Case One: The Additional Constraint is a Less-Than-or-Equal-to Constraint
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4 plus a less-than-or-equal-to constraint 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)+X(2)+X(3)+...+X(2000) <= 5000
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 5000.
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 255, which is 255 X(5001)=5000- SUMNEW.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 5000
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
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
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
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 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 0 -2 4
-514 -32000
-1 1 1 1 1
1 1 1 2 4
-507 -31999
1 1 1 1 1
1 1 1 1 1
-201 -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 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 basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31997 was 12 hours.
Case Two: The Additional Constraint is an Equality Constraint
The computer program below seeks to solve Li and Sun's Problem 14.4 plus an equality constraint 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)+X(2)+X(3)+...+X(5000) = 5000
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 5000.
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 211, which is 211 X(1)=5000-SUMNEW.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 5000
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
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=2 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
200 SUMNEW=0
203 FOR J44=2 TO 5000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
211 X(1)=5000-SUMNEW
222 IF X(1)>5 THEN X(1 )=A(1 )
225 IF X(1)<-5 THEN X(1 )=A(1 )
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
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-444 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 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=-31999 is shown below:
1 1 1 1 1
1 1 1 1 1
0 -31999
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 basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was 14 hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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