Nonlinear Integer/Continuous/Discrete Programming Solver: November 2014

Sunday, November 30, 2014

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.3 but with Five Equality Constraints, Two Less-Than-or-Equal-to Constraints, and 10000 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 14.3, [11, p. 414], but with 10000 unknowns instead of 100 unknowns and with five added equality constraints and two less-than-or-equal-to constraints. The original version of this problem is due to S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically, the computer program below tries to solve the following:

Minimize

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

9*X(61) + 7 *X(77)^3 < = 16

X(52) + 2*X(59 ) + 3*X(99) =6

X(1)+ 3* X(2)+ X(3 ) + 6 *X(4 )+ X(5) = 12

X(5996)+ X(5997)+ 5* X(5998 )+ X(5999) + 3* X(8000) = 11

X(6)^5 + 4 *X(7)^5 +X(8 )^5 +X(9)^5 +X(10)^5 = 8

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) < = 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 175 through line 181, which are as follows:

175 S=16 -9*X(61) -7*X(77)^3
176 IF S<0 THEN S=S ELSE S=0
177 X(52)=6-2*X(59 )-3*X(99)
178 X(1)=12-3*X(2)-X(3 )-6*X(4 )-X(5)
179 X(5996)=11-X(5997)-5*X(5998 )-X(5999)-3*X(8000)
180 IF (8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )<0 THEN 1670
181 X(6)=(8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )^(1/5)

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
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=1 ELSE A(J44)=0
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 10000
171 IF X(J44)>UB THEN X(J44 )=A(J44 )
172 IF X(J44)<-5 THEN X(J44 )=A(J44 )
173 NEXT J44
175 S=16 -9*X(61) -7*X(77)^3
176 IF S<0 THEN S=S ELSE S=0
177 X(52)=6-2*X(59 )-3*X(99)
178 X(1)=12-3*X(2)-X(3 )-6*X(4 )-X(5)
179 X(5996)=11-X(5997)-5*X(5998 )-X(5999)-3*X(8000)
180 IF (8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )<0 THEN 1670
181 X(6)=(8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )^(1/5)
182 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +50000!*X(10001) -50000!*ABS(U) +50000!*S
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1559 GOTO 128
1670 NEXT I
1959 PRINT M,JJJJ,UU,A(1),A(8000)
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:

0 -32000 0 1 1

-4.43708E+09 -31999 0 1 2

0 -31998 0 1 1

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 414].

Of the 10000 A's, only the 2 A's of line 1959 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 12 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Saturday, November 29, 2014

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.3 but with 10000 General Integer Variables and Seven Added Constraints

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, [11, p. 414], but with 10000 unknowns instead of 100 unknowns and with six added equality constraints and one inequality constraint. The original version of this problem is due to S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically, the computer program below tries to solve the following:

Minimize

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(61) + 3*X(77)^3 = 4

X(52) + 2*X(59 ) + 3*X(99) =6

X(1)+ 3* X(2)+ X(3 ) + 6 *X(4 )+ X(5) = 12

X(5996)+ X(5997)+ 5* X(5998 )+ X(5999) + 3* X(8000) = 11

X(6)^5 + 4 *X(7)^5 +X(8 )^5 +X(9)^5 +X(10)^5 = 8

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 175 through line 181, which are as follows:

175 X(61)=4-3*X(77)^3
177 X(52)=6-2*X(59 )-3*X(99)
178 X(1)=12-3*X(2)-X(3 )-6*X(4 )-X(5)
179 X(5996)=11-X(5997)-5*X(5998 )-X(5999)-3*X(8000)
180 IF (8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )<0 THEN 1670
181 X(6)=(8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )^(1/5)

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
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=1 ELSE A(J44)=0
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 10000
171 IF X(J44)>UB THEN X(J44 )=A(J44 )
172 IF X(J44)<-5 THEN X(J44 )=A(J44 )
173 NEXT J44
175 X(61)=4-3*X(77)^3
177 X(52)=6-2*X(59 )-3*X(99)
178 X(1)=12-3*X(2)-X(3 )-6*X(4 )-X(5)
179 X(5996)=11-X(5997)-5*X(5998 )-X(5999)-3*X(8000)
180 IF (8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )<0 THEN 1670
181 X(6)=(8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )^(1/5)
182 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +50000!*X(10001) -50000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 UUB=UB
1559 GOTO 128
1670 NEXT I
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(6),A(7),A(8),A(9),A(10)
1959 PRINT M,JJJJ,UU,UUB,A(8000)
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
0 -32000 0 1 1

1 1 1 1 1
1 1 1 1 1
0 -31999 0 1 1

1 1 1 1 1
1 1 1 1 1
-3.07674E+08 -31998 0 3 0

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 414].

Of the 10000 A's, only the 11 A's of line 1931 through line 1959 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 9 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Friday, November 28, 2014

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.3 but with Six Added Constraints and 10000 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 14.3, [11, p. 414], but with 10000 unknowns instead of 100 unknowns and with five added equality constraints and one inequality constraint. The original version of this problem is due to S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically, the computer program below tries to solve the following:

Minimize

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(52) + 3*X(59 ) + 2*X(99) = 6

X(1)+ 3* X(2)+X(3 ) + 6 *X(4 )+ X(5) = 12

X(5996)+ X(5997)+ 5* X(5998 )+ X(5999) + 3* X(8000) = 11

X(6)^5 + 4 *X(7)^5 +X(8 )^5 +X(9)^5 +X(10)^5 = 8

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes that line 85 is 85 UB=1+FIX(RND*5) and line 181 is 181 X(6)=(8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )^(1/5). One also notes line 171 through line 177, which are as follows:
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT 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 REM
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=1 ELSE A(J44)=0
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
176 NEXT J44
177 X(52)=6-3*X(59 )-2*X(99)
178 X(1)=12-3*X(2)-X(3 )-6*X(4 )-X(5)
179 X(5996)=11-X(5997)-5*X(5998 )-X(5999)-3*X(8000)
180 IF (8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )<0 THEN 1670
181 X(6)=(8-4*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )^(1/5)
182 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +50000!*X(10001) -50000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 UUB=UB
1559 GOTO 128
1670 NEXT I
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(6),A(7),A(8),A(9),A(10)
1959 PRINT M,JJJJ,UU,UUB,A(8000)
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [12, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31999 is shown below:

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

1 1 1 1 1
1 1 1 1 1
0 -31999 0 1 1

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 414].

Of the 10000 A's, only the 11 A's of line 1931 through line 1959 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31999 was 6 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thursday, November 27, 2014

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.3 but with 10000 General Integer Variables and Five Added Constraints, Including X(6)^5+7*X(7)^5 +X(8 )^5 +X(9)^5 +X(10)^5 = 11

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, [11, p. 414], but with 10000 unknowns instead of 100 unknowns and with four added equality constraints and one inequality constraint. The original version of this problem is due to S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically, the computer program below tries to solve the following:

Minimize

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1) +X(2)+X(3 )+X(4 )+X(5) = 5

X(5996)+X(5997)+X(5998 )+X(5999)+X(8000) = 5

X(6)^5+7*X(7)^5 +X(8 )^5 +X(9)^5 +X(10)^5 = 11

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes that line 85 is 85 UB=1+FIX(RND*5) and line 180 is 180 X(6)=(11-7*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )^(1/5) . One also notes line 171 through line 177, which are as follows:
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT 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 REM
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=1 ELSE A(J44)=0
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
178 X(1)=5-X(2)-X(3 )-X(4 )-X(5)
179 X(5996)=5-X(5997)-X(5998 )-X(5999)-X(8000)
180 X(6)=(11-7*X(7)^5 -X(8 )^5 -X(9)^5 -X(10)^5 )^(1/5)
182 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +50000!*X(10001) -50000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 UUB=UB
1559 GOTO 128
1670 NEXT I
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1932 PRINT A(6),A(7),A(8),A(9),A(10)
1935 PRINT A(4996),A(4997),A(4998),A(4999),A(5000)
1936 PRINT A(5996),A(5997),A(5998),A(5999),A(6000)
1938 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1959 PRINT M,JJJJ,UU,UUB
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 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
0 -32000 0 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 -31999 0 1

Illegal function call in line 180

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 414].

Of the 10000 A's, only the 25 A's of line 1931 through line 1938 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output shown above was 6 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Wednesday, November 26, 2014

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.3 but with 10000 General Integer Variables and Five Added Constraints

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, [11, p. 414], but with 10000 unknowns instead of 100 unknowns and with four added equality constraints and one inequality constraint. The original version of this problem is due to S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically, the computer program below tries to solve the following:

Minimize

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1) +X(2)+X(3 )+X(4 )+X(5) = 5

X(6)+X(7) +X(8 ) +X(9) +X(10) = 5

X(5996)+X(5997)+X(5998 )+X(5999)+X(8000) = 5

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes that line 85 is 85 UB=1+FIX(RND*5). One also notes line 171 through line 177, which are as follows:
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT 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 REM
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=1 ELSE A(J44)=0
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
178 X(1)=5-X(2)-X(3 )-X(4 )-X(5)
179 X(6)=5-X(7)-X(8 )-X(9)-X(10)
181 X(5996)=5-X(5997)-X(5998 )-X(5999)-X(8000)
182 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +50000!*X(10001) -50000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 UUB=UB
1559 GOTO 128
1670 NEXT I
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1932 PRINT A(6),A(7),A(8),A(9),A(10)
1935 PRINT A(4996),A(4997),A(4998),A(4999),A(5000)
1936 PRINT A(5996),A(5997),A(5998),A(5999),A(6000)
1938 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1959 PRINT M,JJJJ,UU,UUB
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
1 1 1 1 1
1 1 1 1 1
0 -32000 0 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 -31999 0 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
-2.69164E+09 -31998 0 3

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 414].

Of the 10000 A's, only the 25 A's of line 1931 through line 1938 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=31998 was 10 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.3 but with 10000 General Integer Variables and Four Added Constraints

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, [11, p. 414], but with 10000 unknowns instead of 100 unknowns and with three added equality constraints and one inequality constraint. The original version of this problem is due to S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically, the computer program below tries to minimize the following:

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1) +X(2)+X(3 )+X(4 )+X(5) = 5

X(6)+X(7)+X(8 )+X(9)+X(10) = 5

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes that line 85 is 85 UB=1+FIX(RND*5). One also notes line 171 through line 177, which are as follows:
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT 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 REM
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=1 ELSE A(J44)=0
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
178 X(1)=5-X(2)-X(3 )-X(4 )-X(5)
179 X(6)=5-X(7)-X(8 )-X(9)-X(10)
180 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +50000!*X(10001) -50000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 UUB=UB
1559 GOTO 128
1670 NEXT I
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1932 PRINT A(6),A(7),A(8),A(9),A(10)
1935 PRINT A(4996),A(4997),A(4998),A(4999),A(5000)
1936 PRINT A(5996),A(5997),A(5998),A(5999),A(6000)
1938 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1959 PRINT M,JJJJ,UU,UUB
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
1 1 1 1 1
1 1 1 1 1
0 -32000 0 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 -31999 0 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
-2.32054E+09 -31998 0 3

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 414].

Of the 10000 A's, only the 25 A's of line 1931 through line 1938 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31998 was 10 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Tuesday, November 25, 2014

Mixed Integer Nonlinear Programming (MINLP) Solver Applied to Li and Sun's Problem 14.3 but with 10000 General Integer Variables and Three Added Constraints

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, [11, p. 414], but with 10000 unknowns instead of 100 unknowns and with two added equality constraints and one inequality constraint. The original version of this problem is due to S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically, the computer program below tries to minimize the following:

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1) +X(2)+X(3 )+X(4 )+X(5) = 5

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes that line 128 is 128 FOR I=1 TO 32000 STEP .5. One also notes line 171 through line 177, which are as follows:
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44.

0 REM
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=1 ELSE A(J44)=0
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
179 X(1)=5-X(2)-X(3 )-X(4 )-X(5)
180 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +50000!*X(10001) -50000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 UUB=UB
1559 GOTO 128
1670 NEXT I
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1932 PRINT A(56),A(57),A(58),A(59),A(60)
1935 PRINT A(4996),A(4997),A(4998),A(4999),A(5000)
1936 PRINT A(5996),A(5997),A(5998),A(5999),A(6000)
1938 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1945 PRINT A(9996),A(9997),A(9998),A(9999),A(10000)
1947 PRINT A(10001)
1959 PRINT M,JJJJ,UU,UUB
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
0
0 -32000 0 1

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 414].

Of the 10000 A's, only the 30 A's of line 1931 through line 1945 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 6 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Tuesday, November 18, 2014

A General Nonlinear Integer/Continuous/Discrete Programming Computer Program Applied to Li and Sun's Problem 14.3 but with 10000 General Integer Variables and Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 0, 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 14.3, [11, p. 414], but with 10000 unknowns instead of 100 unknowns and with two added constraints. The original version of this problem is due to S. Walukiewicz--see Schittkowski's Test Problem 282 [15, p. 106]. Specifically the computer program below tries to minimize the following:

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 111, line 114, and line 117, which are as follows:
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT 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.

1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 IF RND<.5 THEN A(J44)=0 ELSE A(J44)=1
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +5000000!*X(10001) -5000000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 UUB=UB
1559 GOTO 128
1670 NEXT I
1889 REM IF M<-33333! THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1932 PRINT A(56),A(57),A(58),A(59),A(60)
1933 PRINT A(76),A(77),A(78),A(79),A(80)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1938 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1939 PRINT M,JJJJ,UU,UUB
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [12, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31999 is shown below:

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000 0 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
-1E+07 -31999 2 1

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 414].

Of the 10000 A's, only the 25 A's of line 1931 and line 1938 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31999 was 6 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Sunday, November 16, 2014

A General Nonlinear Integer/Continuous/Discrete Programming Computer Program Applied to Li and Sun's Problem 14.3 but with 10000 General Integer Variables and Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 0

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, [10, p. 414], but with 10000 unknowns instead of 100 unknowns and with two added constraints. The source is S. Walukiewicz--see Schittkowski's Test Problem 282 [14, p. 106]. Specifically the computer program below tries to minimize the following:

10000-1
(X(1)-1)^2+ ( X(10000)-1)^2 +10000* SIGMA (10000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

X(1)+X(2)+X(3)+...+X(10000) <= 10000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 85, line 114, and line 174, which are 85 UB=1+FIX(RND*5),
114 A(J44)=+FIX(RND*2), and 174 IF X(J44)>UB THEN X(J44 )=A(J44 ), respectively.

1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 UB=1+FIX(RND*5)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 A(J44)=+FIX(RND*2)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>UB THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=1 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
188 U=10000-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=10000- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
200 SUMNEWZ=0
203 FOR J44=1 TO 9999
205 SUMNEWZ=SUMNEWZ+ (10000-J44)* ( X(J44)^2-X(J44+1) )^2
207 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(10000)-1)^2 -10000* SUMNEWZ
689 PD1=SONE +5000000!*X(10001) -5000000!*ABS(U)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1557 UUB=UB
1559 GOTO 128
1670 NEXT I
1889 REM IF M<-33333! THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1932 PRINT A(56),A(57),A(58),A(59),A(60)
1933 PRINT A(76),A(77),A(78),A(79),A(80)
1934 PRINT A(96),A(97),A(98),A(99),A(100)
1938 PRINT A(9996),A(9997),A(9998),A(9999),A(10000)
1939 PRINT M,JJJJ,A(10001),UU,UUB
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
0 -32000 0 0 1

Above there is no rounding by hand.

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

Of the 10000 A's, only the 25 A's of line 1931 through line 1938 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was 9 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. 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. www.sfu.ca/~ssurjano/zakharov.html

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

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

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

Friday, November 14, 2014

A General Nonlinear Integer/Continuous/Discrete Programming Computer Program Applied to Li and Sun's Problem 14.5 but with 10000 General Integer Variables and Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3=0, 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, [11, p. 416], but with 10000 unknowns instead of 100 unknowns and with two added constraints. Specifically the computer program below tries to minimize the following:

10000 10000
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1 i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 0

X(1)+X(2)+X(3)+...+X(10000) <= 0

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 85, line 114, and line 175, which are 85 LB=-FIX(RND*6),
114 A(J44)=FIX(RND*(3)), and 175 IF X(J44)<LB THEN X(J44 )=A(J44 ), respectively.

While the computer program of the earlier edition uses an interpreter, the following computer program uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 REM DEFDBL J-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 A(J44)=FIX(RND*(3))
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
143 REM
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM GOTO 170
169 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
170 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
178 SUMY=0
179 FOR J44=1 TO 10000
180 SUMY=SUMY+X(J44)^3
181 NEXT J44
182 REM
183 U=0-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=0- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
396 SUMYONE=0
397 FOR J44=1 TO 10000
398 SUMYONE=SUMYONE+X(J44)^4
399 NEXT J44
400 SUMNEWTWO=0
403 FOR J44=1 TO 10000
405 SUMNEWTWO=SUMNEWTWO+ X(J44)
407 NEXT J44
411 SONETOTAL= -SUMYONE - ( SUMNEWTWO)^2 +5000000!*X(10001)-5000000!*ABS(U)
689 PD1=SONETOTAL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1556 LLB=LB
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-37000! THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1938 PRINT A(4996),A(4997),A(4998),A(7999),A(8000)
1939 PRINT M,JJJJ,UU,LLB
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:

0 0 0 0 -1
0 -1 0 -1 0
-4096576 -32000 0 -1

0 0 -1 0 0
0 -1 2 0 0
-40064 -31999 0 -3

0 0 0 0 0
0 0 0 0 0
0 -31998 0 0

Above there is no rounding by hand; it is just straight copying by hand from the screen.

M=0 is optimal. See Li and Sun [11, p. 416].

Of the 10000 A's, only the 10 A's of line 1931 and line 1938 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31998 was 22 hours.

Acknowledgment

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

References

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

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

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

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thursday, November 13, 2014

A General Nonlinear Integer/Continuous/Discrete Programming Computer Program Applied to Li and Sun's Problem 14.5 but with 10000 General Integer Variables and Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 0

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 10000 unknowns instead of 100 unknowns and two added constraints. Specifically the computer program below tries to minimize the following:

10000 10000
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1 i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 0

X(1)+X(2)+X(3)+...+X(10000) <= 0

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 10000.

One notes line 85, line 114, and line 175, which are 85 LB=-FIX(RND*6),
114 A(J44)=FIX(RND*(3)), and 175 IF X(J44)<LB THEN X(J44 )=A(J44 ), respectively.

0 REM DEFDBL J-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 A(J44)=FIX(RND*(3))
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
143 REM
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM GOTO 170
169 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
170 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<LB THEN X(J44 )=A(J44 )
177 NEXT J44
178 SUMY=0
179 FOR J44=1 TO 10000
180 SUMY=SUMY+X(J44)^3
181 NEXT J44
182 REM
183 U=0-SUMY
194 SUMNEW=0
195 FOR J44=1 TO 10000
196 SUMNEW=SUMNEW+X(J44)
197 NEXT J44
198 X(10001)=0- SUMNEW
199 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
396 SUMYONE=0
397 FOR J44=1 TO 10000
398 SUMYONE=SUMYONE+X(J44)^4
399 NEXT J44
400 SUMNEWTWO=0
403 FOR J44=1 TO 10000
405 SUMNEWTWO=SUMNEWTWO+ X(J44)
407 NEXT J44
411 SONETOTAL= -SUMYONE - ( SUMNEWTWO)^2 +5000000!*X(10001)-5000000!*ABS(U)
689 PD1=SONETOTAL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1555 UU=U
1556 LLB=LB
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-37000! THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1938 PRINT A(4996),A(4997),A(4998),A(4999),A(10000)
1939 PRINT M,JJJJ,A(10001),UU,LLB
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 -32000 0 0 0

Above there is no rounding by hand.

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

Of the 10000 A's, only the 10 A's of line 1931 and 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 6 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. 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. 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