Jsun Yui Wong
To help produce a usable solution sooner than just using one computer, simultaneously run two or more slightly different computer programs on separate computers. The following computer programs and their outputs illustrate.
The test example used here is Li and Sun's Problem 14.4 but with two added constraints, with 8000 unknowns instead of 100 unknowns, and with lower bounds of -500's and upper bounds of 500's. See Li and Sun [10, p. 415]. Specifically the computer programs below try to minimize the following:
8000-1
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(8000)^3 = 8000
X(1)+X(2)+X(3)+...+X(8000) <= 8000
-500<=X(i)<=500, X(i) integer, i=1, 2, 3,..., 8000.
One notes lines 111 through 117, which are
111 FOR J44=1 TO 8000
114 A(J44)=FIX(RND*(-6))
117 NEXT J44
This 114 A(J44)=FIX(RND*(-6)) can generate only -5, -4, -3, -2, -1, or 0; this makes the starting solutions reasonably away from this test example's optimal solution vector, which is
(1 1 1, ..., 1 1 1)--see Li and Sun [10, p. 415].
One also notes line 174 and line 175, which are as follows:
174 IF X(J44)>500 THEN X(J44 )=A(J44 )
175 IF X(J44)<-500 THEN X(J44 )=A(J44 ).
The objective function is based on the Rosenbrock function [10, p. 415].
The present paper uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.
(i) Line 128 is 128 FOR I=1 TO 32000 STEP .1:
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 8000
114 A(J44)=FIX(RND*(-6))
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 8000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*8000)
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 8000
174 IF X(J44)>500 THEN X(J44 )=A(J44 )
175 IF X(J44)<-500 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 8000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 8000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 8000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 8000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(8001)=8000- SUMNEW
303 IF X(8001)<0 THEN X(8001)=X(8001) ELSE X(8001)=0
401 SONE=0
402 FOR J44=1 TO 7999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(8001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 8001
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)
1934 PRINT A(5996),A(5997),A(5998),A(5999),A(5000)
1936 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
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, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31996 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 2
-509 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2418 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 2
-509 -31998
1 1 1 1 1
1 1 1 1 1
1 1 1 1 2
-307 -31997
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31996
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 [10, p. 415].
Of the 8000 A's, only the 15 A's of line 1931, line 1934, 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=-31996 was 24 hours.
(ii) Line 128 is 128 FOR I=1 TO 32500 STEP .1:
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 8000
114 A(J44)=FIX(RND*(-6))
117 NEXT J44
128 FOR I=1 TO 32500 STEP .1
129 FOR KKQQ=1 TO 8000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*8000)
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 8000
174 IF X(J44)>500 THEN X(J44 )=A(J44 )
175 IF X(J44)<-500 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 8000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 8000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 8000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 8000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(8001)=8000- SUMNEW
303 IF X(8001)<0 THEN X(8001)=X(8001) ELSE X(8001)=0
401 SONE=0
402 FOR J44=1 TO 7999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(8001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 8001
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)
1934 PRINT A(5996),A(5997),A(5998),A(5999),A(5000)
1936 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
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, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31995 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 2
-509 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2016 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 2
-509 -31998
1 1 1 1 1
1 1 1 1 1
1 1 1 2 2
-1315 -31997
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-3024 -31996
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31995
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 [10, p. 415].
Of the 8000 A's, only the 15 A's of line 1931, line 1934, 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=-31995 was 28 hours.
(iii) Line 128 is 128 FOR I=1 TO 32555 STEP .1:
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 8000
114 A(J44)=FIX(RND*(-6))
117 NEXT J44
128 FOR I=1 TO 32555 STEP .1
129 FOR KKQQ=1 TO 8000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*8000)
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 8000
174 IF X(J44)>500 THEN X(J44 )=A(J44 )
175 IF X(J44)<-500 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 8000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 8000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 8000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 8000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(8001)=8000- SUMNEW
303 IF X(8001)<0 THEN X(8001)=X(8001) ELSE X(8001)=0
401 SONE=0
402 FOR J44=1 TO 7999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(8001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 8001
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)
1934 PRINT A(5996),A(5997),A(5998),A(5999),A(5000)
1936 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
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, page iii, Preface]. 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
1 1 1 1 2
-509 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 0
-2319 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 2
-507 -31998
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31997
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 [10, p. 415].
Of the 8000 A's, only the 15 A's of line 1931, line 1934, 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=-31997 was 19 hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] 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.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[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
