670
506 IF X(2) > 1 THEN 1670
507 IF X(3) > 52 THEN 1670
508 IF X(4) > 112 THEN 1670
565 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

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= -31990 is shown below. GW-BASIC also can handle this computer program.
1       .625       51       91       7079.037
-32000
1 .625 51 91 -7079.037
-31999
1 .625 51 91 -7079.037
-31998
1 .625 51 91 -7079.037
-31997
1 .625 51 91 -7079.037
-31996
1 .625 51 91 -7079.037
-31995
1 .625 51 91 -7079.037
-31994
1 .625 51 91 -7079.037
-31993
1 .625 51 91 -7079.037
-31992
1 .625 51 91 -7079.038
-31991
1 .625 51 91 -7079.037
-31990

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 = -31990 was 2 seconds, counting from "Starting program...". One can compare the computational results presented above with the computational results in Table 2 of Tsai [93, Table 2 on p. 840].

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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