Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4 but with two additional constraints and with 5000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:
5000-1
SIGMA [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1
subject to
X(1)^3+X(2)^3+X(3)^3+...+X(5000)^3 = 5000
X(1)+X(2)+X(3)+...+X(5000) <= 5000
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 5000.
Thus, one of the two additional constraints is an equality constraint and the other is a less-than-or-equal-to constraint.
One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).
One also notes line 191, which is 191 X(1)= ( 5000 -(SUMY) )^(1/3).
The objective function here is based on the Rosenbrock function [10, p. 415].
In contrast with the preceding papers, the present paper uses the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 5000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 5000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 5000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 5000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 5000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 5000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 5000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(5001)=5000- SUMNEW
303 IF X(5001)<0 THEN X(5001)=X(5001) ELSE X(5001)=0
401 SONE=0
402 FOR J44=1 TO 4999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(5001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(4996),A(4997),A(4998),A(4999),A(5000)
1939 PRINT 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 [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31993 is shown below:
1 1 1 1 1
1 1 1 1 2
-709 -32000
1 1 1 1 1
1 1 1 2 2
-1315 -31999
1 1 1 1 1
1 1 1 1 2
-509 -31998
1 1 1 1 1
1 1 1 1 0
-2117 -31997
1 1 1 1 1
1 1 1 1 2
-6157 -31996
1 1 1 1 1
1 1 1 1 2
-707 -31995
1 1 1 1 1
1 1 1 1 2
-509 -31994
1 1 1 1 1
1 1 1 1 1
0 -31993
Above there is no rounding by hand.
M=0 is optimal. See Li and Sun [10, p. 415].
Of the 5000 A's, only the 10 A's of line 1931 and line 1936 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 15 hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
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[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] 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/
[17] 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/
[18] 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
