Month: February 2015

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General Mixed Integer Nonlinear Programming (MINLP) Solver with the Help of an Additional Constraint Solving Li and Sun’s Problem 14.3 but with n=15100 General Integer Variables

Jsun Yui Wong

The problem here is Li and Sun’s Problem 14.3 but with 15100 unknowns instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

15100-1
(X(1)-1)^2 + ( X(15100)-1)^2 + 15100* SIGMA (15100-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 15100.

One notes that the problem above is equivalent to minimize

15100-1
(X(1)-1)^2 + ( X(15100)-1)^2 + 15100* SIGMA (15100-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^5 + X(2)^5 + X(3)^5 + … + X(15100)^5 >= 15100

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 15100

and to minimize

15100-1
(X(1)-1)^2 + ( X(15100)-1)^2 + 15100* SIGMA (15100-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^5 + X(2)^5 + X(3)^5 + … + X(15100)^5 <= 15100

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 15100.

Then one takes the best produced.

Generally speaking, while dealing with a given problem directly is advantageous if the given problem is “small,” dealing with a given problem with an additional constraint–twice–is advantageous if the given problem is “large” because the two problems each with a smaller search region are easier.

(1) The Additional Constraint Used Immediately Below Is X(1)^5 + X(2)^5 + X(3)^5 + … + X(15100)^5 >= 15100

The following computer program uses the IBM Personal BASIC Compiler–through A:\>bascom and A:\>link–Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15103),X(15103)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15100
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15100
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15100
228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL=-15100+SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 15099
405 SUMNEWZ=SUMNEWZ+ (15100-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= – (X(1)-1)^2 – ( X(15100)-1)^2 -15100* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15098),A(15099),A(15100),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31992 is shown below:

1 1 1 1 1
-1.028267E+10      -32000

1 1 0 1 0
-7.491621E+11      -31999

0 0 1 1 2
-2.957329E+11      -31998

0 0 0 1 0
-7.564921E+11      -31997

-1 1 0 -1 0
-7.209565E+11      -31996

-1 1 1 1 0
-5.88012E+11      -31995

0 0 0 -1 4
-4.76734E+11      -31994

1 1 1 1 1
-5.150532E+9      -31993

1 1 1 1 1
0      -31992

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, pp. 414-415].

Of the 15100 A’s, only the 5 A’s of of 1778 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=-31992 was 41 hours.

(2) The Additional Constraint Used Below Is X(1)^5 + X(2)^5 + X(3)^5 + … + X(15100)^5 <= 15100

While line 251 above is 251 TSL=-15100+SFE, line 251 below is 251 TSL= 15100-SFE.

The following computer program also uses the IBM Personal BASIC Compiler–through A:\>bascom and A:\>link–Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15103),X(15103)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15100
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15100
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15100
228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL= 15100-SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 15099
405 SUMNEWZ=SUMNEWZ+ (15100-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= – (X(1)-1)^2 – ( X(15100)-1)^2 -15100* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15098),A(15099),A(15100),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31996 is shown below:

0 0 1 1 0
-4.28105E+11      -32000

0 0 0 0 0
-2      -31999

-1 1 0 0 1
-4.853535E+11      -31998

0 0 0 1 2
-4.197145E+11      -31997

0 0 0 0 0
-2      -31996

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

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=-31996 was 22 hours.

(3) The realized solution with M=0 at JJJJ=-31992 and produced through the additional constraint X(1)^5 + X(2)^5 + X(3)^5 + … + X(15100)^5 >= 15100 is the best produced.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

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. http://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. http://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, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[20] 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/

[21] 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

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General Mixed Integer Nonlinear Programming (MINLP) Solver Solving Li and Sun’s Problem 14.3 but of n=15000 General Integer Variables Subject to an Additional Constraint, X(1) + X(2) + X(3) + … + X(15000) >= 15000

Jsun Yui Wong

The problem here is Li and Sun’s Problem 14.3 but of n=15000 general integer variables subject to an additional constraint; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is as follows:

Minimize

15000-1
(X(1)-1)^2 + ( X(15000)-1)^2 + 15000* SIGMA    (15000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1) + X(2) + X(3) + … + X(15000) >= 15000

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 15000.

One notes the starting solution vectors of line 111 through line 117, which are as follows:
111 FOR J44=1 TO 15000
114 A(J44)=-2+FIX(RND*4.98)
117 NEXT J44

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

The following computer program uses the IBM Personal BASIC Compiler–through A:\>bascom and A:\>link–Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15003),X(15003)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15000
114 A(J44)=-2+FIX(RND*4.98)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15003)
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 15000
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15000
228 SFE=SFE+X(J44)
233 NEXT J44
251 TSL=-15000+SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 14999
405 SUMNEWZ=SUMNEWZ+ (15000-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= – (X(1)-1)^2 – ( X(15000)-1)^2 -15000* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(2),A(3),A(4),A(5)
1778 PRINT A(14997),A(14998),A(14999),A(15000),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31996 is shown below:

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

1 1 1 1 2
2 3 2 0 -2.608715E+12
-31999

1 2 1 1 1
-1 2 2 2 -2.369001E+12
-31998

2 1 1 1 1
0 1 1 2 -2.33732E+12
-31997

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

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, pp. 414-415].

Of the 15000 A’s, only the 9 A’s of line 1773 and line 1778 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=-31996 was 21 hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

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. http://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. http://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, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[20] 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/

[21] 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

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General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.5 but Involving 15150 General Integer Variables instead of Their 100 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.5 but with 15150 unknowns instead of their 100 unknowns [12, p. 416]. Specifically, the test example here is as follows:

Minimize

15150                             15150
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,…, 15150.

One notes the starting solution vectors of line 111 through line 117, which are as follows:

111 FOR J44=1 TO 15150
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

The following computer program uses the IBM Personal BASIC Compiler–through A:\>bascom and A:\>link–Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15153),X(15153)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15150
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15150
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15153)
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 15150
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
482 SUMY=0
483 FOR J44=1 TO 15150
485 SUMY=SUMY+X(J44)^4
487 NEXT J44
488 SZ=0
489 FOR J44=1 TO 15150
490 SZ=SZ+ X(J44)
491 NEXT J44
492 PD1=-SUMY-SZ^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15150
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1779 PRINT A(1),A(2),A(15148),A(15149),A(15150),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31983 is shown below:

1 0 1 1 -1
-13756      -32000

-1 1 0 -1 1
-9202      -31999

1 0 0 0 0
-7854      -31998

1 1 1 1 -1
-9158      -31997

0 0 0 0 0
-86      -31996

1 -1 0 0 1
-9350      -31995

-1 -1 1 1 -1
-9036      -31994

1 0 0 1 -1
-8108      -31993

0 0 -1 0 1
-8942      -31992

-1 -1 -1 0 1
-8152      -31991

0 0 0 0 0
-2      -31990

-1 0 -1 0 1
-8414      -31989

1 1 -1 0 0
-9156      -31988

1 -1 1 -1 1
-7928      -31987

-1 1 0 1 0
-7954      -31986

0 -1 1 -1 1
-9826      -31985

0 -1 0 1 0
-8318      -31984

0 0 0 0 0
0      -31983

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 15150 A’s, only the 5 A’s of 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=-31983 was 38 hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

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. http://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. http://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, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[20] 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/

[21] 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

by

General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.5 but Involving 15160 General Integer Variables instead of Their 100 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.5 but with 15160 unknowns instead of their 100 unknowns [12, p. 416]. Specifically, the test example here is as follows:

Minimize

15160                             15160
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,…, 15160.

One notes the starting solution vectors of line 111 through line 117, which are as follows:

111 FOR J44=1 TO 15160
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

The following computer program uses the IBM Personal BASIC Compiler–through A:\>bascom and A:\>link–Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15163),X(15163)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15160
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15160
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15163)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15160
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
482 SY=0
483 FOR J44=1 TO 15160
485 SY=SY+X(J44)^4
487 NEXT J44
488 SZ=0
489 FOR J44=1 TO 15160
490 SZ=SZ+ X(J44)
491 NEXT J44
492 PD1=-SY-SZ^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15160
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1779 PRINT A(1),A(2),A(15158),A(15159),A(15160),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31996 is shown below:

1 0 -1 -1 1
-13762      -32000

0 -1 1 1 0
-9524      -31999

-1 1 0 0 1
-7418       -31998

0 0 0 0 0
0      -31997

0 0 0 1 0
-9996      -31996

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 15160 A’s, only the 5 A’s of 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 Perso
nal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31996 was 9 hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

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. http://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. http://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, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[20] 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/

[21] 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

by

Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.4 but Involving 15160 General Integer Variables instead of Their 100 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 15160 unknowns instead of their 100 unknowns [12, p. 415]. The function is based on the widely known Rosenbrock function–see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:

Minimize

15160-1
SIGMA      100* ( X(i+1) – X(i)^2 )^2 + ( 1-X(i) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 15160.

One notes the starting solution vectors of line 111 through line 118, which are as follows:
111 FOR J44=1 TO 15160
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

The following computer program uses the IBM Personal BASIC Compiler–through A:\>bascom and A:\>link–Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15163),X(15163)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15160
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15160
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15163)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 15160
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
178 REM
207 REM
401 SONE=0
402 FOR J44=1 TO 15159
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 15160
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(15157),A(15158),A(15159),A(15160)
1777 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 [13, page iii, Preface]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31987 is shown below:

1 1 1 1 1
0      -32000

0 0 0 0 0
-15159      -31999

-1 1 1 1 1
-4      -31998

1 1 1 1 1
-201      -31997

0 0 0 0 0
-15159      -31996

0 0 0 0 0
-17167      -31995

-1 1 1 1 1
-4      -31994

-1 1 1 1 1
-4      -31993

0 0 0 0 0
-17979      -31992

-1 1 1 1 1
-4      -31991

0 0 0 0 0
-15362      -31990

1 1 1 1 1
-402      -31989

-1 1 1 1 1
-4      -31988

1 1 1 1 1
0      -31987

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 15160 A’s, only the 5 A’s of line 1773 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=-319987 was 57 hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

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. http://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. http://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, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[20] 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/

[21] 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

by

Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 15170 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 15170 general integer variables instead of their 100 general integer variables [12, p. 415]. The function is based on the widely known Rosenbrock function–see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:

Minimize

15170-1
SIGMA    100* ( X(i+1) – X(i)^2 )^2 + ( 1-X(i) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 15170.

One notes the starting solution vectors of line 111 through line 118, which are as follows:
111 FOR J44=1 TO 15170
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44.

For a computer program involving continuous variables and integer variables, see Wong [19].

The following computer program uses the IBM Personal BASIC Compiler–through A:\>bascom and A:\>link–Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15173),X(15173)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15170
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15170
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15173)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 15170
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
401 SONE=0
402 FOR J44=1 TO 15169
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 15170
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(15167),A(15168),A(15169),A(15170),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31997 is shown below:

1      1      1      1      1
0      -32000

1      1      1      1      1
-1001      -31999

0      0      0      0      0
-15169      -31998

1      1      1      1      1
-201      -31997

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 15170 A’s, only the 5 A’s of line 1773 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=-31997 was 15 hours.

For a computer program involving continuous variables and integer variables, see Wong [19].

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. http://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. http://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, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[20] 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/

[21] 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