Jsun Yui Wong
Numerous problems can be formulated as nonlinear 0-1 integer programming problems; see Dantzig [4] and Balas [1]. The test example used here is based on Li and Sun's Problem 14.3, [11, pp. 414-415]; the source is S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically the computer program below tries to minimize the following:
10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000
X(1)+X(2)+X(3)+...+X(10000) <= 10000
0<=X(i)<=1, X(i) integer, i=1, 2, 3,..., 10000.
The following computer program uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--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 10000
114 A(J44)=+FIX(RND*2)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
168 IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>1 THEN X(J44 )=A(J44 )
175 IF X(J44)<0 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +5000000!*X(10001)-5000000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-33333! THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1932 PRINT A(56),A(57),A(58),A(59),A(60)
1933 PRINT A(76),A(77),A(78),A(79),A(80)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1938 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1939 PRINT M,JJJJ,UU
1999 NEXT JJJJ
This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [12, page iii, Preface]. 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
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2.1801E+08 -32000 11
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31999 0
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 [11, pp. 414-415].
Of the 10000 A's, only the 25 A's of line 1931 through line 1938 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=-31999 was 8 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] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[8] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[9] 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.
[10] 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.
[11] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[12] 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.
[13] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[14] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[15] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[16] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[17] 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/
[18] 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/
[19] 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
Thursday, October 30, 2014
Wednesday, October 22, 2014
A General Nonlinear Integer/Continuous/Discrete Programming Computer Program Applied to Li and Sun's Problem 14.4 but with 10000 Zero-One Variables
Jsun Yui Wong
Numerous problems can be formulated as nonlinear 0-1 integer programming problems; see Dantzig [4] and Balas [1]. The test example used here is based on Li and Sun's Problem 14.4, [11, p. 415]. Specifically the computer program below tries to minimize the following:
10000-1
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(10000)^3 = 10000
X(1)+X(2)+X(3)+...+X(10000) <= 10000
0<=X(i)<=1, X(i) integer, i=1, 2, 3,..., 10000.
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.
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 10000
114 A(J44)=FIX(RND*2)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
143 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 IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>1 THEN X(J44 )=A(J44 )
175 IF X(J44)<0 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1931 PRINT A(1),A(2),A(3),A(4),A(5)
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 [12, page iii, Preface]. 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 -32000
1 1 1 1 1
1 1 1 1 1
0 -31999
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 [11, p. 415].
Of the 10000 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=-31999 was 9 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] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[8] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[9] 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.
[10] 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.
[11] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[12] 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.
[13] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[14] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[15] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[16] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[17] 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/
[18] 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/
[19] 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
Numerous problems can be formulated as nonlinear 0-1 integer programming problems; see Dantzig [4] and Balas [1]. The test example used here is based on Li and Sun's Problem 14.4, [11, p. 415]. Specifically the computer program below tries to minimize the following:
10000-1
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(10000)^3 = 10000
X(1)+X(2)+X(3)+...+X(10000) <= 10000
0<=X(i)<=1, X(i) integer, i=1, 2, 3,..., 10000.
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.
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 10000
114 A(J44)=FIX(RND*2)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
143 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 IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>1 THEN X(J44 )=A(J44 )
175 IF X(J44)<0 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1931 PRINT A(1),A(2),A(3),A(4),A(5)
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 [12, page iii, Preface]. 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 -32000
1 1 1 1 1
1 1 1 1 1
0 -31999
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 [11, p. 415].
Of the 10000 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=-31999 was 9 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] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[8] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[9] 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.
[10] 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.
[11] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[12] 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.
[13] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[14] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[15] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[16] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[17] 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/
[18] 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/
[19] 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
Friday, October 17, 2014
A Multi-Computer Approach To Solve Li and Sun's Problem 14.4 but with 8000 Unknowns, Two Added Constraints, Lower Bounds of -500's and Upper Bounds of 500's, and Initial Solution Vectors of A(J44)=FIX(RND*(-6))
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
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
Thursday, October 9, 2014
A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 10000 Unknowns instead of 100 Unknowns and with Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000, Second Edition
Jsun Yui Wong
The present edition includes an addendum with two slightly different computer programs.
Similar to the computer programs of the preceding papers, the first computer program below seeks to solve Li and Sun's Problem 14.4 but with two added constraints and with 10000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:
10000-1
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(10000)^3 = 10000
X(1)+X(2)+X(3)+...+X(10000) <= 10000
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.
Thus, one of the two added 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)= ( 10000 -(SUMY) )^(1/3).
The objective function here 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.
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 10000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
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 10000
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 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
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(5001),A(5002),A(5003),A(5004),A(5005)
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=-31989 is shown below:
1 1 1 1 1
1 1 1 1 1
-1517 -32000
1 1 1 1 1
1 1 1 1 1
-7575 -31999
1 1 1 1 1
1 1 1 1 1
-4856 -31998
1 1 1 1 1
1 1 1 1 1
-917 -31997
1 1 1 1 1
1 1 1 1 1
-1519 -31996
1 1 1 1 1
1 1 1 2 1
-707 -31995
1 1 1 1 1
1 1 1 1 1
-507 -31994
1 1 1 1 1
1 1 1 1 2
-709 -31993
1 1 1 1 1
1 1 1 1 1
-1317 -31992
1 1 1 1 1
1 1 1 1 1
-2218 -31991
1 1 1 1 1
1 1 1 1 1
-2327 -31990
1 1 1 1 1
1 1 1 1 1
0 -31989
Above there is no rounding by hand; above is just straight copying from the screen.
M=0 is optimal. See Li and Sun [10, p. 415].
Of the 10000 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=-31989 was about 85 hours.
Addendum
But one might not have 85 hours. To alleviate this time issue in future problems, simultaneously run slightly different computer programs. In conjunction with the computer program above and its output, the two computer programs below and their respective outputs demonstrate. One notes that while line 128 above is 128 FOR I=1 TO 32000 STEP .1, lines 128 below are 128 FOR I=1 TO 32500 STEP .1 and 128 FOR I=1 TO 32555 STEP .1, respectively.
(i) 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 10000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32500 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
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 10000
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 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
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(5001),A(5002),A(5003),A(5004),A(5005)
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
-1517 -32000
1 1 1 1 1
1 1 1 1 1
-2325 -31999
1 1 1 1 1
1 1 1 1 1
-5252 -31998
1 1 1 1 1
1 1 1 1 1
-3125 -31997
1 1 1 1 1
1 1 1 1 1
-509 -31996
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 10000 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=-31995 was 42 hours.
(ii) 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 10000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32555 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
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 10000
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 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
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(5001),A(5002),A(5003),A(5004),A(5005)
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=-31993 is shown below:
1 1 1 1 1
1 1 1 1 1
-1517 -32000
1 1 1 1 1
1 1 1 1 1
-7575 -31999
1 1 1 1 1
1 1 1 1 1
-1315 -31998
1 1 1 1 1
1 1 1 1 1
-5347 -31997
1 1 1 1 1
1 1 1 1 1
-1319 -31996
1 1 1 1 1
1 1 1 1 1
-1818 -31995
1 1 1 1 1
1 1 1 1 1
-309 -31994
1 1 1 1 1
1 1 1 1 1
0 -31993
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 10000 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 66 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
The present edition includes an addendum with two slightly different computer programs.
Similar to the computer programs of the preceding papers, the first computer program below seeks to solve Li and Sun's Problem 14.4 but with two added constraints and with 10000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:
10000-1
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(10000)^3 = 10000
X(1)+X(2)+X(3)+...+X(10000) <= 10000
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.
Thus, one of the two added 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)= ( 10000 -(SUMY) )^(1/3).
The objective function here 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.
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 10000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
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 10000
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 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
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(5001),A(5002),A(5003),A(5004),A(5005)
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=-31989 is shown below:
1 1 1 1 1
1 1 1 1 1
-1517 -32000
1 1 1 1 1
1 1 1 1 1
-7575 -31999
1 1 1 1 1
1 1 1 1 1
-4856 -31998
1 1 1 1 1
1 1 1 1 1
-917 -31997
1 1 1 1 1
1 1 1 1 1
-1519 -31996
1 1 1 1 1
1 1 1 2 1
-707 -31995
1 1 1 1 1
1 1 1 1 1
-507 -31994
1 1 1 1 1
1 1 1 1 2
-709 -31993
1 1 1 1 1
1 1 1 1 1
-1317 -31992
1 1 1 1 1
1 1 1 1 1
-2218 -31991
1 1 1 1 1
1 1 1 1 1
-2327 -31990
1 1 1 1 1
1 1 1 1 1
0 -31989
Above there is no rounding by hand; above is just straight copying from the screen.
M=0 is optimal. See Li and Sun [10, p. 415].
Of the 10000 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=-31989 was about 85 hours.
Addendum
But one might not have 85 hours. To alleviate this time issue in future problems, simultaneously run slightly different computer programs. In conjunction with the computer program above and its output, the two computer programs below and their respective outputs demonstrate. One notes that while line 128 above is 128 FOR I=1 TO 32000 STEP .1, lines 128 below are 128 FOR I=1 TO 32500 STEP .1 and 128 FOR I=1 TO 32555 STEP .1, respectively.
(i) 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 10000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32500 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
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 10000
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 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
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(5001),A(5002),A(5003),A(5004),A(5005)
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
-1517 -32000
1 1 1 1 1
1 1 1 1 1
-2325 -31999
1 1 1 1 1
1 1 1 1 1
-5252 -31998
1 1 1 1 1
1 1 1 1 1
-3125 -31997
1 1 1 1 1
1 1 1 1 1
-509 -31996
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 10000 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=-31995 was 42 hours.
(ii) 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 10000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32555 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
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 10000
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 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
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(5001),A(5002),A(5003),A(5004),A(5005)
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=-31993 is shown below:
1 1 1 1 1
1 1 1 1 1
-1517 -32000
1 1 1 1 1
1 1 1 1 1
-7575 -31999
1 1 1 1 1
1 1 1 1 1
-1315 -31998
1 1 1 1 1
1 1 1 1 1
-5347 -31997
1 1 1 1 1
1 1 1 1 1
-1319 -31996
1 1 1 1 1
1 1 1 1 1
-1818 -31995
1 1 1 1 1
1 1 1 1 1
-309 -31994
1 1 1 1 1
1 1 1 1 1
0 -31993
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 10000 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 66 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
Saturday, October 4, 2014
A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 8000 Unknowns instead of 100 Unknowns and with Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(8000)^3 = 8000, Second Edition
Jsun Yui Wong
The present edition includes an addendum with a slightly different computer program.
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 added constraints and with 8000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries 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
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 8000.
Thus, one of the two added 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)= ( 8000 -(SUMY) )^(1/3).
The objective function here 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.
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)=-1+FIX(RND*3)
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)>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 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)
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=-31968 is shown below:
1 1 1 1 1
1 1 1 1 2
-509 -32000
1 1 1 1 1
1 1 1 2 2
-1317 -31999
1 1 1 1 1
1 1 1 1 2
-507 -31998
1 1 1 1 1
1 1 1 1 2
-307 -31997
1 1 1 1 1
1 1 1 2 2
-1119 -31996
1 1 1 1 1
1 1 1 1 2
-507 -31995
1 1 1 1 1
1 1 1 1 2
-709 -31994
1 1 1 1 1
1 1 1 1 2
-507 -31993
1 1 1 1 1
1 1 1 1 1
-2218 -31992
1 1 1 1 1
1 1 1 1 2
-309 -31991
1 1 1 1 1
1 1 1 1 2
-709 -31990
1 1 1 1 1
1 1 1 1 2
-707 -31989
1 1 1 1 1
1 1 1 1 2
-309 -31988
1 1 1 1 1
1 1 1 2 2
-1515 -31987
1 1 1 1 1
1 1 1 1 2
-6163 -31986
1 1 1 1 1
1 1 1 1 2
-509 -31985
1 1 1 1 1
1 1 1 1 2
-707 -31984
1 1 1 1 1
1 1 1 1 2
-509 -31983
1 1 1 1 1
1 1 1 1 2
-507 -31982
1 1 1 1 1
1 1 1 1 0
-2117 -31981
1 1 1 1 1
1 1 1 2 2
-1119 -31980
1 1 1 1 1
1 1 1 1 2
-707 -31979
1 1 1 1 1
1 1 1 2 4
-3781 -31978
1 1 1 1 1
1 1 1 2 2
-1319 -31977
1 1 1 1 1
1 0 0 0 0
-2117 -31976
1 1 1 1 1
1 1 1 1 2
-707 -31975
1 1 1 1 1
1 1 1 1 1
-4036 -31974
1 1 1 1 1
1 1 1 1 2
-507 -31973
1 1 1 1 1
1 1 1 2 2
-1117 -31972
1 1 1 1 1
1 1 1 1 2
-707 -31971
1 1 1 1 1
1 1 1 1 2
-509 -31970
1 1 1 1 1
1 1 1 1 2
-507 -31969
1 1 1 1 1
1 1 1 1 1
0 -31968
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 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=-31968 was 130 hours.
Addendum
But 130 hours is a long time. To alleviate this issue in future problems, simultaneously run slightly different computer programs on separate computers. In conjunction with the computer program above, the following input and its output illustrate. One notes that while line 128 above is 128 FOR I=1 TO 32000 STEP .1, line 128 below is 128 FOR I=1 TO 32000 STEP 9.000001E-02.
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)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP 9.000001E-02
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)>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 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)
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 2
-509 -32000
1 1 1 1 1
1 1 0 0 0
-3735 -31999
1 1 1 1 1
1 1 1 1 2
-4745 -31998
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 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=-31997 was 16 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
The present edition includes an addendum with a slightly different computer program.
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 added constraints and with 8000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries 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
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 8000.
Thus, one of the two added 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)= ( 8000 -(SUMY) )^(1/3).
The objective function here 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.
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)=-1+FIX(RND*3)
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)>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 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)
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=-31968 is shown below:
1 1 1 1 1
1 1 1 1 2
-509 -32000
1 1 1 1 1
1 1 1 2 2
-1317 -31999
1 1 1 1 1
1 1 1 1 2
-507 -31998
1 1 1 1 1
1 1 1 1 2
-307 -31997
1 1 1 1 1
1 1 1 2 2
-1119 -31996
1 1 1 1 1
1 1 1 1 2
-507 -31995
1 1 1 1 1
1 1 1 1 2
-709 -31994
1 1 1 1 1
1 1 1 1 2
-507 -31993
1 1 1 1 1
1 1 1 1 1
-2218 -31992
1 1 1 1 1
1 1 1 1 2
-309 -31991
1 1 1 1 1
1 1 1 1 2
-709 -31990
1 1 1 1 1
1 1 1 1 2
-707 -31989
1 1 1 1 1
1 1 1 1 2
-309 -31988
1 1 1 1 1
1 1 1 2 2
-1515 -31987
1 1 1 1 1
1 1 1 1 2
-6163 -31986
1 1 1 1 1
1 1 1 1 2
-509 -31985
1 1 1 1 1
1 1 1 1 2
-707 -31984
1 1 1 1 1
1 1 1 1 2
-509 -31983
1 1 1 1 1
1 1 1 1 2
-507 -31982
1 1 1 1 1
1 1 1 1 0
-2117 -31981
1 1 1 1 1
1 1 1 2 2
-1119 -31980
1 1 1 1 1
1 1 1 1 2
-707 -31979
1 1 1 1 1
1 1 1 2 4
-3781 -31978
1 1 1 1 1
1 1 1 2 2
-1319 -31977
1 1 1 1 1
1 0 0 0 0
-2117 -31976
1 1 1 1 1
1 1 1 1 2
-707 -31975
1 1 1 1 1
1 1 1 1 1
-4036 -31974
1 1 1 1 1
1 1 1 1 2
-507 -31973
1 1 1 1 1
1 1 1 2 2
-1117 -31972
1 1 1 1 1
1 1 1 1 2
-707 -31971
1 1 1 1 1
1 1 1 1 2
-509 -31970
1 1 1 1 1
1 1 1 1 2
-507 -31969
1 1 1 1 1
1 1 1 1 1
0 -31968
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 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=-31968 was 130 hours.
Addendum
But 130 hours is a long time. To alleviate this issue in future problems, simultaneously run slightly different computer programs on separate computers. In conjunction with the computer program above, the following input and its output illustrate. One notes that while line 128 above is 128 FOR I=1 TO 32000 STEP .1, line 128 below is 128 FOR I=1 TO 32000 STEP 9.000001E-02.
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)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP 9.000001E-02
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)>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 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)
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 2
-509 -32000
1 1 1 1 1
1 1 0 0 0
-3735 -31999
1 1 1 1 1
1 1 1 1 2
-4745 -31998
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 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=-31997 was 16 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
Friday, October 3, 2014
A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 8000 Unknowns instead of 100 Unknowns and with Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(8000)^3 = 8000
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 added constraints and with 8000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries 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
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 8000.
Thus, one of the two added 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)= ( 8000 -(SUMY) )^(1/3).
The objective function here 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.
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)=-1+FIX(RND*3)
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)>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 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)
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=-31968 is shown below:
1 1 1 1 1
1 1 1 1 2
-509 -32000
1 1 1 1 1
1 1 1 2 2
-1317 -31999
1 1 1 1 1
1 1 1 1 2
-507 -31998
1 1 1 1 1
1 1 1 1 2
-307 -31997
1 1 1 1 1
1 1 1 2 2
-1119 -31996
1 1 1 1 1
1 1 1 1 2
-507 -31995
1 1 1 1 1
1 1 1 1 2
-709 -31994
1 1 1 1 1
1 1 1 1 2
-507 -31993
1 1 1 1 1
1 1 1 1 1
-2218 -31992
1 1 1 1 1
1 1 1 1 2
-309 -31991
1 1 1 1 1
1 1 1 1 2
-709 -31990
1 1 1 1 1
1 1 1 1 2
-707 -31989
1 1 1 1 1
1 1 1 1 2
-309 -31988
1 1 1 1 1
1 1 1 2 2
-1515 -31987
1 1 1 1 1
1 1 1 1 2
-6163 -31986
1 1 1 1 1
1 1 1 1 2
-509 -31985
1 1 1 1 1
1 1 1 1 2
-707 -31984
1 1 1 1 1
1 1 1 1 2
-509 -31983
1 1 1 1 1
1 1 1 1 2
-507 -31982
1 1 1 1 1
1 1 1 1 0
-2117 -31981
1 1 1 1 1
1 1 1 2 2
-1119 -31980
1 1 1 1 1
1 1 1 1 2
-707 -31979
1 1 1 1 1
1 1 1 2 4
-3781 -31978
1 1 1 1 1
1 1 1 2 2
-1319 -31977
1 1 1 1 1
1 0 0 0 0
-2117 -31976
1 1 1 1 1
1 1 1 1 2
-707 -31975
1 1 1 1 1
1 1 1 1 1
-4036 -31974
1 1 1 1 1
1 1 1 1 2
-507 -31973
1 1 1 1 1
1 1 1 2 2
-1117 -31972
1 1 1 1 1
1 1 1 1 2
-707 -31971
1 1 1 1 1
1 1 1 1 2
-509 -31970
1 1 1 1 1
1 1 1 1 2
-507 -31969
1 1 1 1 1
1 1 1 1 1
0 -31968
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 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=-31968 was 130 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
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 added constraints and with 8000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries 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
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 8000.
Thus, one of the two added 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)= ( 8000 -(SUMY) )^(1/3).
The objective function here 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.
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)=-1+FIX(RND*3)
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)>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 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)
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=-31968 is shown below:
1 1 1 1 1
1 1 1 1 2
-509 -32000
1 1 1 1 1
1 1 1 2 2
-1317 -31999
1 1 1 1 1
1 1 1 1 2
-507 -31998
1 1 1 1 1
1 1 1 1 2
-307 -31997
1 1 1 1 1
1 1 1 2 2
-1119 -31996
1 1 1 1 1
1 1 1 1 2
-507 -31995
1 1 1 1 1
1 1 1 1 2
-709 -31994
1 1 1 1 1
1 1 1 1 2
-507 -31993
1 1 1 1 1
1 1 1 1 1
-2218 -31992
1 1 1 1 1
1 1 1 1 2
-309 -31991
1 1 1 1 1
1 1 1 1 2
-709 -31990
1 1 1 1 1
1 1 1 1 2
-707 -31989
1 1 1 1 1
1 1 1 1 2
-309 -31988
1 1 1 1 1
1 1 1 2 2
-1515 -31987
1 1 1 1 1
1 1 1 1 2
-6163 -31986
1 1 1 1 1
1 1 1 1 2
-509 -31985
1 1 1 1 1
1 1 1 1 2
-707 -31984
1 1 1 1 1
1 1 1 1 2
-509 -31983
1 1 1 1 1
1 1 1 1 2
-507 -31982
1 1 1 1 1
1 1 1 1 0
-2117 -31981
1 1 1 1 1
1 1 1 2 2
-1119 -31980
1 1 1 1 1
1 1 1 1 2
-707 -31979
1 1 1 1 1
1 1 1 2 4
-3781 -31978
1 1 1 1 1
1 1 1 2 2
-1319 -31977
1 1 1 1 1
1 0 0 0 0
-2117 -31976
1 1 1 1 1
1 1 1 1 2
-707 -31975
1 1 1 1 1
1 1 1 1 1
-4036 -31974
1 1 1 1 1
1 1 1 1 2
-507 -31973
1 1 1 1 1
1 1 1 2 2
-1117 -31972
1 1 1 1 1
1 1 1 1 2
-707 -31971
1 1 1 1 1
1 1 1 1 2
-509 -31970
1 1 1 1 1
1 1 1 1 2
-507 -31969
1 1 1 1 1
1 1 1 1 1
0 -31968
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 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=-31968 was 130 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
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