The computer program listed below seeks to solve Li and Sun's Problem 14.5 but with 15170 unknowns instead of their 100 unknowns [12, p. 416]. Specifically, the test example here is as follows:
Minimize
15170 15170
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,..., 15170.
One notes the starting solution vectors of line 111 through line 117, which are as follows:
111 FOR J44=1 TO 15170
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.
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(15173),X(15173)
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 15170
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15170
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15173)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15170
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 15170
485 SY=SY+X(J44)^4
487 NEXT J44
488 SZ=0
489 FOR J44=1 TO 15170
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 15170
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1779 PRINT A(1),A(15170),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=-31998 is shown below:
1 1 -13776 -32000
-1 -1 -13644 -31999
0 0 0 -31998
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, p. 416].
Of the 15170 A's, only the 2 A's of line 1779 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 3 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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[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
