Tuesday, February 3, 2015

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

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[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