Nonlinear Integer/Continuous/Discrete Programming Solver: September 2014

Monday, September 29, 2014

A Unified Computer Program for Solving Schittkowski's Test Problem 395 but with 7000 Unknowns instead of 50 Unknowns

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's last test problem but with 7000 unknowns instead of 50 unknowns; see Schittkowski [14, page 213, Test Problem 395]. The source of this Test Problem 395 is S. Walukiewicz--see Schittkowski [14, p. 213]. Specifically the computer program below tries to minimize the following:

7000
SIGMA i*(X(i)^2+X(i)^4 )
i=1

subject to

7000
SIGMA X(i)^2.
i=1

One notes line 406, which is 406 X(1)=( 1+SONE )^(1/2).

The present paper uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(7000),X(7000)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 7000
112 A(J44)=-.01+RND*.02
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 7000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*7000)
144 REM GOTO 167
145 IF RND<.33 THEN 150 ELSE IF RND<.5 THEN 163 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
170 GOTO 400
396 FOR J44=1 TO 50
397 REM IF X(J44)<-1 THEN X(J44)=A(J44)
398 REM IF X(J44)>1 THEN X(J44)=A(J44)
399 NEXT J44
400 SONE=0
401 FOR J44=2 TO 7000
403 SONE=SONE-X(J44)^2
404 NEXT J44
405 IF ( 1+SONE )<.0000001 THEN 1670
406 X(1)=( 1+SONE )^(1/2)
410 STWO=0
411 FOR J44=1 TO 7000
413 STWO=STWO+J44*(X(J44)^2+X(J44)^4 )
415 NEXT J44
457 PD1= - STWO
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 7000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999999999# THEN 1999
1911 GOTO 1935
1922 LPRINT A(1),A(2),A(3),A(4),A(5)
1923 LPRINT A(6),A(7),A(8),A(9),A(10)
1924 LPRINT A(11),A(12),A(13),A(14),A(15)
1925 LPRINT A(16),A(17),A(18),A(19),A(20)
1926 LPRINT A(21),A(22),A(23),A(24),A(25)
1927 LPRINT A(26),A(27),A(28),A(29),A(30)
1928 LPRINT A(31),A(32),A(33),A(34),A(35)
1929 LPRINT A(36),A(37),A(38),A(39),A(40)
1930 LPRINT A(41),A(42),A(43),A(44),A(45)
1931 LPRINT A(46),A(47),A(48),A(49),A(50)
1935 PRINT A(1),A(2),A(3),A(4),A(5)
1937 PRINT A(6996),A(6997),A(6998),A(6999),A(7000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-32000 is shown below:

.9129748 .4080154 4.223556E-04 -8.769279E-05
8.233389E-05
-1.720836E-06 1.80706E-06 -5.414935E-07
2.130225E-06 9.461994E-07
-1.916716 -32000

Above there is no rounding by hand.

M=-1.91667 is optimal. See Schittkowski [14, p. 213].

Of the 7000 A's, only the 10 A's of line 1935 and line 1937 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=-32000 was 14 hours and a half.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Sunday, September 28, 2014

A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 10000 Unknowns instead of 100 Unknowns and with Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(10000)^3 = 10000

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4 but with two added constraints and with 10000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

10000-1
SIGMA [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

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

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

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

Thus, one of the two added constraints is an equality constraint and the other is a less-than-or-equal-to constraint.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).

One also notes line 191, which is 191 X(1)= ( 10000 -(SUMY) )^(1/3).

The objective function here is based on the Rosenbrock function [10, p. 415].

The present paper uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 10000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 10000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 10000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 10000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF (10000 -(SUMY) )<0 THEN 1670
191 X(1)= (10000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 10000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(10001)=10000- SUMNEW
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0
401 SONE=0
402 FOR J44=1 TO 9999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(10001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(5001),A(5002),A(5003),A(5004),A(5005)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31989 is shown below:

1 1 1 1 1
1 1 1 1 1
-1517 -32000

1 1 1 1 1
1 1 1 1 1
-7575 -31999

1 1 1 1 1
1 1 1 1 1
-4856 -31998

1 1 1 1 1
1 1 1 1 1
-917 -31997

1 1 1 1 1
1 1 1 1 1
-1519 -31996

1 1 1 1 1
1 1 1 2 1
-707 -31995

1 1 1 1 1
1 1 1 1 1
-507 -31994

1 1 1 1 1
1 1 1 1 2
-709 -31993

1 1 1 1 1
1 1 1 1 1
-1317 -31992

1 1 1 1 1
1 1 1 1 1
-2218 -31991

1 1 1 1 1
1 1 1 1 1
-2327 -31990

1 1 1 1 1
1 1 1 1 1
0 -31989

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

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

Of the 10000 A's, only the 10 A's of line 1931 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31989 was about 85 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thursday, September 25, 2014

A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 7000 Unknowns instead of 100 Unknowns and with Two Added Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(7000)^3 = 7000

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4 but with two added constraints and with 7000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

7000-1
SIGMA [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(7000)^3 = 7000

X(1)+X(2)+X(3)+...+X(7000) <= 7000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 7000.

Thus, one of the two added constraints is an equality constraint and the other is a less-than-or-equal-to constraint.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).

One also notes line 191, which is 191 X(1)= ( 7000 -(SUMY) )^(1/3).

The objective function here is based on the Rosenbrock function [10, p. 415].

The present paper uses the IBM Personal Computer BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 7000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 7000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*7000)
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 7000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 7000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 7000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 7000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 7000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(7001)=7000- SUMNEW
303 IF X(7001)<0 THEN X(7001)=X(7001) ELSE X(7001)=0
401 SONE=0
402 FOR J44=1 TO 6999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(7001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 7001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(6996),A(6997),A(6998),A(6999),A(7000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31974 is shown below:

1 1 1 1 1
1 1 1 1 2
-309 -32000

1 1 1 1 1
1 1 1 1 2
-511 -31999

1 1 1 1 1
1 1 1 1 2
-511 -31998

1 1 1 1 1
1 1 1 1 2
-709 -31997

1 1 1 1 1
1 1 1 1 2
-507 -31996

1 1 1 1 1
1 1 1 1 2
-3129 -31995

1 1 1 1 1
1 1 1 1 2
-509 -31994

1 1 1 1 1
1 1 1 1 1
-2826 -31993

1 1 1 1 1
1 1 1 2 2
-1517 -31992

1 1 1 1 1
1 1 1 1 2
-307 -31991

1 1 1 1 1
1 1 1 1 2
-507 -31990

1 1 1 1 1
1 1 1 1 2
-509 -31989

1 1 1 1 1
1 1 1 1 2
-709 -31988

1 1 1 1 1
1 1 1 1 2
-507 -31987

1 1 1 1 1
1 1 1 1 2
-509 -31986

1 1 1 1 1
1 1 1 1 2
-1315 -31985

1 1 1 1 1
1 1 1 1 2
-3333 -31984

1 1 1 1 1
1 1 1 1 2
-707 -31983

1 1 1 1 1
1 1 1 2 2
-1719 -31982

1 1 1 1 1
1 1 1 1 2
-509 -31981

1 1 1 1 1
1 1 1 2 2
-1317 -31980

1 1 1 1 1
1 1 1 1 1
-2418 -31979

1 1 1 1 1
1 1 1 1 2
-507 -31978

1 1 1 1 1
1 1 1 2 2
-1317 -31977

1 1 1 1 1
1 1 1 2 2
-1319 -31976

1 1 1 1 1
1 1 1 1 2
-3545 -31975

1 1 1 1 1
1 1 1 1 1
0 -31974

Above there is no rounding by hand.

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

Of the 7000 A's, only the 10 A's of line 1931 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31974 was about 80 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Monday, September 22, 2014

A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 9000 Unknowns instead of 100 Unknowns and with Two Additional Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(9000)^3 = 9000

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4 but with two additional constraints and with 9000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

9000-1
SIGMA [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(9000)^3 = 9000

X(1)+X(2)+X(3)+...+X(9000) <= 9000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 9000.

Thus, one of the two additional constraints is an equality constraint and the other is a less-than-or-equal-to constraint.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).

One also notes line 191, which is 191 X(1)= ( 9000 -(SUMY) )^(1/3).

The objective function here is based on the Rosenbrock function [10, p. 415].

The present paper uses the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 9000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*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)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 9000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 9000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 9000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 9000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(9001)=9000- SUMNEW
303 IF X(9001)<0 THEN X(9001)=X(9001) ELSE X(9001)=0
401 SONE=0
402 FOR J44=1 TO 8999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(9001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(5996),A(5997),A(5998),A(5999),A(6000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-32000 is shown below:

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

Above there is no rounding by hand.

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

Of the 9000 A's, only the 10 A's of line 1931 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-32000 was 6 hours and a half.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 6000 Unknowns instead of 100 Unknowns and with Two Additional Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(6000)^3 = 6000

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4 but with two additional constraints and with 6000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

6000-1
SIGMA [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(6000)^3 = 6000

X(1)+X(2)+X(3)+...+X(6000) <= 6000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 6000.

Thus, one of the two additional constraints is an equality constraint and the other is a less-than-or-equal-to constraint.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).

One also notes line 191, which is 191 X(1)= ( 6000 -(SUMY) )^(1/3).

The objective function here is based on the Rosenbrock function [10, p. 415].

The present paper uses the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 6000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 6000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*6000)
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 6000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 6000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 6000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 6000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 6000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(6001)=6000- SUMNEW
303 IF X(6001)<0 THEN X(6001)=X(6001) ELSE X(6001)=0
401 SONE=0
402 FOR J44=1 TO 5999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(6001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 6001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(5996),A(5997),A(5998),A(5999),A(6000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31999 is shown below:

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

1 1 1 1 1
1 1 1 1 2
-509 -31999

Above there is no rounding by hand.

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

Of the 6000 A's, only the 10 A's of line 1931 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31999 was 5 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Friday, September 19, 2014

A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 5000 Unknowns instead of 100 Unknowns and with Two Additional Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(5000)^3 = 5000

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun's Problem 14.4 but with two additional constraints and with 5000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

5000-1
SIGMA [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(5000)^3 = 5000

X(1)+X(2)+X(3)+...+X(5000) <= 5000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 5000.

Thus, one of the two additional constraints is an equality constraint and the other is a less-than-or-equal-to constraint.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).

One also notes line 191, which is 191 X(1)= ( 5000 -(SUMY) )^(1/3).

The objective function here is based on the Rosenbrock function [10, p. 415].

In contrast with the preceding papers, the present paper uses the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 5000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 5000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*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)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 5000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 5000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 5000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 5000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(5001)=5000- SUMNEW
303 IF X(5001)<0 THEN X(5001)=X(5001) ELSE X(5001)=0
401 SONE=0
402 FOR J44=1 TO 4999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(5001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(4996),A(4997),A(4998),A(4999),A(5000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31993 is shown below:

1 1 1 1 1
1 1 1 1 2
-709 -32000

1 1 1 1 1
1 1 1 2 2
-1315 -31999

1 1 1 1 1
1 1 1 1 2
-509 -31998

1 1 1 1 1
1 1 1 1 0
-2117 -31997

1 1 1 1 1
1 1 1 1 2
-6157 -31996

1 1 1 1 1
1 1 1 1 2
-707 -31995

1 1 1 1 1
1 1 1 1 2
-509 -31994

1 1 1 1 1
1 1 1 1 1
0 -31993

Above there is no rounding by hand.

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

Of the 5000 A's, only the 10 A's of line 1931 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31993 was 15 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Wednesday, September 17, 2014

A Unified Computer Program Solving Li and Sun's Problem 14.4 but with 3000 Unknowns instead of 100 Unknowns and with Two Additional Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(3000)^3 = 3000

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 with two additional constraints and with 3000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

3000-1
SIGMA [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(3000)^3 = 3000

X(1)+X(2)+X(3)+...+X(3000) <= 3000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 3000.

Thus, one of the two additional constraints is an equality constraint and the other is a less-than-or-equal-to constraint.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).

One also notes line 191, which is 191 X(1)= ( 3000 -(SUMY) )^(1/3).

The objective function here is based on the Rosenbrock function [10, p. 415].

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 3000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 3000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*3000)
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 3000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 3000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 3000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 3000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 3000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(3001)=3000- SUMNEW
303 IF X(3001)<0 THEN X(3001)=X(3001) ELSE X(3001)=0
401 SONE=0
402 FOR J44=1 TO 2999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(3001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 3001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(2996),A(2997),A(2998),A(2999),A(3000)
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=-31992 is shown below:

1 1 1 1 1
1 1 1 1 2
-709 -32000

1 1 1 1 1
1 1 1 2 2
-1317 -31999

1 1 1 1 1
1 1 1 1 2
-507 -31998

1 1 1 1 1
1 1 1 1 2
-2727 -31997

1 1 1 1 1
1 1 1 1 2
-309 -31996

1 1 1 1 1
1 1 1 1 2
-509 -31995

1 1 1 1 1
1 1 1 1 0
-2115 -31994

1 1 1 1 1
1 1 1 1 2
-709 -31993

1 1 1 1 1
1 1 1 1 1
0 -31992

Above there is no rounding by hand.

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

Of the 3000 A's, only the 10 A's of line 1931 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31992 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] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Monday, September 15, 2014

A Unified Computer Program Solving Li and Sun's Problem 14.4 Plus Two Additional Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(1000)^3 = 1000, and with 1000 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 14.4 plus two additional constraintsand with 1000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

1000-1
SIGMA [ 100* ( X(i+1) - X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(1000)^3 = 1000

X(1)+X(2)+X(3)+...+X(1000) <= 1000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 1000.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).

One also notes line 191, which is 191 X(1)= ( 1000 -(SUMY) )^(1/3).

While line 114 of the preceding paper is 114 A(J44 )=-5+FIX( RND*11), line 114 here is
114 A(J44)=-1+FIX(RND*3).

While line 128 of the preceding paper is 128 FOR I=1 TO 32000 STEP .5, line 128 here is
128 FOR I=1 TO 32000 STEP .1.

The objective function here is based on the Rosenbrock function [10, p. 415].

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 1000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 1000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*1000)
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 1000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 1000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 1000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 1000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 1000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(1001)=1000- SUMNEW
303 IF X(1001)<0 THEN X(1001)=X(1001) ELSE X(1001)=0
401 SONE=0
402 FOR J44=1 TO 999
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(1001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 1001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(996),A(997),A(998),A(999),A(1000)
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=-31996 is shown below:

1 1 1 1 1
1 1 1 1 2
-4941 -32000

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

1 1 1 1 1
1 1 1 1 2
-507 -31998

1 1 1 1 1
1 1 1 1 2
-507 -31997

1 1 1 1 1
1 1 1 1 1
0 -31996

Above there is no rounding by hand.

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

Of the 1000 A's, only the 10 A's of line 1931 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31996 was 2 hours and 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. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

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

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

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

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

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

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

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

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

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

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

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

A Unified Computer Program Solving Li and Sun's Problem 14.4 Plus Two Additional Constraints, Including X(1)^3+X(2)^3+X(3)^3+...+X(100)^3 = 100

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 plus two additional constraints; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

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

subject to

X(1)^3+X(2)^3+X(3)^3+...+X(100)^3 = 100

X(1)+X(2)+X(3)+...+X(100) <= 100

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,..., 100.

One notes line 114, line 174, and line 175, which are as follows:
114 A(J44)=-1+FIX(RND*3)
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 ).

One also notes line 191, which is 191 X(1)= ( 100 -(SUMY) )^(1/3).

While line 114 of the preceding paper is 114 A(J44 )=-5+FIX( RND*11), line 114 here is
114 A(J44)=-1+FIX(RND*3).

The objective function here is based on the Rosenbrock function [10, p. 415].

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10001),X(10001)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 100
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*100)
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 100
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 100
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 100 -(SUMY) )<0 THEN 1670
191 X(1)= ( 100 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 100
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(101)=100- SUMNEW
303 IF X(101)<0 THEN X(101)=X(101) ELSE X(101)=0
401 SONE=0
402 FOR J44=1 TO 99
411 SONE=SONE+ 100* ( X(J44+1) - X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(101)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 101
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 REM PRINT A(1),A(10000),A(10001),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(96),A(97),A(98),A(99),A(100)
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=-31978 is shown below:

1 1 1 1 1
0 0 0 0 0
-3741 -32000

1 1 1 1 1
1 1 1 1 2
-507 -31999

1 1 1 1 1
0 0 0 0 0
-2323 -31998

1 1 1 1 1
1 1 1 0 0
-3331 -31997

1 1 1 1 1
1 1 1 1 2
-707 -31996

1 1 1 1 1
1 0 0 0 0
-2723 -31995

1 1 1 1 1
1 1 1 1 2
-507 -31994

1 1 1 1 1
1 1 1 1 2
-307 -31993

1 1 1 1 1
1 1 1 1 2
-507 -31992

1 1 1 1 1
1 1 1 2 2
-1315 -31991

1 1 1 1 1
1 0 0 0 0
-2723 -31990

0 0 1 1 1
0 0 -1 2 3
-1336 -31989

1 1 1 1 1
1 1 0 0 0
-4139 -31988

1 1 1 1 1
1 1 1 2 2
-915 -31987

1 1 1 1 1
0 0 0 1 2
-3541 -31986

1 1 1 1 1
1 0 0 0 0
-3131 -31985

1 1 1 1 0
1 1 0 -2 4
-1963 -31984

1 1 1 1 1
1 1 1 1 1
0 -31983

1 1 1 1 1
1 1 1 1 2
-707 -31982

1 1 1 1 1
1 1 1 1 2
-507 -31981

1 1 0 0 0
0 1 2 2 2
-1723 -31980

1 1 1 1 2
0 0 0 1 2
-3131 -31979

1 1 1 1 1
1 1 1 1 1
0 -31978

Above there is no rounding by hand.

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

Of the 100 A's, only the 10 A's of line 1931 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31978 was 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. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

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

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

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

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

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

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

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

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

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

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

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