Jsun Yui Wong
The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 32760 unknowns instead of their 100 unknowns [12, pp. 415-417]. The function is based on the widely known Rosenbrock function–-see, for example, Li and Sun [12, p. 415] and Schittkowski [17, Test Problems 294-299]. Specifically, the test example here is as follows:
Minimize
32760-1
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,…, 32760.
One notes the starting solution vectors of line 111 through line 118, which are
111 FOR J44=1 TO 32760
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44.
While line 128 of the third edition is 128 FOR I = 1 TO 100000, here line 128 is 128 FOR I = 1 TO 90000.
The following computer program uses QB64 [19, 20].
0 DEFINT J, K, B, X, A
2 DIM A(32763), X(32763)
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 32760
116 A(J44) = -5 + FIX(RND * 11)
118 NEXT J44
128 FOR I = 1 TO 90000
129 FOR KQ = 1 TO 32760
130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)
167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
171 FOR J9 = 1 TO 32760
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 32759
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 32760
1455 A(KX) = X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1), A(32167), A(32168), A(32709), A(32760), M, JJJJ
1783 PRINT A(111), A(30167), A(31168), A(32758), A(32759)
1999 NEXT JJJJ
Modelled after the computer program in Wong [25], this BASIC computer program was run with QB64 [19, 20]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31991 is shown below:
-1 1 1 1 3
-205 -32000
1 1 1 1 2
-1 1 1 1 1
--205 -31999
1 1 1 1 1
-1 1 1 1 1
-4 -31998
1 1 1 1 1
0 0 0 0 0
-32759 -31997
0 0 0 0 0
0 0 0 0 0
-32759 -31996
0 0 0 0 0
-1 1 1 1 1
-4 -31995
1 1 1 1 1
1 1 1 1 1
0 -31994
1 1 1 1 1
-1 1 1 1 1
-4 -31993
1 1 1 1 1
0 0 0 0 0
-32759 -31992
0 0 0 0 0
1 1 1 1 1
0 -31991
1 1 1 1 1
Above there is no rounding by hand; it is just straight copying by hand from the screen.
At JJJJ=-31994 and JJJJ=-31991, M=0, which is optimal. See Li and Sun [12, p. 415].
Of the 32760 A's, only the 10 A's of line 1773 and line 1783 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 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31991 was 24 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
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[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.
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[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] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operatons Research, Volume 14 (1988), Issue 1, pp.. 291-304.
[15] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[16] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[17] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[18] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[19] E.K. Virtanen (2008-05-26). "Interview With Galleon",
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview
[20] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[21] 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/
[22] 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/
[23] 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/
[24] 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
[25] 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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