General Mixed Integer Nonlinear Programming (MINLP) Solver Solving Li and Sun's Problem 14.3 but of n=15100 General Integer Variables Subject to an Additional Constraint, X(1)^3 + X(2)^3 + X(3)^3 + ... + X(15100)^3 >= 15100

Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but of n=15100 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

                                                                      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)^3 + X(2)^3 + X(3)^3 + ... + X(15100)^3 >= 15100

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

One notes the starting solution vectors of line 111 through line 117, which are as follows:
111 FOR J44=1 TO 15100
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(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)^3
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:

1   1   1   1   1
-1.028267E+10        -32000

1   1   1   1   0
-1.657285E+12        -31999

1   1   1   0   1  
-1.512755E+12        -31998

1   1   1   1   1  
0           -31997

0   0   0   0   0
-7.55E+13        -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 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=-31996  was 18 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.  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, 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

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)=-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(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)=-5+FIX(RND*11)
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
1778 PRINT A(1),A(2),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=-31999 is shown below:

1     1     1     1     1
0      -32000

1     1     3     2     -2
-3.157331E+12      -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, pp. 414-415].

Of the 15000 A's, only the 5 A's of 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=-31999 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.  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, 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

General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun's Problem 14.5 but Involving 15170 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 15170 unknowns instead of their 100 unknowns [12, p. 416].  Specifically, the test example here is as follows:

Minimize

15170                                   15170
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,..., 15170.

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

111 FOR J44=1 TO 15170
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(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
114 A(J44)=-5+FIX(RND*11)
117 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
170 FOR J44=1 TO 15170
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 15170
485 SY=SY+X(J44)^4
487 NEXT J44
488 SZ=0
489 FOR J44=1 TO          15170
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 15170
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1779 PRINT A(1),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=-31998 is shown below:

1        1        -13776        -32000

-1        -1        -13644        -31999

0         0        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. 416].

Of the 15170 A's, only the 2 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=-31998 was 3 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.  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, 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

Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 15180 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15180 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

15180-1
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,..., 15180.

One notes the starting solution vectors of line 111 through line 118, which are as follows:
111 FOR J44=1 TO 15180
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(15183),X(15183)
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 15180
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15180
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15183)
167 IF RND<.5 THEN X(B)=(A(B)-1)   ELSE X(B)=(A(B)   +1  )
169 NEXT IPP
171 FOR J9=1 TO 15180
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
401 S=0
402 FOR J44=1 TO 15179
411 S=S+  100*  ( X(J44+1) - X(J44)^2  )^2  +  (  1-X(J44)   )^2
421 NEXT J44
689 PD1=-S
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 15180
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(15180),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=-31995 is shown below:

1        -402        -32000
3        -411        -31999
1        -402        -31998
1        0        -31997
0        -15382        -31996
1        0        -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 15180 A's, only the A of line 1773, A(15180), is 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 34 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.  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, 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

Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 15150 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15150 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

15150-1
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,..., 15150.

One notes the starting solution vectors of line 111 through line 118, which are 111 FOR J44=1 TO 15150, 116 A(J44)=-5+FIX(RND*11), and 118 NEXT J44.  The starts so generated are colder than the starts of the several papers immediately preceding the present paper.

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(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
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15150
130 X(KQ)=A(KQ)
131 NEXT KQ
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  )
169 NEXT IPP
171 FOR J9=1 TO 15150
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 15149
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 15150
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(15147),A(15148),A(15149),A(15150)
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 [12, 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
-406        -32000

0        0        0        0        0
-15352         -31999

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 15150 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=-31998 was 11 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.  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, 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

Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 15000 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15000 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

15000-1
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,..., 15000.

One notes the starting solution vectors of line 111 through line 118, which are 111 FOR J44=1 TO 15000,
116 A(J44)=-5+FIX(RND*11), and 118 NEXT J44.  The starts so generated are colder than the starts of the several papers immediately preceding the present paper.

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(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
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15000
130 X(KQ)=A(KQ)
131 NEXT KQ
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  )
169 NEXT IPP
171 FOR J9=1 TO 15000
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 14999
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 15000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(14998),A(14999),A(15000)
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 [12, page iii, Preface].  Copied by hand from the screen, the computer program's complete output through JJJJ=-31985 is shown below:

1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-403   -32000

0   0   0   0   0
0   0   0   0   0
0   0   0   0   0
-15202   -31999

1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-1001   -31998

-1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-205   -31997

-1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-4   -31996

1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-1001   -31995

1   1   1   1   1
1   1   1   1   1
1   1   0   -2   4
-510   -31994
 
0   0   0   0   0
0   0   0   0   0
0   0   0   0   0
-14999   -31993

0   0   0   0   0
0   0   0   0   0
0   0   0   0   0
-14999   -31992

-1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-1408   -31991

1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-201   -31990

-1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-205   -31989

-1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-4   -31988

0   0   0   0   0
0   0   0   0   0
0   0   0   0   0
-15405   -31987

-1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-410   -31986

1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
0   -31985

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 15000 A's, only the 15 A's of line 1771 through  line 1774 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=-31985 was 65 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.  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, 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

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.4 but with 15000 General Integer Variables, Second Edition

Jsun Yui Wong  

The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15000 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

15000-1
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,..., 15000.

While line 111 through line 118 of the preceding paper are
111 FOR J44=1 TO 15000
116 A(J44)=FIX(RND*2)
118 NEXT J44,
here line 111 through line 118 are
111 FOR J44=1 TO 15000
116 A(J44)=-1+FIX(RND*3)
118 NEXT J44.
The starting solution vectors of the latter will involve in general more searches.

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(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
116 A(J44)=-1+FIX(RND*3)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15000
130 X(KQ)=A(KQ)
131 NEXT KQ
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  )
169 NEXT IPP
171 FOR J9=1 TO 15000
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 14999
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 15000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(14998),A(14999),A(15000)
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 [12, 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
1   1   1   1   1
-402        -32000                
 
0   0   0   0   0
0   0   0   0   0
0   0   0   0   0
-14999        -31999
 
1   1   1   1   1
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 15000 A's, only the 15 A's of line 1771 through  line 1774 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 eight 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.  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, 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