Substance

Absolute values in computer programs for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations

Absolute values in computer programs for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following system of nonlinear equations from Brainfuse [13]:
X(1)^2 + X(2)^2 = 25
ABS(X(1)) + X(2) = 5.

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
117 FOR J44 = 1 TO 2
121 A(J44) = -200 + RND * 400

124 NEXT J44
127 FOR i = 1 TO 3000

129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 5)

140 B = 1 + FIX(RND * 2)
156 IF RND < .5 THEN 160 ELSE GOTO 169

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

164 X(B) = A(B) + (RND ^ (RND * 15)) * r
165 REM
166 GOTO 188
169 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

188 NEXT IPP
211 FOR J44 = 1 TO 2

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

219 NEXT J44
233 X(2) = 5 - ABS(X(1))

421 FOR J44 = 1 TO 2

433 IF X(J44) < -200 THEN 1670
435 IF X(J44) > 200 THEN 1670
455 NEXT J44
666 PD1 = -ABS(X(1) ^ 2 + X(2) ^ 2 - 25)
679 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

1778 IF M < -1 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= -31994 is shown below. GW-BASIC also can handle this computer program.

-5 0 0 -32000

5 0 0 -31999

-5 0 0 -31998

5 0 0 -31997

-5 0 0 -31996

5 0 0 -31995

-6.87427E-10    5    0    -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…”. Please see the computational results in Brainfuse Inc. [13].

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, including nonlinear systems of equations: an illustration using a slightly nonlinear system of seven equations

Computer program for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations: an illustration using a slightly nonlinear system of seven equations

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following system, which is based on a system of equations from La Cruz et al. [49, p. 25, 25. Brown almost function]:

X(1) * (X(2) * X(3) * X(4) * X(5) * X(6) * X(7) -1=0
2 * X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) -8=0
X(1) + 2 * X(2) + X(3) + X(4) + X(5) + X(6) + X(7) -8=0
X(1) + X(2) + 2 * X(3) + X(4) + X(5) + X(6) + X(7) -8=0
X(1) + X(2) + X(3) + 2 * X(4) + X(5) + X(6) + X(7) -8=0
X(1) + X(2) + X(3) + X(4) + 2 * X(5) + X(6) + X(7) -8=0
X(1) + X(2) + X(3) + X(4) + X(5) + 2 * X(6) + X(7) -8=0.
-2 <= X(i) <= 2, i=1, ... , 7.

 

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
117 FOR J44 = 1 TO 7
121 A(J44) = -2 + RND * 4
124 NEXT J44
127 FOR i = 1 TO 100000

128 REM 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 169

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

164 X(B) = A(B) + (RND ^ (RND * 15)) * R
165 REM X(B) = A(B)
166 GOTO 188
169 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

188 NEXT IPP
211 FOR J44 = 1 TO 7
REM
213 REM X(J44) = INT(X(J44))

219 NEXT J44

222 REM X(2) = -2 * X(1) - X(3) - X(4) - X(5) + 6
226 X(1) = 1 / (X(2) * X(3) * X(4) * X(5) * X(6) * X(7))

414 REM X(1) = (-X(2) * 3 ^ X(4) + 42) / 2 ^ X(3)
421 FOR J44 = 1 TO 7

433 IF X(J44) < -2 THEN 1670
435 IF X(J44) > 2 THEN 1670
455 NEXT J44
566 REM PD1 = -(2 + X(5)) / ((X(1) * X(2)) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4)

567 REM PD1 = -ABS(2 * X(1) + X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + 2 * X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + 2 * X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + X(3) + 2 * X(4) + X(5) - 6) - ABS(X(1) * X(2) * X(3) * X(4) * X(5) - 1)
571 REM X(1) = 1 / (X(2) * X(3) * X(4) * X(5) * X(6) * X(7))

576 PD1 = -ABS(2 * X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) - 8) - ABS(X(1) + 2 * X(2) + X(3) + X(4) + X(5) + X(6) + X(7) - 8) - ABS(X(1) + X(2) + 2 * X(3) + X(4) + X(5) + X(6) + X(7) - 8) - ABS(X(1) + X(2) + X(3) + 2 * X(4) + X(5) + X(6) + X(7) - 8) - ABS(X(1) + X(2) + X(3) + X(4) + 2 * X(5) + X(6) + X(7) - 8) - ABS(X(1) + X(2) + X(3) + X(4) + X(5) + 2 * X(6) + X(7) - 8)

679 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 REM IF M < 22.84042 THEN 1999
1778 IF M < -.0001 THEN 1999
1779 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 output of one run through
JJJJ= -31986 is summarized below. GW-BASIC also can handle this computer program.

.
.
.
.9999997    .9999967    .9999999    1    .9999998
.9999995    1.000004    -3.33786E-06    -31987

.9576598    .9576584    .9576529    .9576527    .9576584
.9576527    1.296412    -1.811981E-05    -31986

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 = -31986 was 4 seconds, counting from “Starting program…”.

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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Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Template for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations: an illustration with a slightly nonlinear system of five equations, second edition

Template for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations: an illustration with a slightly nonlinear system of five equations, 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 nonlinear system of equations from Maranas and Floudas [59, Example 5, p. 24]:
2 * X(1) + X(2) + X(3) + X(4) + X(5) - 6 = 0
X(1) + 2 * X(2) + X(3) + X(4) + X(5) - 6 = 0
X(1) + X(2) + 2 * X(3) + X(4) + X(5) - 6 = 0
X(1) + X(2) + X(3) + 2 * X(4) + X(5) - 6 = 0
X(1) * X(2) * X(3) * X(4) * X(5) - 1 = 0
-2 <= X(i) <= 2, i=1, ... , 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

87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 5
121 A(J44) = -2 + RND * 4
124 NEXT J44
127 FOR i = 1 TO 100000

128 REM FOR i = 1 TO 10000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 5)

140 B = 1 + FIX(RND * 5)
156 IF RND < .5 THEN 160 ELSE GOTO 169

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

164 X(B) = A(B) + (RND ^ (RND * 15)) * r
165 REM X(B) = A(B)
166 GOTO 188
169 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

188 NEXT IPP
211 FOR J44 = 1 TO 5

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

219 NEXT J44

222 REM X(2) = -2 * X(1) - X(3) - X(4) - X(5) + 6
226 X(1) = 1 / (X(2) * X(3) * X(4) * X(5))

414 REM X(1) = (-X(2) * 3 ^ X(4) + 42) / 2 ^ X(3)
421 FOR J44 = 1 TO 5

433 IF X(J44) < -2 THEN 1670
435 IF X(J44) > 2 THEN 1670
455 NEXT J44
566 REM PD1 = -(2 + X(5)) / ((X(1) * X(2)) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4)
567 REM PD1 = -ABS(2 * X(1) + X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + 2 * X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + 2 * X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + X(3) + 2 * X(4) + X(5) - 6) - ABS(X(1) * X(2) * X(3) * X(4) * X(5) - 1)

570 PD1 = -ABS(2 * X(1) + X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + 2 * X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + 2 * X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + X(3) + 2 * X(4) + X(5) - 6)

679 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
1777 REM IF M < 22.84042 THEN 1999
1778 IF M < -.0001 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5), 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.

.9163545825344733    .9163545825333481    .9163545825325766
.9163545825324777    1.418227087334656    -3.002043058586423D-12
-32000
1.00000000009394    1.000000001476913    1.00000000011803
1.00000000130429    .9999999970068272    -2.993172820708878D-09
-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 2 seconds, counting from “Starting program…”. One can compare the computational results presented above with the computational results in Maranas and Floudas [59, Example 5, p. 24, Table 5, p. 39].

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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Errata in: Template for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations

Errata in: Template for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations

Jsun Yui Wong

The output of the preceding paper should read as follows:

38.72983346207417       25.81988897471611       77.45966692414834
-6.4170890823334054D-14       -32000       8166.666666666666

38.72983346207417 25.81988897471611 77.45966692414834
-6.4170890823334054D-14 -31999 8166.6666666666666

38.72983346207417 25.81988897471611 77.45966692414834
-6.4170890823334054D-14 -31998 8166.6666666666666

Above the number 77.45966692414834 appears three times, not one time.

Template for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations

Template for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following nonlinear system of equations from Goel [34]:

AB = 1000
BC = 2000
CA = 3000 .

 

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
87 RANDOMIZE JJJJ
88 M = -3D+30
90 REM GOTO 117
117 FOR J44 = 1 TO 5

121 A(J44) = 0 + RND * 9.9

126 NEXT J44

127 FOR i = 1 TO 100000

128 REM FOR i = 1 TO 10000

129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 5)

140 B = 1 + FIX(RND * 5)
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 5

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

219 NEXT J44

413 REM X(1) = ((-1 / X(5) ^ 3 + 2) * (1 / 8)) / ((X(2) + 3 * X(4)) ^ 2)

414 X(1) = 1000 / X(2)

415 REM IF -2 * X(1) + X(3) - X(4) > 10 THEN 1670
416 X(3) = 2000 / X(2)

417 REM IF X(3) + X(1) + .5 * X(5) > 8 THEN 1670

422 FOR J44 = 1 TO 5

433 REM IF X(J44) < .1 THEN 1670
435 REM IF X(J44) < 0 THEN 1670
455 NEXT J44
566 REM PD1 = -(2 + X(5)) / ((X(1) * X(2)) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4)

569 REM PD1 = -ABS(8 ^ X(1) + 2 ^ X(1) - 130)

571 REM PD1 = -ABS(8 ^ (X(1) + 1) - 8 ^ (X(1) - 1) - 3024)

573 REM PD1 = -ABS(X(1) + X(2) - X(1) * X(2) * X(3)) - ABS(X(2) + X(3) - X(1) * X(2) * X(3)) - ABS(X(3) + X(1) - X(1) * X(2) * X(3))

577 REM PD1 = -ABS(X(1) * X(2) - 1000) - ABS(X(2) * X(3) - 2000) - ABS(X(3) * X(1) - 3000)

579 PD1 = -ABS(X(3) * X(1) - 3000)

669 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
1777 REM IF M < 22.84042 THEN 1999
1778 IF M < -.01 THEN 1999

1779 REM 23.40042
1904 PRINT A(1), A(2), A(3), M, JJJJ, A(1) ^ 2 + A(2) ^ 2 + A(3) ^ 2
1999 NEXT JJJJ

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

38.72983346207417       25.81988897471611       77.45966692414834
-6.4170890823334054D-14       -32000       8166.666666666666

38.72983346207417 25.81988897471611 27.45966692414834
-6.4170890823334054D-14 -31999 8166.6666666666666

38.72983346207417 25.81988897471611 27.45966692414834
-6.4170890823334054D-14 -31998 8166.6666666666666

 

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 Goel [34].

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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Template for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations

Template for solving mixed-integer nonlinear programming (MINLP) problems, including nonlinear systems of equations

Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following nonlinear system of equations from Learncommunolizer [50]:

1/a + 1/(b+c) = 1/2
1/b + 1/(c+a) = 1/3
1/c + 1/(a+b) = 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
87 RANDOMIZE JJJJ
88 M = -3D+30
90 REM GOTO 117
117 FOR J44 = 1 TO 5

121 A(J44) = 0 + RND * 9.9

126 NEXT J44

127 FOR i = 1 TO 100000

128 REM FOR i = 1 TO 10000

129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 5)

140 B = 1 + FIX(RND * 5)
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 5

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

219 NEXT J44

413 REM X(1) = ((-1 / X(5) ^ 3 + 2) * (1 / 8)) / ((X(2) + 3 * X(4)) ^ 2)

414 REM X(1) = (8 / (2 - 1 / X(5) ^ 3)) * (1 / (X(2) + 3 * X(4)) ^ 2)

415 REM IF -2 * X(1) + X(3) - X(4) > 10 THEN 1670

417 REM IF X(3) + X(1) + .5 * X(5) > 8 THEN 1670

422 FOR J44 = 1 TO 5

433 REM IF X(J44) < .1 THEN 1670
435 REM IF X(J44) < 0 THEN 1670
455 NEXT J44
566 REM PD1 = -(2 + X(5)) / ((X(1) * X(2)) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4)

569 REM PD1 = -ABS(8 ^ X(1) + 2 ^ X(1) - 130)

571 REM PD1 = -ABS(8 ^ (X(1) + 1) - 8 ^ (X(1) - 1) - 3024)

573 PD1 = -ABS(1 / X(1) + 1 / (X(2) + X(3)) - 1 / 2) - ABS(1 / X(2) + 1 / (X(3) + X(1)) - 1 / 3) - ABS(1 / X(3) + 1 / (X(1) + X(2)) - 1 / 4)

669 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
1777 REM IF M < 22.84042 THEN 1999
1778 IF M < -.0000001 THEN 1999

1779 REM 23.40042

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= -31883 is shown below. GW-BASIC also can handle this computer program.

2.299999605470452 3.833331896064856 11.50001902221562
-9.511902092845326D-08 -31991

2.299999799199326       3.833332603929451       11.50000965388264
-4.826937610067316D-08       -31883

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 = -31883 was 18 seconds, counting from “Starting program…”. One can compare the computational results presented above with the computational results in Learncommunolizer [50].

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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http://bazziahmad.com/
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https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab

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A general-purpose computer program for solving nonlinear fractional programming problems, sixth edition

A general-purpose computer program for solving nonlinear fractional programming problems, sixth edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following nonlinear fractional programming problem from Tsai [94, Example 3, p. 408].

Minimize (2 + X(5)) / ((X(1) * X(2)) * (2 * X(3) + X(4))) – X(5) ^ .5 * X(3) ^ 1.5 + 2 * X(2) + X(4)

subject to

8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) + 1 / X(5) ^ 3 <= 2
-2 * X(1) + X(3) – X(4) <= 10
X(1) + X(3) + .5 * X(5) <= 8
.1 <= X(1), X(2), X(3), X(4), X(5) <= 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
87 RANDOMIZE JJJJ
88 M = -3D+30
90 REM GOTO 117
117 FOR J44 = 1 TO 5

121 A(J44) = .1 + RND * 9.9
126 NEXT J44
127 REM
128 FOR i = 1 TO 100000

129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 5)

140 B = 1 + FIX(RND * 5)
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 5

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

219 NEXT J44

355 X(3) = 8 - X(1) - .5 * X(5)

413 IF (8 / (X(1) * (X(2) + 3 * X(4)) ^ 2)) + 1 / X(5) ^ 3 > 2 THEN 1670

415 IF -2 * X(1) + X(3) - X(4) > 10 THEN 1670
422 FOR J44 = 1 TO 5

433 IF X(J44) < .1 THEN 1670
435 IF X(J44) > 10 THEN 1670

455 NEXT J44
566 PD1 = -(2 + X(5)) / ((X(1) * X(2)) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4)
669 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
177 IF M < 22.84140 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its output of one run through
JJJJ= -31840 is shown below. GW-BASIC also can handle this computer program.

.2973201 .9230682 5.934283 .9219145 3.536794
22.8414 -31967

.2969714 .9260215 5.933192 .9216344 3.539673
22.8414 -31891

.2973345       .9247497       5.933402       .9213125       3.538528
22.84141       -31840   

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 = -31840 was 30 seconds, counting from “Starting program…”. One can compare the computational results presented above with the computational results in Tsai [94, Example 3, p. 408].

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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