The problem here is based on Li and Sun's Problem 14.3, which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize
15120-1
(X(1)-1)^2 + ( X(15120)-1)^2 + 15120* SIGMA (15120-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^5 + X(2)^5 + X(3)^5 + ... + X(15120)^5 = 15120
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15120.
One notes line 111 through line 117, which are
111 FOR J44=1 TO 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.
The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.
0 DEFINT J,K,B,X,A
2 DIM A(15123),X(15123)
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 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15120
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15123)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15120
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 15120
228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL=-15120+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 15119
405 SZ=SZ+ (15120-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15120)-1)^2 -15120* SZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15120
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15118),A(15119),A(15120),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31993 is shown below:
1 1 1 1 1
-5.063338E+09 -32000
0 0 0 0 0
-7.56E+13 -31999
0 0 0 0 0
-7.560546E+13 -31998
-2 -2 3 -2 -2
-5.084827E+13 -31997
-1 -1 -1 -1 -1
-7.471533E+12 -31996
-3 2 -1 1 0
-2.084663E+13 -31995
0 0 0 0 0
-7.56E+13 -31994
1 1 1 1 1
0 -31993
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 15120 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 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=-31993 was 27 hours.
For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[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.
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[18] 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/
[19] 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/
[20] 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/
[21] 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
