A Unified Computer Program Solving Li and Sun's Problem 14.4 Plus Two Less-Than-or-Equal-to Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(100)^3 <= 100

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 plus two less-than-or-equal-to constraints; see Li and Sun [10, p. 415].  Specifically the computer program below tries to minimize the following:

100-1
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(100)^3 <=  100

X(1)+X(2)+X(3)+...+X(100)  <=  100

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 100.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44 )=-5+FIX(   RND*11)
174 IF X(J44)>5 THEN X(J44 )=A(J44  )
175 IF X(J44)<-5 THEN X(J44 )=A(J44  ).

One also notes line 180 through line 195, especially line 185, which is 185 SUMY=SUMY+X(J44)^3.

The objective function here is based on the Rosenbrock function [10, p. 415].

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 100
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 3200
129 FOR KKQQ=1 TO 100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*100)
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 100
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=1 TO 100
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
193 X(102)=  100    -SUMY
195 IF X(102)<0 THEN X(102)=X(102) ELSE X(102)=0
200 SUMNEW=0
203 FOR J44=1 TO 100
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255  X(101)=100- SUMNEW
303 IF X(101)<0 THEN X(101)=X(101) ELSE X(101)=0
401 SONE=0
402 FOR J44=1 TO 99
411 SONE=SONE+  100*  ( X(J44+1) - X(J44)^2  )^2  +  (  1-X(J44)   )^2
421 NEXT J44
689 PD1=-SONE  +5000000!*X(101)    +5000000!*X(102)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 102
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-50000! THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(96),A(97),A(98),A(99),A(100)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  See the BASIC manual [11].  Copied by hand from the screen, the computer program's complete output through JJJJ=-31997 is shown below:

-1   1   1   1   1
1   1   1   1   1
-4        -32000

1   1   1   1   1
1   1   1   1   1
0        -31999

-1   1   1   1   1
1   1   1   1   1
-4        -31998

1   1   1   1   1
1   1   1   1   1
0        -31997

Above there is no rounding by hand.

M=0 is optimal.  See Li and Sun [10, p. 415].

Of the 100 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 basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output through JJJJ=-31997 was 35 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

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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