Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but with **32510 unknowns** instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski Test Problem 282 [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

32510-1

(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2

i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510.

One notes that the problem above is equivalent to minimize

32510-1

(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2

i=1

subject to

X(1) + X(2) + X(3) + ... + X(32510) >= 32500

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510

**and** to minimize

32510-1

(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2

i=1

subject to

X(1) + X(2) + X(3) + ... + X(32510) <= 32500

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510.

One notes that here the artificial constraint is X(1) + X(2) + X(3) + ... + X(32510) <= 32500 while the corresponding artificial constraint of the preceding paper is

X(1)^5 + X(2)^5 + X(3)^5 + ... + X(32710)^5 <= 32710. The former is simpler to use.

**And then one takes the better of the two.**

Generally speaking, while directly dealing with an initial problem is advantageous if the initial problem is "small," dealing with the initial problem plus an additional constraint--twice--is advantageous if the initial problem is "large" because each of these two problems has a smaller penalty-free region than that of the initial problem; see line 251, line 257, and line 492 of each of the two computer programs below.

One notes line 221 through line 257 of the first computer program below, which are

221 SFE = 0

225 FOR J44 = 1 TO 32510

228 SFE = SFE + X(J44)

233 NEXT J44

251 TSL = -32500 + SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0

(1) The Additional Constraint Used Immediately Below Is X(1) + X(2) + X(3) + ... + X(32510) >= 32500

The following computer program uses QB64 [18, 19].

0 DEFINT J, K, B, X, A

2 DIM A(32522), X(32522)

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 32510

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

117 NEXT J44

128 FOR I = 1 TO 100000

129 FOR KKQQ = 1 TO 32510

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * .3)

140 B = 1 + FIX(RND * 32513)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)

169 NEXT IPP

170 FOR J44 = 1 TO 32510

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 32510

228 SFE = SFE + X(J44)

233 NEXT J44

251 TSL = -32500 + SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0

400 SUMNEWZ = 0

403 FOR J44 = 1 TO 32509

405 SUMNEWZ = SUMNEWZ + (32510 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44

411 SONE = -(X(1) - 1) ^ 2 - (X(32510) - 1) ^ 2 - 32510 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL

1111 IF PD1 <= M THEN 1670

1452 M = PD1

1454 FOR KLX = 1 TO 32510

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 GOTO 128

1670 NEXT I

1778 PRINT A(1), A(2), A(32108), A(32509), A(32510), M, JJJJ

1788 PRINT A(1111), A(11111), A(23333), A(27777), A(28888)

1999 NEXT JJJJ

Based on the computer programs 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=-31997 is shown below:

-1 1 1 3 3

-1.668491E+13 -32000

1 1 1 1 2

1 1 1 1 1

-1.84491E+09 -31999

1 1 1 1 1

0 0 0 0 0

-1.625E+14 -31998

0 0 0 0 0

1 1 1 1 1

0 -31997

1 1 1 1 1

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

At JJJJ=-31997, M=0 is optimal. See Li and Sun [12, pp. 414-415].

Of the 32510 A's, only the ten A's of line 1778 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 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31997 was nine hours.

(2) The Additional Constraint Used Immediately Below Is X(1) + X(2) + X(3) + ... + X(32510) <= 32500

One notes that line 251 above is 251 TSL=-32500+SFE and that line 251 below is 251 TSL=32500-SFE.

The following computer program uses QB64 [18, 19].

0 DEFINT J, K, B, X, A

2 DIM A(32522), X(32522)

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 32510

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

117 NEXT J44

128 FOR I = 1 TO 100000

129 FOR KKQQ = 1 TO 32510

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * .3)

140 B = 1 + FIX(RND * 32513)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)

169 NEXT IPP

170 FOR J44 = 1 TO 32510

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 32510

228 SFE = SFE + X(J44)

233 NEXT J44

251 TSL = 32500 - SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0

400 SUMNEWZ = 0

403 FOR J44 = 1 TO 32509

405 SUMNEWZ = SUMNEWZ + (32510 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44

411 SONE = -(X(1) - 1) ^ 2 - (X(32510) - 1) ^ 2 - 32510 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL

1111 IF PD1 <= M THEN 1670

1452 M = PD1

1454 FOR KLX = 1 TO 32510

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 GOTO 128

1670 NEXT I

1778 PRINT A(1), A(2), A(32108), A(32509), A(32510), M, JJJJ

1788 PRINT A(1111), A(11111), A(23333), A(27777), A(28888)

1999 NEXT JJJJ

Based on the computer programs 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=-31997 is shown below:

-1 1 1 2 3

-4.678518 -32000

0 1 1 1 1

0 0 0 0 0

-2 -31999

0 0 0 0 0

0 0 0 0 0

-2 -31998

0 0 0 0 0

0 0 0 0 0

-2 -31997

0 0 0 0 0

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

Of the 32510 A's, only the ten A's of line 1778 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 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31997 was six hours.

(3) The realized solution with M=0 at JJJJ=-31997, which was produced through the artificial constraint X(1) + X(2) + X(3) + ... + X(32510) >= 32500

of the first computer program, is the best produced.

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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[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.

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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. 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 Researc*h, 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. 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, March 08). Nonconvex Mixed Integer Nonlinear Programming (MINLP) Computer Programs with a Divide-and-Conquer Strategy To Solve Li and Sun's Problem 14.3 but with n=15110 General Integer Variables. http://myblogsubstance.typepad.com/substance/2015/03/

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