A Unified Computer Program Solving Li and Sun’s Problem 14.3 but with 1500 Unknowns instead of 100 Unknowns

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun’s Problem 14.3 but with 1500 unknowns instead of 100 unknowns, [10, p. 414]. Specifically the computer program below tries to minimize the following:

1500-1
(X(1)-1)^2 + (X(1500) -1 )^2 + 1500* SIGMA (1500-i) * ( X(i)^2 – X(i+1)) ^2
i=1

subject to -5<=X(i)<=5, X(i) integer, i=1, 2, 3,…, 1500.

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

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 1500
114 A(J44 )=-5+FIX( RND*11)
115 REM IF RND<.5 THEN A(J44 )=2 +FIX(RND*4 ) ELSE A(J44)=-5+FIX(RND*6)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 1500
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*1500)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 1500
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 1499
411 SONE=SONE+ (1500-J44) * ( X(J44)^2 – X(J44+1)) ^2
421 NEXT J44
551 SUMX=(X(1)-1)^2 + (X(1500) -1 )^2+1500*SONE
689 REM PD1=-SONE
699 PD1=-SUMX
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 1500
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-100 THEN 1999
1936 PRINT A(1),A(2),A(1498),A(1499),A(1500)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31999 is shown below:

0 0 0 0 0
-2 -32000

1 1 1 1 1
0 -31999

Above there is no rounding by hand.

M=0 is optimal; see Li and Sun [10, p. 414].

Of the 1500 A’s, only the 5 A’s of line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was 15 hours and 10 minutes.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program Solving Li and Sun’s Problem 14.4, Which is Based on the Rosenbrock Function; the Case of 8000 Unknowns

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun’s Problem 14.4, [10, p. 415], but with 8000 unknowns instead of 100 unknowns. Specifically the computer program below tries to minimize the following:

8000-1
SIGMA      [ 100* ( X(i+1) – X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to -5<=X(i)<=5, X(i) integer, i=1, 2, 3,…, 8000.

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

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 8000
114 A(J44 )=-5+FIX( RND*11)
115 REM IF RND<.5 THEN A(J44 )=2 +FIX(RND*4 ) ELSE A(J44)=-5+FIX(RND*6)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 8000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*8000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 8000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 7999
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 8000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-100 THEN 1999
1936 PRINT A(1),A(2),A(7998),A(7999),A(8000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31999 is shown below:

-1 1 1 1 1
-205      -32000

1 1 1 1 1
0      -31999

Above there is no rounding by hand.

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

Of the 8000 A’s, only the 5 A’s of line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was 7 hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program Solving Li and Sun’s Problem 14.4, Which is Based on the Rosenbrock Function

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun’s Problem 14.4, [10, p. 415], but with 200 unknowns instead of 100 unknowns. Specifically the computer program below tries to minimize the following:

200-1
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,…, 200.

One notes line 85, line 86, line 174, and line 175.

For the purpose of testing the algorithm, none of the 200 elements of the starting solution vector of each JJJJ is made one, the optimal; see line 111 through line 117, which are as follows:
111 FOR J44=1 TO 200
114 IF RND<.5 THEN A(J44 )=2 +FIX(RND*4 ) ELSE A(J44)=-5+FIX(RND*6)
117 NEXT J44.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
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 200
114 IF RND<.5 THEN A(J44 )=2 +FIX(RND*4 ) ELSE A(J44)=-5+FIX(RND*6)
117 NEXT J44
128 FOR I=1 TO 3200
129 FOR KKQQ=1 TO 200
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*200)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 200
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 199
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 200
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-100 THEN 1999
1936 PRINT A(96),A(97),A(198),A(199),A(200)
1939 PRINT M,JJJJ,LB,UB
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31909 is shown below:

1 1 1 1 1
0      -31986      -4      4

1 1 1 1 1
-4      -31978      -4      5

1 1 1 1 1
0      -31924      -4      4

1 1 1 1 1
0      -31909      -4      4

Above there is no rounding by hand.

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

Of the 200 A’s, only the 5 A’s of line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31909 was 16 minutes.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program Solving Li and Sun’s Problem 14.5 but with 9000 Unknowns instead of 100 Unknowns, Second Edition

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, [10, p. 416], but with 9000 unknowns instead of 100 unknowns. Specifically the computer program below tries to minimize the following:

9000                            9000
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,…, 9000.

One notes line 85 and line 175, which are 85 LB=-FIX(RND*6) and
175 IF X(J44)<LB THEN X(J44 )=A(J44 ).

For the purpose of testing the algorithm, none of the 3000 elements of the starting solution vector of each JJJJ is made zero, the optimal; see line 111 through line 117, which are as follows:
111 FOR J44=1 TO 9000
114 A(J44)=1+FIX(RND*5)
117 NEXT J44.

The present edition shows more computational results than the earlier edition.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 9000
114 A(J44)=1+FIX(RND*5)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*9000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 9000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 9000
411 SONE=SONE+ (X(J44))^4
421 NEXT J44
501 STWO=0
502 FOR J44=1 TO 9000
511 STWO=STWO+ (X(J44))
521 NEXT J44
688 PD1=-SONE-STWO^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-2 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1935 PRINT A(996),A(997),A(998),A(999),A(1000)
1936 PRINT A(8996),A(8997),A(8998),A(8999),A(9000)
1939 PRINT M,JJJJ,LB
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31996 is shown below:

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0      -32000      0

-1 -1 1 0 0
-1 0 0 -1 1
1 1 0 1 0
-1 0 1 0 0
-4556      -31999      -4

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-20      -31998      0

-1 -1 1 -1 0
-1 0 1 0 0
0 1 0 0 0
0 1 1 1 0
-4558      -31997      -4

-1 0 -1 0 1
1 -1 0 -1 1
1 0 1 0 -1
0 0 -1 1 0
-4526      -31996      -4

Above there is no rounding by hand.

M=0 is optimal; see Li and Sun [10, p. 416].

Of the 9000 A’s, only the 10 A’s of line 1923 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31996 was 16 hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program Solving Li and Sun’s Problem 14.5 but with 9000 Unknowns instead of 100 Unknowns

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, [10, p. 416], but with 9000 unknowns instead of 100 unknowns. Specifically the computer program below tries to minimize the following:

9000                             9000
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,…, 9000.

One notes line 85 and line 175, which are 85 LB=-FIX(RND*6) and
175 IF X(J44)<LB THEN X(J44 )=A(J44 ).

For the purpose of testing the algorithm, none of the 3000 elements of the starting solution vector of each JJJJ is made zero, the optimal; see line 111 through line 117, which are as follows:
111 FOR J44=1 TO 9000
114 A(J44)=1+FIX(RND*5)
117 NEXT J44.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 9000
114 A(J44)=1+FIX(RND*5)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*9000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 9000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 9000
411 SONE=SONE+ (X(J44))^4
421 NEXT J44
501 STWO=0
502 FOR J44=1 TO 9000
511 STWO=STWO+ (X(J44))
521 NEXT J44
688 PD1=-SONE-STWO^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-2 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1935 PRINT A(996),A(997),A(998),A(999),A(1000)
1936 PRINT A(8996),A(8997),A(8998),A(8999),A(9000)
1939 PRINT M,JJJJ,LB
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31999 is shown below:

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0      -32000      0

-1 -1 1 0 0
-1 0 0 -1 1
1 1 0 1 0
-1 0 1 0 0
-4556      -31999      -4

Above there is no rounding by hand.

M=0 is optimal; see Li and Sun [10, p. 416].

Of the 9000 A’s, only the 20 A’s of line 1923 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM b*asica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was 7 hours and 20 minutes.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program Solving Li and Sun’s Problem 14.5 but with 9000 Unknowns instead of 100 Unknowns

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, [10, p. 416], but with 9000 unknowns instead of 100 unknowns. Specifically the computer program below tries to minimize the following:

9000                           9000
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,…, 9000. See line 111 through line 117 and line 171 through line 177.

One notes line 85 and line 175, which are 85 LB=-FIX(RND*6) and 175 IF X(J44)<LB THEN X(J44 )=A(J44 ).

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 9000
114 A(J44)=LB+FIX(RND*6)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*9000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 9000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 9000
411 SONE=SONE+ (X(J44))^4
421 NEXT J44
501 STWO=0
502 FOR J44=1 TO 9000
511 STWO=STWO+ (X(J44))
521 NEXT J44
688 PD1=-SONE-STWO^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-2 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1935 PRINT A(996),A(997),A(998),A(999),A(1000)
1936 PRINT A(8996),A(8997),A(8998),A(8999),A(9000)
1939 PRINT M,JJJJ,LB
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31999 is shown below:

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0      -32000      0

0 -1 0 0 0
1 0 -1 0 -1
1 0 -1 1 0
-1 -1 0 0 0
-4678      -31999      -4

Above there is no rounding by hand.

M=0 is optimal; see Li and Sun [10, p. 416].

Of the 9000 A’s, only the 20 A’s of line 1923 through line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was six hours.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program Solving Li and Sun’s Problem 14.5 but with 5000 Unknowns instead of 100 Unknowns

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, [10, p. 416], but with 5000 unknowns instead of 100 unknowns. Specifically the computer program below tries to minimize the following:

5000                             5000
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,…, 5000. See line 111 through line 117 and line 171 through line 177.

One notes line 85 and line 175, which are 85 LB=-FIX(RND*6) and 175 IF X(J44)<LB THEN X(J44 )=A(J44 ).

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 5000
114 A(J44)=LB+FIX(RND*6)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 5000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*5000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 5000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 5000
411 SONE=SONE+ (X(J44))^4
421 NEXT J44
501 STWO=0
502 FOR J44=1 TO 5000
511 STWO=STWO+ (X(J44))
521 NEXT J44
688 PD1=-SONE-STWO^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-2 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1935 PRINT A(996),A(997),A(998),A(999),A(1000)
1936 PRINT A(4996),A(4997),A(4998),A(4999),A(5000)
1939 PRINT M,JJJJ,LB
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-32000 is shown below:

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0      -32000     0

Above there is no rounding by hand.

M=0 is optimal; see Li and Sun [10, p. 416].

Of the 5000 A’s, only the 20 A’s of line 1923 through line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was one hour and fifteen minutes.

Acknowledgment

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

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program Solving Li and Sun’s Problem 14.5 but with 2000 Unknowns instead of 100 Unknowns

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, [10, p. 416], but with 2000 unknowns instead of 100 unknowns. Specifically the computer program below tries to minimize the following:

2000                             2000
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,…, 2000. See line 111 through line 117 and line 171 through line 177.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 2000
114 A(J44)=LB+FIX(RND*6)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 2000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*2000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 2000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 2000
411 SONE=SONE+ (X(J44))^4
421 NEXT J44
501 STWO=0
502 FOR J44=1 TO 2000
511 STWO=STWO+ (X(J44))
521 NEXT J44
688 PD1=-SONE-STWO^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 2000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-2 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1935 PRINT A(996),A(997),A(998),A(999),A(1000)
1936 PRINT A(1996),A(1997),A(1998),A(1999),A(2000)
1939 PRINT M,JJJJ,LB
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31998 is shown below:

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0      -32000      0

1 0 1 0 -1
0 0 0 1 0
0 -1 0 0 0
-1 1 -1 0 0
-1084      -31999      -3

0 0 -1 0 0
1 0 0 1 -1
1 0 0 -1 0
0 0 0 1 -1
-1186      -31998      -2

Above there is no rounding by hand.

M=0 is optimal; see Li and Sun [10, p. 416].

Of the 3000 A’s, only the 20 A’s of line 1923 through line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31998 was 35 minutes.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program for Solving a General-Integer Version of Schittkowski’s Test Problem 281 but with 9000 Unknowns instead of 10 Unknowns

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski’s Test Problem 281 [14, p. 105] but with 9000 unknowns instead of 10 unknowns. The source of this Test Problem 281 is S. Walukiewicz; see Schittkowski [14, p. 105]. Specifically the computer program below tries to minimize the following:

9000
[ SIGMA    i ^3 *(X(i)-1)^2 ] ^(1/3)
i=1

subject to X(i)=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, i=1, 2, 3,…, 9000. See line 111 through line 117 and line 171 through line 177.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(9000),X(9000)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 9000
114 A(J44)=FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*9000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 9000
174 IF X(J44)>10 THEN X(J44)=A(J44)
175 IF X(J44)<0 THEN X(J44)=A(J44)
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 9000
411 SONE=SONE+ J44 ^3 *(X(J44)-1)^2
421 NEXT J44
699 PD1=-SONE^(1/3)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<0 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
1924 PRINT A(6),A(7),A(8),A(9),A(10)
1933 PRINT A(591),A(592),A(593),A(594),A(595)
1935 PRINT A(1496),A(1497),A(1498),A(1499),A(1500)
1936 PRINT A(6496),A(6497),A(6498),A(6499),A(6500)
1937 PRINT A(8991),A(8992),A(8993),A(8994),A(8995)
1938 PRINT A(8996),A(8997),A(8998),A(8999),A(9000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-32000 is shown below:

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000

Above there is no rounding by hand.

M=0 is optimal; see Schittkowski [14, p. 105].

Of the 9000 A’s, only the 35 A’s of line 1923 through line 1938 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was six hours and ten minutes.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html

A Unified Computer Program Solving a Problem with 9000 General Integer Variables That Is Based on Li and Sun’s Problem 14.5

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve a problem based on Li and Sun’s Problem 14.5, [10, p. 416]. Specifically the computer program below tries to minimize the following:

9000                              9000
SIGMA (X(i))^4 + [  SIGMA X(i)  ]^2
i=1                                   i=1

subject to X(i)=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, i=1, 2, 3,…, 9000. See line 111 through line 117 and line 171 through line 177.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(9000),X(9000)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 9000
114 A(J44)=+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+.3)
140 B=1+FIX(RND*9000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 9000
174 IF X(J44)>10 THEN X(J44)=A(J44)
175 IF X(J44)<0 THEN X(J44)=A(J44)
177 NEXT J44
401 SONE=0
402 FOR J44=1 TO 9000
411 SONE=SONE+ (X(J44))^4
421 NEXT J44
501 STWO=0
502 FOR J44=1 TO 9000
511 STWO=STWO+ (X(J44))
521 NEXT J44
588 REM SUMT=SONE+(STWO )^2
688 PD1=-SONE-STWO^2
699 REM PD1=-SONE^(1/3)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-2 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
1924 PRINT A(6),A(7),A(8),A(9),A(10)
1926 PRINT A(21),A(22),A(23),A(24),A(25)
1933 REM PRINT A(91),A(92),A(93),A(94),A(95)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1935 PRINT A(2996),A(2997),A(2998),A(8999),A(9000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-32000 is shown below:

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -32000

Above there is no rounding by hand.

M=0 is optimal; see Li and Sun [10, p. 416].

Of the 9000 A’s, only the 25 A’s of line 1923 through line 1935 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was 4 hours and 10 minutes..

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html