Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.5, [11, p. 416], but with 10000 unknowns instead of 100 unknowns and with two added constraints. Specifically the computer program below tries to minimize the following:
10000 10000
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1 i=1
subject to
X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 0
X(1)+X(2)+X(3)+...+X(10000) <= 0
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.
One notes line 85, line 114, and line 175, which are 85 LB=-FIX(RND*6),
114 A(J44)=FIX(RND*(3)), and 175 IF X(J44)<LB THEN X(J44 )=A(J44 ), respectively.
While the computer program of the earlier edition uses an interpreter, the following computer program uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.
0 REM DEFDBL J-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 A(J44)=FIX(RND*(3))
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
143 REM
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM GOTO 170
169 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
170 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
178 SUMY=0
179 FOR J44=1 TO 10000
180 SUMY=SUMY+X(J44)^3
181 NEXT J44
182 REM
183 U=0-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=0- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
396 SUMYONE=0
397 FOR J44=1 TO 10000
398 SUMYONE=SUMYONE+X(J44)^4
399 NEXT J44
400 SUMNEWTWO=0
403 FOR J44=1 TO 10000
405 SUMNEWTWO=SUMNEWTWO+ X(J44)
407 NEXT J44
411 SONETOTAL= -SUMYONE - ( SUMNEWTWO)^2 +5000000!*X(10001)-5000000!*ABS(U)
689 PD1=SONETOTAL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1556 LLB=LB
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-37000! THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1938 PRINT A(4996),A(4997),A(4998),A(7999),A(8000)
1939 PRINT M,JJJJ,UU,LLB
1999 NEXT JJJJ
This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [12, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31998 is shown below:
0 0 0 0 -1
0 -1 0 -1 0
-4096576 -32000 0 -1
0 0 -1 0 0
0 -1 2 0 0
-40064 -31999 0 -3
0 0 0 0 0
0 0 0 0 0
0 -31998 0 0
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 [11, p. 416].
Of the 10000 A's, only the 10 A's of line 1931 and line 1938 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31998 was 22 hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[8] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[9] 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.
[10] 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.
[11] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[12] 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.
[13] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[14] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[15] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[16] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[17] 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/
[18] 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/
[19] 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
