Substance

Using the MINLP Solver Presented Here, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve Example 3 on pp. 47-48 of Tsai and Lin [7].

The X(5) and X(6) of the following computer program are slack variables. Line 233 and line 403 below refer to the first given constraint and the given objective function;
these are 233 X(5) = 4 - X(1) * X(4) ^ 1.5 + X(2) + X(2) ^ .5 * X(3) ^ .4 and 403 POBA2 = -(X(1) * X(4) ^ 3 - X(3) - .5 * X(1) ^ 2 * X(2) ^ 4), respectively.

One notes that here line 128 is 128 FOR I = 1 TO 8000. .

0 REM DEFDBL A-Z

2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32111

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 4
65 A(J44) = RND * 30

66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 8000

129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 4)

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
224 GOTO 233

225 FOR J23 = 1 TO 4

227 X(J23) = INT(X(J23))

229 NEXT J23

233 X(5) = 4 - X(1) * X(4) ^ 1.5 + X(2) + X(2) ^ .5 * X(3) ^ .4

244 X(6) = -2 + X(1) + 2 * X(2) - X(3)

283 IF X(1) < 0 THEN 1670

284 IF X(1) > 6 THEN 1670

286 IF X(2) < 1 THEN 1670

287 IF X(2) > 10 THEN 1670

288 IF X(3) < 1 THEN 1670

289 IF X(3) > 6 THEN 1670

291 IF X(4) < 20 THEN 1670

292 IF X(4) > 30 THEN 1670

331 FOR J44 = 5 TO 6

335 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0

339 NEXT J44
403 POBA2 = -(X(1) * X(4) ^ 3 - X(3) - .5 * X(1) ^ 2 * X(2) ^ 4)

411 POBA = POBA2 - 1000000 * PS(5) - 1000000 * PS(6)

422 REM POBA = POBA2 + (PS(4))
459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 6

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128

1670 NEXT I
1889 IF M < 4 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)

1904 PRINT A(5), A(6), M, JJJJ, PPOBA2

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [8]. The complete output through JJJJ=-31949 is shown below:

1.930865E-11       8.195931       5.999997       21.7256
18.05812       8.391867       5.999997       -31992       5.999997

2.702062E-11       5.886934       6       20.61941
14.8552       3.773869       6       -31961       6

1.730769E-11       6.638494       6       20.45676
15.91438       5.276988       6       -31960       6

1.519407E-11       4.327571       6       22.89165
12.58731       .6551418       6       -31949       6

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [8], the wall-clock time for obtaining the output through JJJJ=-31949 was 10 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp.882-891.

[3] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[4] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[5] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.

[6] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.

[7] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.

[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[9] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/

 

Using the MINLP Solver Presented Here, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the nonconvex example on pp. 45-47 of Tsai and Lin [7].

The X(6), X(7), and X(8) of the following computer program are slack variables. Line 403 below refers to the given objective function.

0 REM DEFDBL A-Z

2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32111

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 5
65 A(J44) = 1 + RND * 9

66 NEXT J44

77 A(1) = -7 + RND * 12

126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 8000

129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 J = 1 + FIX(RND * 5)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
223 REM GOTO 233

225 REM FOR J23 = 1 TO 4

227 X(3) = INT(X(3))

229 REM NEXT J23

233 X(6) = 2 - X(1) - 6 * X(2) + X(3) + 5 * X(4)

244 X(7) = -10 - (X(3) ^ 1.5) * X(4) - .5 * X(2) - 3 * X(1)

249 X(8) = 6 + X(1) + .5 * X(4) - X(5)

283 IF X(1) < -7 THEN 1670

284 IF X(1) > 5 THEN 1670

286 IF X(2) < 1 THEN 1670

287 IF X(2) > 10 THEN 1670

288 IF X(3) < 1 THEN 1670

289 IF X(3) > 5 THEN 1670

291 IF X(4) < 2 THEN 1670

292 IF X(4) > 8 THEN 1670

297 IF X(5) < 2 THEN 1670

298 IF X(5) > 9 THEN 1670

331 FOR J44 = 6 TO 8

335 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0

339 NEXT J44
403 POBA2 = -(X(1) ^ 2 * X(2) ^ -2 * X(3) - 2 * X(2) ^ .7 * X(3) ^ .2 + X(4) * X(5) ^ -2 - 2 * X(1) - 4 * X(3))

411 POBA = POBA2 - 1000000 * PS(6) - 1000000 * PS(7) - 1000000 * PS(8)

422 REM POBA = POBA2 + (PS(4))
459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 8

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128

1670 NEXT I
1889 IF M < -2.906 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)

1904 PRINT A(5), A(6), A(7), A(8), M, JJJJ, PPOBA2

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [8]. The complete output through JJJJ=-30043 is shown below:

-5.364902          4.569214          1          3.810099
2.540147          1.137257E-04         1.66893E-06          1.192093E-07
-2.905704          -31395          -2.905704

-5.338768          4.520022          1          3.756273
2.539359          0          1.811981E-05          1.049042E-05
-2.90566          -31303          -2.90566

-5.377561          4.593031          1          3.836126
2.540477          3.33786E-06 4.053116E-05          2.527237E-05
-2.905963          -31022          -2.905963

-5.372931          4.584244          1          3.826506
2.540321          1.66893eE-06 1.661777E-04          5.960464E-07
-2.905968          -30043          -2.905968

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [8], the wall-clock time for obtaining the output through JJJJ=-30043 was 40 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp.882-891.

[3] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[4] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[5] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.

[6] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.

[7] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.

[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[9] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/

An Application of the MINLP Solver Presented Here, Second Editio

Jsun Yui Wong

The computer program listed below seeks to solve the following insulated steel tank design example on page 18 of Tsai, Lin, and Hu [6] and on page 564 of Ryoo and Sahinidis [5].

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) .

The X(5) below is an added variable.    One notes that here line 128 is 128 FOR I = 1 TO 8000.

0 REM DEFDBL A-Z

2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32111

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 4
65 A(J44) = RND

66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 8000

129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 J = 1 + FIX(RND * 3)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

233 X(2) = EXP((-3950 / (X(3) + 460)) + 11.86)

244 X(5) = 144 * (80 - X(3)) - X(1) * X(4)

283 IF X(1) < 0 THEN 1670

284 IF X(1) > 15.1 THEN 1670

286 IF X(2) < 14.7 THEN 1670

287 IF X(2) > 94.2 THEN 1670

288 IF X(3) < -459.67 THEN 1670

289 IF X(3) > 80 THEN 1670

291 IF X(4) < 0 THEN 1670

292 IF X(4) > 999 THEN 1670

331 FOR J44 = 4 TO 4
335 PS(J44) = -1000000 * ABS(X(5))

339 NEXT J44
403 POBA2 = -(400 * X(1) ^ .9 + 1000 + 22 * (X(2) - 14.7) ^ 1.2 + X(4))

411 REM POBA = POBA2 - 1000000 * (PS(4))

422 POBA = POBA2 + (PS(4))
459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 5

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128

1670 NEXT I
1889 IF M < -5195 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5)

1904 PRINT M, JJJJ, PPOBA2
1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [7]. The complete output through JJJJ=-28537 is shown below:

7.737477E-08          94.17786          80          1.812935E-03
-1.402754E-10
-5194.868          -30811          -5194.868

1.21274E-08          94.17786          80          2.377033E-04
-2.882723E-12
-5194.866          -30049          -5194.866

2.840714E-08          94.17786          80          2.596378E-04
-7.375569E-12
-5194.866          -29243          -5194.866

1.103056E-07          94.17786          80          5.87143E-05
-6.416933E-12
-5194.866          -28537          -5194.866

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [7], the wall-clock time for obtaining the output through JJJJ=-28537 was 50 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp.882-891.

[3] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[4] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[5] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.

[6] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.

[7] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[8] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/

A Computer Program about Markowitz Portfolio Selection, Third Edition

Jsun Yui Wong

The computer program listed below seeks to solve the small example about Markowitz [3] portfolio selection on pp. 888-890 of Li and Tsai [2].

Minimize .0108 * X(1) ^ 2 + .0584 * X(2) ^ 2 + .0942 * X(3) ^ 2 + .0248 * X(1) * X(2) + .0262 * X(1) * X(3) + .1108 * X(2) * X(3)

subject to 1.089 * X(1) + 1.214 * X(2) + 1.235 * X(3) >= 115

X(1) + X(2) + X(3) = 100

where X(1), X(2), and X(3) are integers.

Below X(1) through X(3) are integer variables, and X(4) is a slack variable.

Whereas line 1557 of the earlier editions is 1557 GOTO 128, here line 1557 is 1557 REM GOTO 128.

 

0 REM DEFDBL A-Z

2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32111

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 3
65 A(J44) = RND * 10

66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 8000

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 J = 1 + FIX(RND * 3)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

223 X(1) = 100 - X(2) - X(3)

224 FOR J33 = 1 TO 3

226 X(J33) = INT(X(J33))

227 NEXT J33

229 REM X(5) = 1
233 X(4) = -115 + 1.089 * X(1) + 1.214 * X(2) + 1.235 * X(3)
281 FOR J44 = 1 TO 3

283 IF X(J44) < 0 THEN 1670

284 IF X(J44) > 100 THEN 1670

285 NEXT J44

331 FOR J44 = 4 TO 4

332 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0
333 NEXT J44
403 POBA2 = -(.0108 * X(1) ^ 2 + .0584 * X(2) ^ 2 + .0942 * X(3) ^ 2 + .0248 * X(1) * X(2) + .0262 * X(1) * X(3) + .1108 * X(2) * X(3))

411 POBA = POBA2 - 1000000 * (PS(4))

459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128

1670 NEXT I
1889 IF M < -224.647 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)

1904 PRINT M, JJJJ, PPOBA2
1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [5]. The complete output through JJJJ=-30582 is shown below:

53       35       12       .027
-224.6452       -31704       -224.6452

53       36       11       .006
-223.8916       -31522       -223.8916

53       36       11       .006
-223.8916       -31485       -223.8916

53       36        11       .006
-223.8916       -30582       -223.8916

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [5], the wall-clock time for obtaining the output through JJJJ=-30582 was 30 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp.882-891.

[3] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[4] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[5] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[6] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/

A Computer Program about Markowitz Portfolio Selection, Second Edition

A Computer Program about Markowitz Portfolio Selection, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the small example about Markowitz portfolio selection [3] on pp. 888-890 of Li and Tsai [2].

Minimize    .0108 * X(1) ^ 2 + .0584 * X(2) ^ 2 + .0942 * X(3) ^ 2 + .0248 * X(1) * X(2) + .0262 * X(1) * X(3) + .1108 * X(2) * X(3)

subject to    1.089 * X(1) + 1.214 * X(2) + 1.235 * X(3) >= 115

X(1) + X(2) + X(3) = 100

where X(1), X(2), and X(3) are integers.

Below X(1) through X(3) are integer variables, and X(4) is a slack variable.

Whereas line 128 of the earlier edition is 128 FOR I = 1 TO 10000, here line 128 is 128 FOR I = 1 TO 8000.

0 REM DEFDBL A-Z

2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 3
65 A(J44) = RND * 10

66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 8000

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 J = 1 + FIX(RND * 3)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

223 X(1) = 100 - X(2) - X(3)

224 FOR J33 = 1 TO 3

226 X(J33) = INT(X(J33))

227 NEXT J33

229 REM X(5) = 1
233 X(4) = -115 + 1.089 * X(1) + 1.214 * X(2) + 1.235 * X(3)

281 FOR J44 = 1 TO 3

283 IF X(J44) < 0 THEN 1670

284 IF X(J44) > 100 THEN 1670

285 NEXT J44

331 FOR J44 = 4 TO 4

332 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0
333 NEXT J44
403 POBA2 = -(.0108 * X(1) ^ 2 + .0584 * X(2) ^ 2 + .0942 * X(3) ^ 2 + .0248 * X(1) * X(2) + .0262 * X(1) * X(3) + .1108 * X(2) * X(3))

411 POBA = POBA2 - 1000000 * (PS(4))

459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -224.647 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)

1904 PRINT M, JJJJ, PPOBA2
1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [5]. The complete output through JJJJ=-16563 is shown below:

53       35       12       .027
-224.6452       -29251       -224.6452

53       36       11       .006
-223.8916       -19156       -223.8916

53       35       12       .027
-224.6452       -18228       -224.6452

53       36       11       .006
-223.8916       -16563       -223.8916

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [5], the wall-clock time for obtaining the output through JJJJ=-16563 was 3 minutes and 20 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp.882-891.

[3] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[4] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[5] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[6] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/

A Computer Program for Integer Fractional Programming, Fourth Edition

Jsun Yui Wong

The computer program listed below seeks to solve the integer fractional programming example on pp. 55-57 of Anzai [1].

Here X(1) through X(5) are integer variables, and X(6) through X(11) are slack variables.

Whereas line 128 and line 1454 of the Second Edition are 128 FOR I = 1 TO 100 and 1454 FOR KLX = 1 TO 8, here line 128 and line 1454 are 128 FOR I = 1 TO 25 and 1454 FOR KLX = 1 TO 11, respectively.

0 REM DEFDBL A-Z

2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 5
65 A(J44) = RND * 10

66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 25

129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))
181 J = 1 + FIX(RND * 5)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

224 FOR J33 = 1 TO 5

226 X(J33) = INT(X(J33))

227 NEXT J33

229 X(5) = 1

240 X(6) = X(1) - 2 * X(2) + 9

259 X(7) = -4 * X(1) - X(2) + 21

261 X(8) = -X(1) + 3 * X(2)

266 X(9) = 4 * X(1) + 5 * X(2) - 12

268 X(10) = 2 * X(3) - 6 * X(4) + 21

270 X(11) = -7 * X(3) - 4 * X(4) + 39

281 FOR J44 = 1 TO 5

283 IF X(J44) < 0 THEN 1670

284 IF X(J44) > 100 THEN 1670

285 NEXT J44

331 FOR J44 = 6 TO 11
332 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0
333 NEXT J44
403 POBA2 = -(-X(1) - 3 * X(2) - X(3) + X(4) - X(5)) / (2 * X(1) + X(2) + 3 * X(3) + 5 * X(4) + X(5))
411 POBA = POBA2 - 1000000 * (PS(6) + PS(7) + PS(8) + PS(9) + PS(10) + PS(11))

459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<0 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4)
1901 PRINT A(5), A(6), A(7), A(8)

1902 PRINT A(9), A(10), A(11)

1904 PRINT M, JJJJ, PPOBA2
1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [3]. The complete output through JJJJ=-31996 is shown below:

0      3       0       0
1       3       18       9
3       21       39
2.5       -32000       2.5

2       1       0       0
1       9       12       1
1       21       39
1       -31999       1

0       4       0       0
1       1       17       12
8       21       39
2.6       -31998       2.6

0       4       0       0
1       1       17       12
8       21       39
2.6       -31997       2.6

3       6       0       0
1       0       3       15
30       21       39
1.692308       -31996       1.692308

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [3], the wall-clock time for obtaining the output through JJJJ=-31996 was 2 seconds, not including "Creating .EXEC file" time.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[3] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[4] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/

The Mixed Integer Nonlinear Programming Solver Presented Here Applied to a Mixed Integer Fractional Posynomial Programming Example from the Literature, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the mixed integer fractional posynomial programming problem/example on pp. 383-385 of Chang [2].

In the fpllowing computer program X(1) and X(2) are binary variables, X(3) through X(5) are continuous variables, and X(6) through X(9) are slack variables.

One notes that line 128 here is 128 FOR I = 1 TO 750.

0 REM DEFDBL A-Z

2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 5

65 A(J44) = RND * 7

66 NEXT J44

128 FOR I = 1 TO 750

129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))
181 J = 3 + FIX(RND * 3)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
222 NEXT IPP

224 FOR J33 = 1 TO 2

226 X(J33) = INT(X(J33))
227 IF X(J33) > 1 THEN GOTO 1670

228 NEXT J33

229 REM X(5) = 1

240 X(6) = 2 * X(1) + X(1) * X(3) ^ 1.6 - 5

259 X(7) = X(1) + X(2) - 1

261 X(8) = -X(2) - X(4) + 6

266 X(9) = X(1) + X(2) + X(5) - 3

268 REM X(10) = 2 * X(3) - 6 * X(4) + 21

270 REM X(11) = -7 * X(3) - 4 * X(4) + 39

281 FOR J44 = 3 TO 5

283 IF X(J44) < 1 THEN 1670

284 IF X(J44) > 7 THEN 1670

285 NEXT J44

331 FOR J44 = 6 TO 9

332 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0
333 NEXT J44
403 POBA2 = (-ABS(X(5) ^ 1.8) * X(1) * X(3) ^ .3 * X(4) ^ 1.5 - X(2) * X(3) ^ 1.8 * X(4) ^ 1.3 * X(5) - X(1) * X(2) * X(3) ^ 1.7) / (X(1) * X(3) * X(4) + X(2) * X(3) ^ 2.3 * X(4) ^ 1.4)

411 POBA = POBA2 - 9000000 * (PS(6) + PS(7) + PS(8) + PS(9))

459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 9

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128

1670 NEXT I
1889 IF M < -.9 THEN 1999
1892 REM IF M > -.2 THEN GOTO 1999

1893 FOR J47 = 6 TO 9

1894 IF A(J47) < 0 THEN GOTO 1999

1895 NEXT J47

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1901 PRINT A(6), A(7), A(8), A(9)

1902 REM PRINT A(10), A(11)

1904 PRINT M, JJJJ, PPOBA2

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [5]. The complete output through JJJJ=-31945 is shown below:

1       1       6.999617       4.999769       1.000049
19.4967       1       2.31266E-04       4.935265E-05
-.3632627       -31957       -.3632627

1       0       6.995616       1.001267       2.000136
19.47613       0       4.998733       1.363754E-04
-.8929027       -31945      -.8929027

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [5], the wall-clock time for obtaining the output through JJJJ=-31945 was 2 seconds, not including "Creating .EXEC file" time.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www.orsj.or.jp/~archive/pdf/e_mag/Vol.17_01_049.pdf.

[2] Ching-Ter Chang (2006). Formulating the Mixed Integer Fractional Posynomial Programming. European Journal of Operational Research 173 (2006) 370-386.

[3] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[4] Jung-Fa Tsai (2005): Global Optimization of Nonlinear Fractional Programming Problems in Engineering Design, Engineering Optimization, 37:4, 399-409.

[5] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[6] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/ http://myblogsubstance.typepad.com/substance/2012/04/12/

Erratum: A Computer Program for Integer Fractional Programming, Second Edition

Jsun Yui Wong

Reference 1 should read as follows:

[1] Y. Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www.orsj.or.jp/~archive/pdf/e_mag/Vol.17_01_049.pdf.

A Computer Program for Integer Fractional Programming, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the integer fractional programming example on pages 55-57 of Anzai [1].

Here X(1) through X(5) are integer variables, and X(6) through X(11) are slack variables.

0 DEFDBL A-Z
2 DEFINT I, J, K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 5
65 A(J44) = RND * 10

66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 100

129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))
181 J = 1 + FIX(RND * 5)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

224 FOR J33 = 1 TO 5

226 X(J33) = INT(X(J33))

227 NEXT J33

229 X(5) = 1

240 X(6) = X(1) - 2 * X(2) + 9

259 X(7) = -4 * X(1) - X(2) + 21

261 X(8) = -X(1) + 3 * X(2)

266 X(9) = 4 * X(1) + 5 * X(2) - 12

268 X(10) = 2 * X(3) - 6 * X(4) + 21

270 X(11) = -7 * X(3) - 4 * X(4) + 39

281 FOR J44 = 1 TO 5

283 IF X(J44) < 0 THEN 1670

284 IF X(J44) > 50 THEN 1670
285 NEXT J44

331 FOR J44 = 6 TO 11
332 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0
333 NEXT J44
403 POBA2 = -(-X(1) - 3 * X(2) - X(3) + X(4) - X(5)) / (2 * X(1) + X(2) + 3 * X(3) + 5 * X(4) + X(5))
411 POBA = POBA2 - 999999999# * (PS(6) + PS(7) + PS(8) + PS(9) + PS(10) + PS(11))
459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 8
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<0 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4)
1901 PRINT A(5), A(6), A(7), A(8)

1902 PRINT A(9), A(10), A(11)

1904 PRINT M, JJJJ, PPOBA2
1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [3]. The complete output through JJJJ=-31991 is shown below:

0          4          0          0
1          1          17          12
0          0          0
2.6          -32000          2.6

0 4 0 0
1 1 17 12
0 0 0
2.6 -31999 2.6

0 4 0 0
1 1 17 12
0 0 0
2.6 -31998 2.6

0 4 0 0
1 1 17 12
0 0 0
2.6 -31997 2.6

0          4          0          0
1          1          17          12
0          0          0
2.6          -31996          2.6

0          3          0          0
1          3          18          9
0          0          0
2.5          -31995          2.5

2          1          0          0
1          9          12          1
0          0          0
1          -31994          1

0 4 0 0
1 1 17 12
0 0 0
2.6 -31993 2.6

0 4 0 0
1 1 17 12
0 0 0
2.6 -31992 2.6

3          0          0          0
1          12          9          -3
0          0          0
-2999999996.428571          -31991          .5714285714285714

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [3], the wall-clock time for obtaining the output through JJJJ=-31991 was 3 seconds, not including "Creating .EXEC file" time..

Acknowledgment

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

References

[1] Y. Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.

[2] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[3] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[4] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/ http://myblogsubstance.typepad.com/substance/2012/04/12/

The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature--Revisited

Jsun Yui Wong

Based on the computer program of the 2012 April 12 paper [21], the computer program listed below seeks to solve the nonlinefar fractional programming problem on page 408 of Tsai [16].

Here X(1) through X(5) are continuous variables, and X(6) through X(8) are slack variables.

One notes line 2 and line 128, which are 2 DEFINT J, K and 128 FOR I = 1 TO 1000, respectively.

0 DEFDBL A-Z
2 DEFINT J, K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 5
65 A(J44) = .1 + RND * 9.899999
66 NEXT J44
126 REM IMAR = 10 + FIX(RND * 32000)
128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))
181 J = 1 + FIX(RND * 5)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
231 FOR J44 = 1 TO 5
233 IF X(J44) < .1 THEN 1670
234 IF X(J44) > 10 THEN 1670
235 NEXT J44
240 X(6) = 8 - X(1) - X(3) - .5 * X(5)
259 X(7) = 10 + 2 * X(1) - X(3) + X(4)
261 X(8) = 2 - 8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) - 1 / X(5) ^ 3
331 FOR J44 = 6 TO 8
332 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0
333 NEXT J44
403 POBA2 = -(2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4)
411 POBA = POBA2 - 999999999# * (PS(4) + PS(5) + PS(6) + PS(7) + PS(8))
459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 8
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 22.84 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)
1901 PRINT A(5), A(6), A(7), A(8)
1904 PRINT M, JJJJ, PPOBA2
1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [19]. The complete output through JJJJ=-30813 is shown below:

.2947472515329632 .9175500226527604 5.93906159902053
.9291337963870181
3.532382296796923 1.04804609435405D-09 5.579566700432416
2.232539395946713D-08
22.84111394811393 -31801 22.84111394811393

.2967501008561616       .9343622710082007       5.925629617892431
.9192197212955193
3.555240561915777       2.935185428043496D-10       5.587090305115412
1.746166857009607D-10
22.8411314212559       -31594       22.8411314212559

.2982068380293917 .9309152995910263 5.91960546415551
.9173061559946026
3.564375395629317 4.403144515663371D-13 5.594114367897878
2.895050089544985D-09
22.84092830959955 -30813 22.84092830959955

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [19], the wall-clock time for obtaining the output through JJJJ=-30813 was 3 minutes.

Acknowledgment

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

References

[1] M. Avriel, J. D. Barrette, Optimal Design of Pitched Laminated Wood Beams. Journal of Optimization Theory and Applications, Vol. 26, Number 2 (October 1978), pp. 291-203.

[2] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.

[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[5] Jing-Fan Fu, Robert G. Fenton, William L. Cleghorn (1991): A Mixed Integer-Discrete-Continuous Programming Method and its Applicaion to Engineering Design Optimization, Engineering Optimization, 17:4. 263-280.

[6] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.

[10] Han-Lin Li, Chih-Tan Chou (1993): A Global Approach for Nonlinear Mixed Discrete Programming in Design Optimization, Engineering Optimization, 22:2, 109-122.

[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.

[13] Tapabrata Ray, Pankaj Saini (2001), Engineering Design Optimization Using a Swarm with an Intelligent Information Sharing among Individuals, Engineering Optimization, 33:6, 735-748.

[14] M. J. Rijckaert, X. M. Martens, Comparison of Generalized Geometric Programming Algorithms. Journal of Optimization Theory and Applications, Vol. 26, Number 2 (October 1978), pp. 205-242.

[15] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.

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[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

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