Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun's Problem 14.5 but with **32760** **unknowns** instead of their 100 unknowns [12, pp. 416-417]. Specifically, the test example here is as follows:

Minimize

32760 32760

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,..., 32760.

One notes the starting solution vectors of line 111 through line 117, which are as follows:

111 FOR J44=1 TO 32760

114 A(J44)=-5+FIX(RND*11)

117 NEXT J44.

While line 128 of the third edition is 128 FOR I = 1 TO 90000, here line 128 is 128 FOR I = 1 TO 150000.

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

114 A(J44) = -5 + FIX(RND * 11)

117 NEXT J44

128 FOR I = 1 TO 150000

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

170 FOR J44 = 1 TO 32760

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 32760

485 SY = SY + X(J44) ^ 4

487 NEXT J44

488 SZ = 0

489 FOR J44 = 1 TO 32760

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 32760

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(32760), M, JJJJ

1788 PRINT A(1111), A(11111), A(32757), 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=-31990 is shown below:

-1 1 0 1 0

-17730 -32000

0 1 -1 -1 0

0 0 0 0 0

0 -31999

0 0 0 0 0

1 0 -1 1 0

-19210 -31998

0-1010

1 1 -1 1 -1

-23528 -31997

-1 -1 -1 1 1

0 0 0 0 0

0 -31996

0 0 0 0 0

0 1 0 -1 1

-16988 -31995

-1 0 -1 -1 0

0 1 1 1 1

-21630 -31994

1 1 1 -1 1

0 -1 -1 -1 1

-23636 -31993

1 0 -1 1 1

0 0 0 0 0

0 -31992

0 0 0 0 0

0 0 0 0 0

0 -31991

0 0 0 0 0

1 -1 0 1 0

-17772 -31990

0 0 1 -1 -1

Above there is no rounding by hand; it is just straight copying by hand from the screen.

At JJJJ=-31999, JJJJ=-31996, JJJJ=-31992, and JJJJ=-31991, M=0, which is optimal. See Li and Sun [12, p. 416].

Of the 32760 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 QB64 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31990 was 14 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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[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] 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 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/

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