Jsun Yui Wong
The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15000 general integer variables instead of their 100 general integer variables [12, p. 415]. The function is based on the widely known Rosenbrock function--see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:
Minimize
15000-1
SIGMA 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15000.
One notes the starting solution vectors of line 111 through line 118, which are 111 FOR J44=1 TO 15000,
116 A(J44)=-5+FIX(RND*11), and 118 NEXT J44. The starts so generated are colder than the starts of the several papers immediately preceding the present paper.
For a computer program involving continuous variables and integer variables, see Wong [19].
The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.
0 DEFINT J,K,B,X,A
2 DIM A(15003),X(15003)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15000
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15000
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15003)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 15000
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
178 REM
207 REM
401 SONE=0
402 FOR J44=1 TO 14999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 15000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(14998),A(14999),A(15000)
1777 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=-31985 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-403 -32000
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-15202 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1001 -31998
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-205 -31997
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-4 -31996
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1001 -31995
1 1 1 1 1
1 1 1 1 1
1 1 0 -2 4
-510 -31994
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-14999 -31993
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-14999 -31992
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1408 -31991
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-201 -31990
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-205 -31989
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-4 -31988
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-15405 -31987
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-410 -31986
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31985
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 [12, p. 415].
Of the 15000 A's, only the 15 A's of line 1771 through line 1774 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=-31985 was 65 hours.
For a computer program involving continuous variables and integer variables, see Wong [19].
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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, January 31, 2015
Wednesday, January 28, 2015
Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.4 but with 15000 General Integer Variables, Second Edition
Jsun Yui Wong
The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15000 general integer variables instead of their 100 general integer variables [12, p. 415]. The function is based on the widely known Rosenbrock function--see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:
Minimize
15000-1
SIGMA 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15000.
While line 111 through line 118 of the preceding paper are
111 FOR J44=1 TO 15000
116 A(J44)=FIX(RND*2)
118 NEXT J44,
here line 111 through line 118 are
111 FOR J44=1 TO 15000
116 A(J44)=-1+FIX(RND*3)
118 NEXT J44.
The starting solution vectors of the latter will involve in general more searches.
For a computer program involving continuous variables and integer variables, see Wong [19].
The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.
0 DEFINT J,K,B,X,A
2 DIM A(15003),X(15003)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15000
116 A(J44)=-1+FIX(RND*3)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15000
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15003)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 15000
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
178 REM
207 REM
401 SONE=0
402 FOR J44=1 TO 14999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 15000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(14998),A(14999),A(15000)
1777 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=-31998 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-402 -32000
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-14999 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31998
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 [12, p. 415].
Of the 15000 A's, only the 15 A's of line 1771 through line 1774 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=-31998 was eight hours.
For a computer program involving continuous variables and integer variables, see Wong [19].
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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 computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15000 general integer variables instead of their 100 general integer variables [12, p. 415]. The function is based on the widely known Rosenbrock function--see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:
Minimize
15000-1
SIGMA 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15000.
While line 111 through line 118 of the preceding paper are
111 FOR J44=1 TO 15000
116 A(J44)=FIX(RND*2)
118 NEXT J44,
here line 111 through line 118 are
111 FOR J44=1 TO 15000
116 A(J44)=-1+FIX(RND*3)
118 NEXT J44.
The starting solution vectors of the latter will involve in general more searches.
For a computer program involving continuous variables and integer variables, see Wong [19].
The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.
0 DEFINT J,K,B,X,A
2 DIM A(15003),X(15003)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15000
116 A(J44)=-1+FIX(RND*3)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15000
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15003)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 15000
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
178 REM
207 REM
401 SONE=0
402 FOR J44=1 TO 14999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 15000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(14998),A(14999),A(15000)
1777 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=-31998 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-402 -32000
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-14999 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31998
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 [12, p. 415].
Of the 15000 A's, only the 15 A's of line 1771 through line 1774 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=-31998 was eight hours.
For a computer program involving continuous variables and integer variables, see Wong [19].
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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
Tuesday, January 27, 2015
Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.4 but with 15000 General Integer Variables
Jsun Yui Wong
The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15000 general integer variables instead of their 100 general integer variables [12, p. 415]. The function is based on the widely known Rosenbrock function--see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:
Minimize
15000-1
SIGMA 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15000.
For a computer program involving continuous variables and integer variables, see Wong [19].
The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.
0 DEFINT J,K,B,X,A
2 DIM A(15003),X(15003)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15000
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15000
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15003)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 15000
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
178 REM
207 REM
401 SONE=0
402 FOR J44=1 TO 14999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 15000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(14998),A(14999),A(15000)
1777 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=-31995 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-201 -31998
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-14999 -31997
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-201 -31996
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-4 -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 [12, p. 415].
Of the 15000 A's, only the 15 A's of line 1771 through line 1774 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 17 hours.
For a computer program involving continuous variables and integer variables, see Wong [19].
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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 computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 15000 general integer variables instead of their 100 general integer variables [12, p. 415]. The function is based on the widely known Rosenbrock function--see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:
Minimize
15000-1
SIGMA 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15000.
For a computer program involving continuous variables and integer variables, see Wong [19].
The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.
0 DEFINT J,K,B,X,A
2 DIM A(15003),X(15003)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15000
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15000
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15003)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 15000
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
178 REM
207 REM
401 SONE=0
402 FOR J44=1 TO 14999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 15000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(14998),A(14999),A(15000)
1777 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=-31995 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-201 -31998
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-14999 -31997
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-201 -31996
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-4 -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 [12, p. 415].
Of the 15000 A's, only the 15 A's of line 1771 through line 1774 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 17 hours.
For a computer program involving continuous variables and integer variables, see Wong [19].
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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
Monday, January 26, 2015
Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.4 but with 11000 General Integer Variables
Jsun Yui Wong
The computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 11000 general integer variables instead of their 100 general integer variables [12, p. 415]. The function used is based on the widely known Rosenbrock function--see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:
Minimize
11000-1
SIGMA 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 11000.
For a computer program involving continuous variables and integer variables, see Wong [19].
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.
0 DEFINT J,K,B,X,A
2 DIM A(11003),X(11003)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 11000
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 11000
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*11003)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 11000
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
178 REM
207 REM
401 SONE=0
402 FOR J44=1 TO 10999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 11000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(10098),A(10999),A(11000)
1777 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=-31995 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-205 -31998
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-205 -31997
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-10999 -31996
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-203 -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 [12, p. 415].
Of the 11000 A's, only the 15 A's of line 1771 through line 1774 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 eleven hours.
For a computer program involving continuous variables and integer variables, see Wong [19].
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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 computer program listed below seeks to solve Li and Sun's Problem 14.4 but with 11000 general integer variables instead of their 100 general integer variables [12, p. 415]. The function used is based on the widely known Rosenbrock function--see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:
Minimize
11000-1
SIGMA 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 11000.
For a computer program involving continuous variables and integer variables, see Wong [19].
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.
0 DEFINT J,K,B,X,A
2 DIM A(11003),X(11003)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*6)
86 UB= FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 11000
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 11000
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*11003)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 11000
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
178 REM
207 REM
401 SONE=0
402 FOR J44=1 TO 10999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 11000
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(10098),A(10999),A(11000)
1777 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=-31995 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-205 -31998
-1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-205 -31997
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-10999 -31996
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-203 -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 [12, p. 415].
Of the 11000 A's, only the 15 A's of line 1771 through line 1774 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 eleven hours.
For a computer program involving continuous variables and integer variables, see Wong [19].
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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, January 24, 2015
Mixed Integer Nonlinear Programming (MINLP) Solver Applied to a Problem with 10100 General Integer Variables and X(1)+X(2)+X(3)+...+X(10100) = 10100
Jsun Yui Wong
The computer program listed below seeks to solve a nonlinear integer programming problem with 10100 general integer variables. The present problem is based on Li and Sun's Problem 14.3, [12, pp. 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is as follows:
Minimize
10100-1
(X(1)-1)^2 + ( X(10100)-1)^2 +10100* SIGMA (10100-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)+X(2)+X(3)+...+X(10100) = 10100
-4 <= X(i) <= 4, X(i) integer, i=1, 2, 3,..., 10100.
While line 85 and line 86 of the preceding paper are 85 LB=- FIX(RND*4) and 86 UB= FIX(RND*4), respectively, here line 85 and line 86 are 85 LB=- FIX(RND*5) and 86 UB= FIX(RND*5), respectively.
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. For a computer program involving continuous variables and integer variables, see Wong [19], for example.
0 DEFINT J,K,B,X,A
2 DIM A(10103),X(10103)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*5)
86 UB= FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 10100
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 10100
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
181 SJ=0
182 FOR J9=1 TO 10100
183 SJ=SJ+ X(J9)
184 NEXT J9
187 TS= 10100-SJ
200 SZ=0
203 FOR J9=1 TO 10099
205 SZ=SZ+ (10100-J9)* ( X(J9)^2-X(J9+1) )^2
207 NEXT J9
411 SO= - (X(1)-1)^2 - ( X(10100)-1)^2 -10100* SZ
689 PD1=SO -50000!*ABS(TS)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 10100
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(10098),A(10099),A(10100)
1777 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=-31994 is shown below:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-5.05E+08 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 2
-8.47592E+07 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 2 4
-4.551061E+07 -31998
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31997
1 1 1 1 1
1 1 1 1 1
1 1 1 2 4
-1.014848E+08 -31996
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-5.05E+08 -31995
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-5.05E+08 -31994
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 [12, p. 415].
Of the 10100 A's, only the 15 A's of line 1771 through line 1774 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=-31994 was eight hours.
For a computer program involving continuous variables and integer variables, see Wong [19], for example.
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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 computer program listed below seeks to solve a nonlinear integer programming problem with 10100 general integer variables. The present problem is based on Li and Sun's Problem 14.3, [12, pp. 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is as follows:
Minimize
10100-1
(X(1)-1)^2 + ( X(10100)-1)^2 +10100* SIGMA (10100-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)+X(2)+X(3)+...+X(10100) = 10100
-4 <= X(i) <= 4, X(i) integer, i=1, 2, 3,..., 10100.
While line 85 and line 86 of the preceding paper are 85 LB=- FIX(RND*4) and 86 UB= FIX(RND*4), respectively, here line 85 and line 86 are 85 LB=- FIX(RND*5) and 86 UB= FIX(RND*5), respectively.
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. For a computer program involving continuous variables and integer variables, see Wong [19], for example.
0 DEFINT J,K,B,X,A
2 DIM A(10103),X(10103)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*5)
86 UB= FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 10100
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 10100
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
181 SJ=0
182 FOR J9=1 TO 10100
183 SJ=SJ+ X(J9)
184 NEXT J9
187 TS= 10100-SJ
200 SZ=0
203 FOR J9=1 TO 10099
205 SZ=SZ+ (10100-J9)* ( X(J9)^2-X(J9+1) )^2
207 NEXT J9
411 SO= - (X(1)-1)^2 - ( X(10100)-1)^2 -10100* SZ
689 PD1=SO -50000!*ABS(TS)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 10100
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(10098),A(10099),A(10100)
1777 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=-31994 is shown below:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-5.05E+08 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 2
-8.47592E+07 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 2 4
-4.551061E+07 -31998
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31997
1 1 1 1 1
1 1 1 1 1
1 1 1 2 4
-1.014848E+08 -31996
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-5.05E+08 -31995
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-5.05E+08 -31994
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 [12, p. 415].
Of the 10100 A's, only the 15 A's of line 1771 through line 1774 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=-31994 was eight hours.
For a computer program involving continuous variables and integer variables, see Wong [19], for example.
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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] 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/
[21] 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, January 23, 2015
Mixed Integer Nonlinear Programming (MINLP) Solver Applied to a Nonlinear Integer Programming Problem with 10100 General Integer Variables
Jsun Yui Wong
The computer program listed below seeks to solve a nonlinear integer programming problem with 10100 general integer variables. The present problem is based on Li and Sun's Problem 14.3, [12, pp. 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is as follows:
Minimize
10100-1
(X(1)-1)^2 + ( X(10100)-1)^2 +10100* SIGMA (10100-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)+X(2)+X(3)+...+X(10100) = 10100
-3 <= X(i) <= 3, X(i) integer, i=1, 2, 3,..., 10100.
One notes line 85 through line 86 and line 171 through line 177, which are as follows:
85 LB=- FIX(RND*4)
86 UB= FIX(RND*4)
171 FOR J9=1 TO 10100
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
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.
0 DEFINT J,K,B,X,A
2 DIM A(10103),X(10103)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*4)
86 UB= FIX(RND*4)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 10100
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 10100
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
181 SJ=0
182 FOR J9=1 TO 10100
183 SJ=SJ+ X(J9)
184 NEXT J9
187 TS= 10100-SJ
200 SZ=0
203 FOR J9=1 TO 10099
205 SZ=SZ+ (10100-J9)* ( X(J9)^2-X(J9+1) )^2
207 NEXT J9
411 SO= - (X(1)-1)^2 - ( X(10100)-1)^2 -10100* SZ
689 PD1=SO -50000!*ABS(TS)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 10100
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(10098),A(10099),A(10100)
1777 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=-31997 is shown below:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-5.05E+08 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-8.47991E+07 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 2 3
-4.55106E+07 -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 [12, p. 415].
Of the 10100 A's, only the 15 A's of line 1771 through line 1774 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 five 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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] 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/
[20] 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 computer program listed below seeks to solve a nonlinear integer programming problem with 10100 general integer variables. The present problem is based on Li and Sun's Problem 14.3, [12, pp. 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is as follows:
Minimize
10100-1
(X(1)-1)^2 + ( X(10100)-1)^2 +10100* SIGMA (10100-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)+X(2)+X(3)+...+X(10100) = 10100
-3 <= X(i) <= 3, X(i) integer, i=1, 2, 3,..., 10100.
One notes line 85 through line 86 and line 171 through line 177, which are as follows:
85 LB=- FIX(RND*4)
86 UB= FIX(RND*4)
171 FOR J9=1 TO 10100
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
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.
0 DEFINT J,K,B,X,A
2 DIM A(10103),X(10103)
81 FOR JJJJ=-32000 TO 32000
85 LB=- FIX(RND*4)
86 UB= FIX(RND*4)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
116 A(J44)=FIX(RND*2)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 10100
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 10100
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
181 SJ=0
182 FOR J9=1 TO 10100
183 SJ=SJ+ X(J9)
184 NEXT J9
187 TS= 10100-SJ
200 SZ=0
203 FOR J9=1 TO 10099
205 SZ=SZ+ (10100-J9)* ( X(J9)^2-X(J9+1) )^2
207 NEXT J9
411 SO= - (X(1)-1)^2 - ( X(10100)-1)^2 -10100* SZ
689 PD1=SO -50000!*ABS(TS)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 10100
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1771 PRINT A(1),A(2),A(3),A(4),A(5)
1773 PRINT A(10086),A(10087),A(10088),A(10089),A(10090)
1774 PRINT A(10096),A(10097),A(10098),A(10099),A(10100)
1777 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=-31997 is shown below:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-5.05E+08 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-8.47991E+07 -31999
1 1 1 1 1
1 1 1 1 1
1 1 1 2 3
-4.55106E+07 -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 [12, p. 415].
Of the 10100 A's, only the 15 A's of line 1771 through line 1774 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 five 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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] 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/
[20] 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
Sunday, January 11, 2015
Mixed Integer Nonlinear Programming (MINLP) Solver Applied to a Nonlinear Integer Programming Problem with 10100 General Integer Variables
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve a nonlinear integer programming problem with 10100 unknowns. The present problem is based on Li and Sun's Problem 14.3, [12, pp. 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, p. 106]. Specifically, the test example here is as follows:
Minimize
10100-1
(X(1)-1)^2 + ( X(10100)-1)^2 +10100* SIGMA (10100-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)+X(2)+X(3)+...+X(10100) = 10100
-1 <= X(i) >= 1, X(i) integer, i=1, 2, 3,..., 10100.
One notes line 111 through line 117 and line 171 through line 177.
The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.
0 DEFINT J,K,B,X
2 DIM A(10103),X(10103)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
116 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 10100
173 IF X(J9)<-1 THEN X(J9)=A(J9)
175 IF X(J9)>1 THEN X(J9)=A(J9)
177 NEXT J9
181 SJ=0
182 FOR J9=1 TO 10100
183 SJ=SJ+ X(J9)
184 NEXT J9
187 TS= 10100-SJ
200 SZ=0
203 FOR J9=1 TO 10099
205 SZ=SZ+ (10100-J9)* ( X(J9)^2-X(J9+1) )^2
207 NEXT J9
411 SONE= - (X(1)-1)^2 - ( X(10100)-1)^2 -10100* SZ
689 PD1=SONE -5000000!*ABS(TS)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1777 PRINT A(10100),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [13]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31995 is shown below:
1 -1E+07 -32000
1 -1E+07 -31999
1 -1E+07 -31998
1 -7.847851E+07 -31997
1 -2.799528E+08 -31996
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 [12, p. 415].
Of the 10100 A's, only the A of line 1777, A(10100), is shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31995 was 38 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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] 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/
[20] 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 a nonlinear integer programming problem with 10100 unknowns. The present problem is based on Li and Sun's Problem 14.3, [12, pp. 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, p. 106]. Specifically, the test example here is as follows:
Minimize
10100-1
(X(1)-1)^2 + ( X(10100)-1)^2 +10100* SIGMA (10100-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)+X(2)+X(3)+...+X(10100) = 10100
-1 <= X(i) >= 1, X(i) integer, i=1, 2, 3,..., 10100.
One notes line 111 through line 117 and line 171 through line 177.
The following computer program uses Microsoft's GW-BASIC 3.11 interpreter for DOS.
0 DEFINT J,K,B,X
2 DIM A(10103),X(10103)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
116 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 10100
173 IF X(J9)<-1 THEN X(J9)=A(J9)
175 IF X(J9)>1 THEN X(J9)=A(J9)
177 NEXT J9
181 SJ=0
182 FOR J9=1 TO 10100
183 SJ=SJ+ X(J9)
184 NEXT J9
187 TS= 10100-SJ
200 SZ=0
203 FOR J9=1 TO 10099
205 SZ=SZ+ (10100-J9)* ( X(J9)^2-X(J9+1) )^2
207 NEXT J9
411 SONE= - (X(1)-1)^2 - ( X(10100)-1)^2 -10100* SZ
689 PD1=SONE -5000000!*ABS(TS)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1777 PRINT A(10100),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [13]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31995 is shown below:
1 -1E+07 -32000
1 -1E+07 -31999
1 -1E+07 -31998
1 -7.847851E+07 -31997
1 -2.799528E+08 -31996
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 [12, p. 415].
Of the 10100 A's, only the A of line 1777, A(10100), is shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31995 was 38 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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] 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/
[20] 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, January 1, 2015
Mixed Integer Nonlinear Programming (MINLP) Solver Using Hot Starts To Solve Li and Sun's Problem 14.5 but with 10100 General Integer Variables instead of Their 100 General Integer Variables
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.5, [12, p. 416], but with 10100 unknowns instead of their 100 unknowns. Specifically, the test example here is as follows:
Minimize
10100 10100
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1 i=1
subject to
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10100.
One notes line 81 through line 86, line 111 through line 117, and line 170 through line 173. Lines 111 through 117 give relatively hot starts. In the following computer program there is no line 172 of the preceding post, which is 172 IF X(J44)>UP THEN X(J44 )=A(J44 ).
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 DEFINT J,K,B,X
2 DIM A(10100),X(10100)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
114 REM A(J44)=-5+FIX(RND*11)
115 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10100)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B)+1 )
169 NEXT IPP
170 FOR J44=1 TO 10100
171 IF X(J44)<LB THEN X(J44 )=A(J44 )
173 NEXT J44
482 SUMY=0
483 FOR J44=1 TO 10100
485 SUMY=SUMY+X(J44)^4
487 NEXT J44
488 SUMNEWZ=0
489 FOR J44=1 TO 10100
490 SUMNEWZ=SUMNEWZ+ X(J44)
491 NEXT J44
492 PD1=-SUMY-SUMNEWZ^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1779 PRINT A(1),A(8000),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31990 is shown below:
1 -1 -5628 -32000
-1 1 -4922 -31999
0 0 -4924 -31998
-1 -1 -5640 -31997
0 0 0 -31996
-1 -1 -4974 -31995
0 0 0 -31994
0 0 -5640 -31993
-1 0 -4988 -31992
0 -1 -5592 -31991
0 0 -4962 -31990
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 [12, p. 416].
Of the 10100 A's, only the two A's of line 1779 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=-31990 was 5 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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] 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/
[20] 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 14.5, [12, p. 416], but with 10100 unknowns instead of their 100 unknowns. Specifically, the test example here is as follows:
Minimize
10100 10100
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1 i=1
subject to
-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10100.
One notes line 81 through line 86, line 111 through line 117, and line 170 through line 173. Lines 111 through 117 give relatively hot starts. In the following computer program there is no line 172 of the preceding post, which is 172 IF X(J44)>UP THEN X(J44 )=A(J44 ).
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 DEFINT J,K,B,X
2 DIM A(10100),X(10100)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10100
114 REM A(J44)=-5+FIX(RND*11)
115 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*10100)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B)+1 )
169 NEXT IPP
170 FOR J44=1 TO 10100
171 IF X(J44)<LB THEN X(J44 )=A(J44 )
173 NEXT J44
482 SUMY=0
483 FOR J44=1 TO 10100
485 SUMY=SUMY+X(J44)^4
487 NEXT J44
488 SUMNEWZ=0
489 FOR J44=1 TO 10100
490 SUMNEWZ=SUMNEWZ+ X(J44)
491 NEXT J44
492 PD1=-SUMY-SUMNEWZ^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1779 PRINT A(1),A(8000),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31990 is shown below:
1 -1 -5628 -32000
-1 1 -4922 -31999
0 0 -4924 -31998
-1 -1 -5640 -31997
0 0 0 -31996
-1 -1 -4974 -31995
0 0 0 -31994
0 0 -5640 -31993
-1 0 -4988 -31992
0 -1 -5592 -31991
0 0 -4962 -31990
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 [12, p. 416].
Of the 10100 A's, only the two A's of line 1779 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=-31990 was 5 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] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] 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.
[11] 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.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] 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.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] 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/
[19] 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/
[20] 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
Erratum: Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.5 but with 10000 Unknowns (instead of 100 Unknowns) and an Additional Constraint,
X(1)^2 + X(2)^2 + X(3)^2 +...+ X(10000)^2 <= 10000
Jsun Yui Wong
Line 172, which is 172 IF X(J44)>UP THEN X(J44 )=A(J44 ), of the 2014 December 22 paper of the present blog is wrong. Thus, the computational results there are questionable. A related paper will be presented in this blog in about one day.
X(1)^2 + X(2)^2 + X(3)^2 +...+ X(10000)^2 <= 10000
Jsun Yui Wong
Line 172, which is 172 IF X(J44)>UP THEN X(J44 )=A(J44 ), of the 2014 December 22 paper of the present blog is wrong. Thus, the computational results there are questionable. A related paper will be presented in this blog in about one day.
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