Jsun Yui Wong
The computer program listed below seeks to solve Li and Sun’s Problem 14.5 but with 32765 unknowns instead of their 100 unknowns [12, pp. 416-417]. Specifically, the test example here is as follows:
Minimize
32765 32765
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1 i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32765.
One notes the starting solution vectors of line 111 through line 117, which are as follows:
111 FOR J44=1 TO 32765
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.
The following computer program uses qb64v1000-win [19, 20].
0 DEFINT J, K, B, X, A
2 DIM A(32765), X(32765)
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 32765
114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 125000
129 FOR KQ = 1 TO 32765
130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32765)
167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32765
171 IF X(J44) < LB THEN X(J44) = LB
172 IF X(J44) > UB THEN X(J44) = UB
173 NEXT J44
482 SY = 0
483 FOR J44 = 1 TO 32765
485 SY = SY + X(J44) ^ 4
487 NEXT J44
488 SZ = 0
489 FOR J44 = 1 TO 32765
490 SZ = SZ + X(J44)
491 NEXT J44
492 PD1 = -SY – SZ ^ 2
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32765
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1781 PRINT A(1), A(2), A(3), A(222), A(32765), M, JJJJ
1788 PRINT A(1111), A(11111), A(32762), A(32763), A(32764)
1999 NEXT JJJJ
Modelled after the computer program in Wong [23], this BASIC computer program was run with qb64v1000-win [19, 20]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31993 is shown below:
-1 1 -1 0 -1
-17742 -32000
0 0 0 1 0
1 -1 1 0 1
-19202 -31999
0 0 0 -1 -1
-1 -1 0 0 0
-17074 -31998
0 1 0 1 0
0 0 0 0 0
-16120 -31997
1 1 0 -1 1
0 0 0 0 0
0 -31996
0 0 0 0 0
1 1 -1 1 1
-19790 -31995
0 0 1 1 0
1 1 1 1 -1
-29824 -31994
1 1 -1 1 -1
0 0 0 0 0
-16178 -31993
-1 0 0 -1 0
Above there is no rounding by hand; it is just straight copying by hand from the screen.
At JJJJ=-31996, M=0, which is optimal. See Li and Sun [12, p. 416].
Of the 32765 A’s, only the 10 A’s of line 1781 and line 1788 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 qb64v1000-win [19, 20], the wall-clock time for obtaining the output through JJJJ=-31993 was eleven 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. 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] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operations 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. http://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 (2015, February 11). General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.5 but Involving 15120 General Integer Variables instead of Their 100 General Integer Variables. http://myblogsubstance.typepad.com/substance/2015/02/
computerprogramsandresults 