Substance

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to a Nonlinear Integer Programming Problem Based on the Rosenbrock Function with 10100 General Integer Variables of Lower Bounds of -12s and Upper Bounds of 12s, Second Edition

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve a problem based on Li and Sun's Problem 14.4, [12, p. 415], but with 10100 unknowns instead of 100 unknowns and with lower bounds of -12s instead of -5s and upper bounds of 12s instead of 5s of Li and Sun [12, p.415].  The function is based on the Rosenbrock function--see Schittkowski [16] and Li and Sun [12, p. 415].  Specifically, the test example here is as follows:

Minimize

10100-1
SIGMA       [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

-12<=X(i)<=12, X(i) integer, i=1, 2, 3,..., 10100.

One notes line 111 through line 117 and line 170 through line 173.

The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.

0 DEFINT J,K,B,X
2 DIM A(10100),X(10100)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
114 A(J44)=-12+FIX(RND*25)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10100)
167 IF RND<.5 THEN X(B)=A(B)-1 ELSE X(B)=A(B) +1
169 NEXT IPP
170 FOR J44=1 TO 10100
171 IF X(J44)<-12 THEN X(J44 )=A(J44 )
172 IF X(J44)>12 THEN X(J44 )=A(J44 )
173 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 10099
205 SUMNEWZ=SUMNEWZ+ 100* ( X(J44+1)-X(J44)^2 )^2 + ( 1-X(J44 ) ) ^2
207 NEXT J44
511 SONE= - SUMNEWZ
689 PD1=SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1899 PRINT A(9996),A(9997),A(9998),A(10099),A(10100),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [13]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31999 is shown below:

1 1 1 1 1
1 1 1 1 1
0       -32000

1 1 1 1 1
1 1 1 1 1
0       -31999

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

M=0 is optimal. See Li and Sun [12, p. 415].

Of the 10100 A's, only the 10 A's of line 1771 and line 1899 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was 26 hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.American Mathematical Monthly

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver Using Hot Starts To Solve a Nonlinear Integer Programming Problem Based on the Rosenbrock Function with 10100 General Integer Variables of Lower Bounds of -30s and Upper Bounds of 30s

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve a problem based on Li and Sun's Problem 14.4, [12, p. 415], but with 10100 unknowns instead of 100 unknowns and with lower bounds of -30s instead of -5s and upper bounds of 30s instead of 5s. The function is based on the Rosenbrock function--see Schittkowski [16] and Li and Sun [12, p. 415]. Specifically, the test example here is as follows:

Minimize

10100-1
SIGMA          [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

-30<=X(i)<=30, X(i) integer, i=1, 2, 3,..., 10100.

One notes line 111 through line 117 and line 170 through line 173; line 115 gives relatively hot starts. Although many real, engineering or not, problems do not exhibit obvious hot starts, one can experiment.

The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.

0 DEFINT J,K,B,X
2 DIM A(10100),X(10100)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
114 REM A(J44)=-10+FIX(RND*21)
115 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10100)
167 IF RND<.5 THEN X(B)=A(B)-1 ELSE X(B)=A(B) +1
169 NEXT IPP
170 FOR J44=1 TO 10100
171 IF X(J44)<-30 THEN X(J44 )=A(J44 )
172 IF X(J44)>30 THEN X(J44 )=A(J44 )
173 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 10099
205 SUMNEWZ=SUMNEWZ+ 100* ( X(J44+1)-X(J44)^2 )^2 + ( 1-X(J44 ) ) ^2
207 NEXT J44
511 SONE= - SUMNEWZ
689 PD1=SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1899 PRINT A(9996),A(9997),A(9998),A(10099),A(10100),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-32000 is shown below:

1    1    1    1    1
1    1    1    1    1
0        -32000

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

M=0 is optimal. See Li and Sun [12, p. 415].

Of the 10100 A's, only the 10 A's of line 1771 and line 1899 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was five hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to a Nonlinear Integer Programming Problem with 10100 General Integer Variables of Lower Bounds of -8s and Upper Bounds of 8s

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve a problem based on Li and Sun's Problem 14.4, [12, p. 415], but with 10100 unknowns instead of 100 unknowns and with lower bounds of -8s instead of -5s and upper bounds of 8s instead of 5s. The function is based on the Rosenbrock function--see Schittkowski [16] and Li and Sun [12, p. 415]. Specifically, the test example here is as follows:

Minimize

10100-1
SIGMA         [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

-8<=X(i)<=8, X(i) integer, i=1, 2, 3,..., 10100.

One notes line 111 through line 117 and line 170 through line 173.

The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.

0 DEFINT J,K,B,X
2 DIM A(10100),X(10100)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
114 A(J44)=-8+FIX(RND*17)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10100)
167 IF RND<.5 THEN X(B)=A(B)-1 ELSE X(B)=A(B) +1
169 NEXT IPP
170 FOR J44=1 TO 10100
171 IF X(J44)<-8 THEN X(J44 )=A(J44 )
172 IF X(J44)>8 THEN X(J44 )=A(J44 )
173 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 10099
205 SUMNEWZ=SUMNEWZ+ 100* ( X(J44+1)-X(J44)^2 )^2 + ( 1-X(J44 ) ) ^2
207 NEXT J44
511 SONE= - SUMNEWZ
689 PD1=SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1899 PRINT A(9996),A(9997),A(9998),A(10099),A(10100),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-32000 is shown below:

1    1    1    1    1
1    1    1    1    1
0       -32000

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

M=0 is optimal. See Li and Sun [12, p. 415].

Of the 10100 A's, only the 10 A's of line 1771 and line 1899 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was nine hours and a half.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to a Nonlinear Integer Programming Problem with 10100 General Integer Variables of Lower Bounds of -7s and Upper Bounds of 7s

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve a problem based on Li and Sun's Problem 14.4, [12, p. 415], but with 10100 unknowns instead of 100 unknowns and with lower bounds of -7s instead of -5s and upper bounds of 7s instead of 5s. The function is based on the Rosenbrock function--see Schittkowski [16] and Li and Sun [12, p. 415]. Specifically, the test example here is as follows:

Minimize

10100-1
SIGMA         [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

-7<=X(i)<=7, X(i) integer, i=1, 2, 3,..., 10100.

One notes line 111 through line 117 and line 170 through line 173.

The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.

0 DEFINT J,K,B,X
2 DIM A(10100),X(10100)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
114 A(J44)=-7+FIX(RND*15)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10100)
167 IF RND<.5 THEN X(B)=A(B)-1 ELSE X(B)=A(B) +1
169 NEXT IPP
170 FOR J44=1 TO 10100
171 IF X(J44)<-7 THEN X(J44 )=A(J44 )
172 IF X(J44)>7 THEN X(J44 )=A(J44 )
173 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 10099
205 SUMNEWZ=SUMNEWZ+ 100* ( X(J44+1)-X(J44)^2 )^2 + ( 1-X(J44 ) ) ^2
207 NEXT J44
511 SONE= - SUMNEWZ
689 PD1=SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1899 PRINT A(9996),A(9997),A(9998),A(10099),A(10100),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-32000 is shown below:

1    1    1    1    1
1    1    1    1    1
0       -32000

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

M=0 is optimal. See Li and Sun [12, p. 415].

Of the 10100 A's, only the 10 A's of line 1771 and line 1899 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was nine hours and a half.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to a Nonlinear Integer Programming Problem with 10100 General Integer Variables of Lower Bounds of -2s and Upper Bounds of 2s

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve a problem based on Li and Sun's Problem 14.4, [12, p. 415], but with 10100 unknowns instead of 100 unknowns and with lower bounds of -2s instead of -5s and upper bounds of 2s instead of 5s. The function is based on the Rosenbrock function--see Schittkowski [16] and Li and Sun [12, p. 415]. Specifically, the test example here is as follows:

Minimize

10100-1
SIGMA         [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

-2<=X(i)<=2, X(i) integer, i=1, 2, 3,..., 10100.

One notes line 111 through line 117 and line 170 through line 173.

The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.

0 DEFINT J,K,B,X
2 DIM A(10103),X(10103)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
115 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10103)
167 IF RND<.5 THEN X(B)=A(B)-1 ELSE X(B)=A(B) +1
169 NEXT IPP
170 FOR J44=1 TO 10100
171 IF X(J44)<-2 THEN X(J44 )=A(J44 )
172 IF X(J44)>2 THEN X(J44 )=A(J44 )
173 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 10099
205 SUMNEWZ=SUMNEWZ+ 100* ( X(J44+1)-X(J44)^2 )^2 + ( 1-X(J44 ) ) ^2
207 NEXT J44
511 SONE= - SUMNEWZ
689 PD1=SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1888 PRINT A(1),A(10099),A(10100),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31999 is shown below:

1      1     1     0       -32000

1      1     1     0       -31999

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

M=0 is optimal. See Li and Sun [12, p. 415].

Of the 10100 A's, only the 3 A's of line 1888 are shown above.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver Using Microsoft's GW-BASIC 3.11 Interpreter To Solve Li and Sun's Problem 14.5 but with 10000 Unknowns instead of 100 Unknowns

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.5, [12, p. 416], but with 10000 unknowns instead of 100 unknowns. Specifically, the test example here is as follows:

Minimize

10000                             10000
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1                                      i=1

subject to

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 83 through line 87 and line 170 through line 173.

The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.

0 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
83 LB=-FIX(RND*6)
87 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 10000
171 IF X(J44)<LB THEN X(J44 )=A(J44 )
172 IF X(J44)>UP THEN X(J44 )=A(J44 )
173 NEXT J44
182 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^4
187 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 10000
205 SUMNEWZ=SUMNEWZ+ X(J44)
207 NEXT J44
689 PD1=-SUMY-SUMNEWZ^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1953 PRINT A(1),A(2),A(3),A(4),A(5)
1955 PRINT A(6),A(7),A(8),A(9),A(10)
1957 PRINT A(3991),A(3992),A(3993),A(3994),A(3995)
1960 PRINT A(3996),A(3997),A(3998),A(3999),A(4000)
1965 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1966 PRINT A(9996),A(9997),A(9998),A(9999),A(10000)
1997 PRINT M,JJJJ,A(10001)
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31999 is shown below:

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0       -32000       0

-1      0      0      1      0
0      0      0      0      0
-1      0      0      0      0
0      0      0      -1      0
1      -1      0      -1      -1
-1      -1      0      0      1
-4912       -31999       0

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

M=0 is optimal. See Li and Sun [12, p. 416].

Of the 10000 A's, only the 30 A's of line 1953 through line 1966 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was five hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver Using Microsoft's GW-BASIC 3.11 Interpreter To Solve Li and Sun's Problem 14.5 but with 10000 Unknowns (instead of 100 Unknowns) and an Additional Constraint, X(1)^2+X(2)^2+X(3)^2+...+X(10000)^2 <= 10000

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.5, [12, p. 416], but with an additional less-than-or-equal-to constraint and with 10000 unknowns instead of 100 unknowns. Specifically, the test example here is as follows:

Minimize

 

10000                             10000
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1                                        i=1

subject to

X(1)^2+X(2)^2+X(3)^2+...+X(10000)^2 <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 83 through line 87 and line 170 through line 173.

The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.

0 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
83 LB=-FIX(RND*6)
87 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 10000
171 IF X(J44)<LB THEN X(J44 )=A(J44 )
172 IF X(J44)>UP THEN X(J44 )=A(J44 )
173 NEXT J44
174 SUMNEW=0
175 FOR J44=1 TO 10000
176 SUMNEW=SUMNEW+X(J44) ^2
177 NEXT J44
178 X(10001)=10000- SUMNEW
179 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
182 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^4
187 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 10000
205 SUMNEWZ=SUMNEWZ+ X(J44)
207 NEXT J44
689 PD1=-SUMY-SUMNEWZ^2 +50000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1953 PRINT A(1),A(2),A(3),A(4),A(5)
1960 PRINT A(3996),A(3997),A(4998),A(4999),A(4999)
1962 PRINT A(8996),A(8997),A(8998),A(8999),A(9999)
1965 PRINT A(9996),A(9997),A(9998),A(9999),A(10000)
1966 PRINT M,JJJJ,A(10001)
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31999 is shown below:

 0      0      0      0      0
0      0      0      0      0
0      0      0      0      0
0      0      0      0      0
0     -32000     0       

-1     0      0      1      0
0      0      -1      0      0
-1      0      0      1      0
-1      -1      0      0      1
-4912     -31999     0       

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

M=0 is optimal. See Li and Sun [12, p. 416].

Of the 10000 A's, only the 10 A's of line 1953 through line 1965 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was seven hours and a half.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver with Hot Starts Applied to Li and Sun's Problem 14.4 but with 10000 Unknowns (instead of 100 Unknowns), Lower Bounds of -1000s, and Upper Bounds of 1000s, Second Edition

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4, [12, p. 415], but with 10000 unknowns instead of 100 unknowns, lower bounds of -1000s instead of -5s, and upper bounds of 1000s instead of 5s. Like Schittkowski's Test Problems 294-299, their problem refers to the Rosenbrock function--see Schittkowski [16] and Li and Sun [12, p. 415]. Specifically, the test example here is as follows:

Minimize

10000-1
SIGMA        [ 100* ( X(i+1)-X(i)^2 )^2 + ( 1-X(i) ) ^2 ]
i=1

subject to

-1000<=X(i)<=1000, X(i) integer, i=1, 2, 3,..., 10000.

One notes that line 115 of the following computer program is 115 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1. Compare it to 115 A(J44)=-1000+FIX(RND*2001), for example. With respect to the present test example, the former produces hotter starts.

However, many real, engineering or not, problems do not give out obvious hot starts. That makes using hot starts not a regular approach.

One also notes line 170 through line 173.

The present paper uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 REM A(J44)=-11+FIX(RND*23)
115 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=A(B)-1 ELSE X(B)=A(B) +1
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
170 FOR J44=1 TO 10000
171 IF X(J44)<-1000 THEN X(J44 )=A(J44 )
172 IF X(J44)>1000 THEN X(J44 )=A(J44 )
173 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ 100* ( X(J44+1)-X(J44)^2 )^2 + ( 1-X(J44 ) ) ^2
207 NEXT J44
511 SONE= - SUMNEWZ
689 PD1=SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1775 PRINT A(5996),A(5997),A(5998),A(5999),A(6000)
1779 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1953 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31995 is shown below:

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31999

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31998

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31997

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31996

-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-4 -31995

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

M=0 is optimal. See Li and Sun [12, p. 415].

Of the 10000 A's, only the 15 A's of line 1771 through line 1779 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31995 was nine hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver with Hot Starts Applied to Li and Sun's Problem 14.4 but with 10000 Unknowns (instead of 100 Unknowns), Lower Bounds of -20s, and Upper Bounds of 20s

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4, [12, p. 415], but with 10000 unknowns instead of 100 unknowns, lower bounds of -20s instead of -5s, and upper bounds of 20s instead of 5s. Like Schittkowski's Test Problems 294-299, their problem refers to the Rosenbrock function--see Schittkowski [16] and Li and Sun [12, p. 415]. Specifically, the test example here is as follows:

Minimize

10000-1
SIGMA [ 100* ( X(i+1)-X(i)^2 )^2 + ( 1-X(i) ) ^2 ]
i=1

subject to

-20<=X(i)<=20, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 111 through line 117 of the computer program listed below. While line 111 through line 117 of the preceding paper are 111 FOR J44=1 TO 10000, 114 A(J44)=-10+FIX(RND*21), 117 NEXT J44, here line 111 through line 117 are 111 FOR J44=1 TO 10000, 114 REM A(J44)=-10+FIX(RND*21), 115 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1, 117 NEXT J44. Comparing line 114 there to line 115 here, one sees that with respect to the present test example, the starts here are hotter than the starts there.

However, many real, engineering or not, problems do not give out obvious hot starts.

One notes line 170 through line 173 of the preceding paper and line 170 through line 173 of the present paper.

The present paper uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 REM A(J44)=-10+FIX(RND*21)
115 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=A(B)-1 ELSE X(B)=A(B) +1
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
170 FOR J44=1 TO 10000
171 IF X(J44)<-20 THEN X(J44 )=A(J44 )
172 IF X(J44)>20 THEN X(J44 )=A(J44 )
173 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ 100* ( X(J44+1)-X(J44)^2 )^2 + ( 1-X(J44 ) ) ^2
207 NEXT J44
511 SONE= - SUMNEWZ
689 PD1=SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1779 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1953 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31999 is shown below:

1 1 1 1 1
1 1 1 1 1
0      -32000

-1 1 1 1 1
1 1 1 1 1
-4      -31999

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

M=0 is optimal. See Li and Sun [12, p. 415].

Of the 10000 A's, only the 10 A's of line 1771 and line 1779 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31999 was three hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Arty . Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

Mixed Integer Nonlinear Programming (MINLP) Solver Solving Li and Sun's Problem 14.4 but with 10000 Unknowns (instead of 100 Unknowns), Lower Bounds of -10s, and Upper Bounds of 10s, Second Edition

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4, [12, p. 415], but with 10000 unknowns instead of 100 unknowns, lower bounds of -10s instead of -5s, and upper bounds of 10s instead of 5s. Like Schittkowski's Test Problems 294-299, their problem refers to the Rosenbrock function--see Schittkowski [16] and Li and Sun [12, p. 415]. Specifically, the test example here is as follows:

Minimize

10000-1
SIGMA        [ 100* ( X(i+1)-X(i)^2 )^2 + ( 1-X(i) ) ^2 ]
i=1

subject to

-10<=X(i)<=10, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 111 through line 117 and line 170 through line 173.

The present paper uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 A(J44)=-10+FIX(RND*21)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=A(B)-1 ELSE X(B)=A(B) +1
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
170 FOR J44=1 TO 10000
171 IF X(J44)<-10 THEN X(J44 )=A(J44 )
172 IF X(J44)>10 THEN X(J44 )=A(J44 )
173 NEXT J44
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ 100* ( X(J44+1)-X(J44)^2 )^2 + ( 1-X(J44 ) ) ^2
207 NEXT J44
511 SONE= - SUMNEWZ
689 PD1=SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1779 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1953 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31998 is shown below:

1 1 1 1 1
1 1 1 1 1
0      -32000

-1 1 1 1 1
1 1 1 1 1
-105      -31999

1 1 1 1 1
1 1 1 1 1
0      -31998

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

M=0 is optimal. See Li and Sun [12, p. 415].

Of the 10000 A's, only the 10 A's of line 1771 and line 1779 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31998 was nine hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

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

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] 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.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

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

[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[19] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[20] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html