Monday, January 26, 2015

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.4 but with 11000 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 11000 general integer variables instead of their 100 general integer variables [12, p. 415].  The function used 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

11000-1
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,..., 11000.

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

The following computer program uses the IBM Personal Computer 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(11003),X(11003)
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 11000
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 11000
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*11003)
167 IF RND<.5 THEN X(B)=(A(B)-1)   ELSE X(B)=(A(B)   +1  )
169 NEXT IPP
171 FOR J9=1 TO 11000
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 10999
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 11000
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(10098),A(10999),A(11000)
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=-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
-205        -31998

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

0   0   0   0   0
0   0   0   0   0
0   0   0   0   0
-10999        -31996

1   1   1   1   1
1   1   1   1   1
1   1   1   1   1
-203        -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 11000 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=-31995 was eleven 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