Friday, June 19, 2015

A General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 20170 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 20170 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

20170-1
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,…, 20170.

One notes the starting solution vectors of line 111 through line 118, which are as follows:        
111 FOR J44=1 TO 20170
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44.

The following computer program uses QB64 [18, 19].

0 DEFINT J,K,B,X,A
2 DIM A(33173),X(33173)
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 20170

116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 20170

130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*20173)

167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 20170

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 20169

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 20170

1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(20167),A(20168),A(20169),A(20170),M,JJJJ

1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19].  Copied by hand from the screen, the computer program's complete output through JJJJ=-31986 is shown below:

-1      1      1      1      1
-611      -32000

-1      1      1      1      1
-4     -31999

1      1      1      1      1
-403     -31998

1      1      1      1      1
-201     -31997

0      0      0      0      0
-20778      -31996

-1      1      1      1      1
-611      -31995

1      1      1      2     3
-1611     -31994

-1      1      1      2     4
-306     -31993

-1      1      1      1      1
-406      -31992

-1      1      1      1      1
-205      -31991

1      1      1      1      1
-1001      -31990

-1      1      1      1      1
-414     -31989

0      0      0      0      0
-20169      -31988

1      1      1      1      1
0      -31987

1      1      1      1      1
0      -31986

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 20170 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  with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31986  was 15 hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [22], 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] E.K. Virtanen (2008-05-26).  "Interview With Galleon",      
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, February 2).  Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 15170 General Integer Variables.  https://computerprogramsandresults.wordpress.com/2015/02/02/

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