Jsun Yui Wong
The problem here is based on Li and Sun’s Problem 14.3 [12, 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize
20120-1
(X(1)-1)^2 + ( X(20120)-1)^2 + 20120* SIGMA (20120-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^3 + X(2)^3 + X(3)^3 + … + X(20120)^3 = 20120
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 20120.
The following computer program uses QB64 [18, 19].
0 DEFINT J, K, B, X, A
2 DIM A(33333), X(33333)
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 20120
114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 32000
129 FOR KKQQ = 1 TO 20120
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 20123)
167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 20120
171 IF X(J44) < LB THEN X(J44) = LB
172 IF X(J44) > UB THEN X(J44) = UB
173 NEXT J44
221 SFE = 0
225 FOR J44 = 1 TO 20120
228 SFE = SFE + X(J44) ^ 3
233 NEXT J44
251 TSL = -20120 + SFE
257 REM IF TSL<0 THEN TSL=TSL ELSE TSL=0
267 IF TSL = 0 THEN TSL = TSL ELSE TSL = -ABS(TSL)
400 SZ = 0
403 FOR J44 = 1 TO 20119
405 SZ = SZ + (20120 – J44) * (X(J44) ^ 2 – X(J44 + 1)) ^ 2
407 NEXT J44
411 SONE = -(X(1) – 1) ^ 2 – (X(20120) – 1) ^ 2 – 20120 * SZ
492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 20120
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(20118), A(20119), A(20120), 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=-31990 is shown below:
2 3 3 -1 0
-2.478626E+14 -32000
0 2 0 -2 2
-5.006566E+13 -31999
0 0 0 0 0
-1.006054E+14 -31998
-5 2 0 -2 0
-7.411186E+14 -31997
-3 -3 2 0 -2
-3.459166E+14 -31996
0 0 0 0 0
-1.006E+14 -31995
5 -4 3 -1 5
-5.799339E+14 -31994
0 0 0 0 0
-1.006057E+14 -31993
-1 -1 -1 0 1
-2.005778E+13 -31992
-3 3 2 -1 -3
-2.925154E+14 -31991
1 1 1 1 1
0 -31990
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, pp. 414-415].
Of the 20120 A’s, only the 5 A’s of line 1778 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=-31990 was six 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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[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
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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”.
[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, March 11). Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Computer Program Applied to a Problwm with 15120 General Integer Variables and with X(1)^9 + X(2)^9 + X(3)^9 + … + X(15120)^9 = 15120. http://myblogsubstance.typepad.com/substance/2015/03/
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