Jsun Yui Wong
The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 22170 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
22170-1
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,…, 22170.
One notes line 111 through line 118, which are
111 FOR J44=1 TO 22170
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 22170
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 22170
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*22173)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 22170
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 22169
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 22170
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(22167),A(22168),A(22169),A(22170),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=-31995 is shown below:
-1 1 1 2 3
-608 -32000
1 1 1 2 4
-302 -31999
0 0 0 0 0
-22778 -31998
0 0 0 0 0
-22169 -31997
1 1 1 1 1
0 -31996
1 1 1 2 3
-405 -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 22170 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=-31995 was seven hours.
For a computer program involving a mix of continuous variables and integer variables, see Wong [21], for instance.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[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] 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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