Month: September 2014

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A Unified Computer Program for Solving Schittkowski’s Test Problem 395 but with 6000 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 6000 unknowns instead of 50 unknowns; see Schittkowski [14, page 213, Test Problem 395]. The source of this Test Problem 395 is S. Waluziewicz–see Schittkowski [14, p. 213]. Specifically the computer program below tries to minimize the following:

6000
SIGMA i*(X(i)^2+X(i)^4 )
i=1

subject to

6000
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(6000),X(6000)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 6000
112 A(J44)=-.01+RND*.02
114 NEXT J44
128 FOR I=1 TO 32000
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)
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 6000
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 6000
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 6000
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(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:

.912969      .4080289      -1.309792E-04      -1.038962E-04
-8.346089E-05
2.216979E-06      -1.479326E-06      -3.149312E-06
-3.481227E-07      1.24744E-06
-1.916725      -32000

.9129156      .408148      -4.185103E-04      9.129979E-05
-2.865979E-04
1.329322E-06      -7.766722E-07      -2.418335E-06
-1.454119E-06      -1.169394E-06
-1.916696      -31999

Above there is no rounding by hand.

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

Of the 6000 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=-31999 was 20 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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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].

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(8996),A(8997),A(8998),A(8999),A(9000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

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

1      1      1      1      1
1      1      1      1      0
-2315      -32000

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

Above there is no rounding by hand.

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

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

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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A Unified Computer Program Solving Li and Sun’s Problem 14.4 but with 10000 Unknowns instead of 100 Unknowns and with Two Additional 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 additional 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 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)= ( 10000 -(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 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(9996),A(9997),A(9998),A(9999),A(10000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

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

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

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

Above there is no rounding by hand.

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

Of the 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 basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31999 was about 60 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

by

A Unified Computer Program Solving Li and Sun’s Problem 14.4 but with 4000 Unknowns instead of 100 Unknowns and with Two Additional Constraints, Including X(1)^3+X(2)^3+X(3)^3+…+X(4000)^3 = 4000

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 and with 4000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

4000-1
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(4000)^3 = 4000

X(1)+X(2)+X(3)+…+X(4000) <= 4000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,…, 4000.

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)= ( 4000 -(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 4000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 4000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4000)
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 4000
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 4000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 4000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 4000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 4000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(4001)=4000- SUMNEW
303 IF X(4001)<0 THEN X(4001)=X(4001) ELSE X(4001)=0
401 SONE=0
402 FOR J44=1 TO 3999
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(4001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4001
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(3996),A(3997),A(3998),A(3999),A(4000)
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=-31998 is shown below:

1 1 1 1 1
1 1 1 1 0
-4131      -32000

1 1 1 1 1
1 1 1 1 0
-3933      -31999

1 1 1 1 1
1 1 1 1 1
0      -31998

Above there is no rounding by hand.

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

Of the 4000 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=-31998 was 14 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

by

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(2000)^3 = 2000, and with 2000 Unknowns instead of 100 Unknowns

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun’s Problem 14.4 plus two additional constraintsand with 2000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

2000-1
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(2000)^3 = 2000

X(1)+X(2)+X(3)+…+X(2000) <= 2000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,…, 2000.

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)= ( 2000 -(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 2000
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .1
129 FOR KKQQ=1 TO 2000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*2000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 2000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
180 SUMY=0
183 FOR J44=2 TO 2000
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 2000 -(SUMY) )<0 THEN 1670
191 X(1)= ( 2000 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 2000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(2001)=2000- SUMNEW
303 IF X(2001)<0 THEN X(2001)=X(2001) ELSE X(2001)=0
401 SONE=0
402 FOR J44=1 TO 1999
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(2001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 2001
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(1996),A(1997),A(1998),A(1999),A(2000)
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=-31994 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
-715      -31999

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

1      1      1      1      1
1      1      1      1      2
-509      -31997

1      1      1      1      1
1      1      1      2      2
-3533      -31996

1      1      1      1      1
1      1      1      1      2
-2725      -31995

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

Above there is no rounding by hand.

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

Of the 2000 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=-31994 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] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

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

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

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

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

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

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

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

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

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

by

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(300)^3 = 300, and with 300 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 constraints; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

300-1
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(300)^3 = 300

X(1)+X(2)+X(3)+…+X(300) <= 300

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,…, 300.

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)= ( 300 -(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 300
114 A(J44)=-1+FIX(RND*3)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 300
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*300)
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 300
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 300
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
189 IF ( 300 -(SUMY) )<0 THEN 1670
191 X(1)= ( 300 -(SUMY) )^(1/3)
200 SUMNEW=0
203 FOR J44=1 TO 300
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(301)=300- SUMNEW
303 IF X(301)<0 THEN X(301)=X(301) ELSE X(301)=0
401 SONE=0
402 FOR J44=1 TO 299
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(301)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 301
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(296),A(297),A(298),A(299),A(300)
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=-31983 is shown below:

1 1 1 1 1
1 1 1 1 2
-507 -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
-2016 -31997

1 1 1 1 1
1 1 1 1 2
-509 -31996

1 1 1 1 1
1 1 1 1 2
-309 -31995

1 1 1 1 1
1 1 1 2 2
-1315 -31994

1 1 1 1 1
1 1 1 1 1
-2824 -31993

1 1 1 1 1
1 1 1 1 2
-7165 -31992

1 1 1 1 1
1 1 1 1 1
0 -31991

1 1 1 1 1
1 1 1 1 2
-3331 -31990

1 1 1 1 1
1 1 1 1 2
-2725 -31989

1 1 1 1 1
1 1 1 1 2
-707 -31988

1 1 1 1 1
1 1 1 1 2
-2723 -31987

1 1 1 1 1
1 1 1 1 2
-507 -31986

1 1 1 1 1
1 1 1 2 2
-3533 -31985

1 1 1 1 1
1 1 1 1 2
-509 -31984

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

Above there is no rounding by hand.

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

Of the 300 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=-31983 was 44 minutes.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

by

A Unified Computer Program Solving Li and Sun’s Problem 14.4 Plus Two Less-Than-or-Equal-to Constraints, Including X(1)^3+X(2)^3+X(3)^3+…+X(300)^3 <= 300, and with 300 Unknowns instead 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 less-than-or-equal-to constraints; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

300-1
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(300)^3 <= 300

X(1)+X(2)+X(3)+…+X(300) <= 300

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,…, 300.

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

One also notes line 180 through line 195, especially line 185, which is 185 SUMY=SUMY+X(J44)^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 300
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 3200
129 FOR KKQQ=1 TO 300
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*300)
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 300
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=1 TO 300
185 SUMY=SUMY+X(J44)^3
187 NEXT J44
193 X(302)= 300 -SUMY
195 IF X(302)<0 THEN X(302)=X(302) ELSE X(302)=0
200 SUMNEW=0
203 FOR J44=1 TO 300
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(301)=300- SUMNEW
303 IF X(301)<0 THEN X(301)=X(301) ELSE X(301)=0
401 SONE=0
402 FOR J44=1 TO 299
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(301) +5000000!*X(302)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 302
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-50000! THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(296),A(297),A(298),A(299),A(300)
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=-31997 is shown below:

-1 1 1 1 1
1 1 1 1 1
-4      -32000

-1 1 1 1 1
1 1 1 2 4
-4583     -31999

1 1 1 1 1
1 1 1 1 1
0      -31998

1 1 1 1 1
1 1 1 1 1
0      -31997

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=-31997 was 3 minutes.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

by

A Unified Computer Program Solving Li and Sun’s Problem 14.4 Plus a Less-Than-or-Equal-to Constraint and with 10000 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 a less-than-or-equal-to constraint 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)+X(2)+X(3)+…+X(10000) <= 10000

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

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

One also notes line 255 and line 303, which are:
255 X(10001)=10000- SUMNEW.
303 IF X(10001)<0 THEN X(10001)=X(10001) ELSE X(10001)=0.

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)=-5+FIX(RND*11)
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
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 10001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1459 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(9996),A(9997),A(9998),A(9999),A(10000)
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 output partly through JJJJ=-31999 is summarized below:

.
.
.
1 1 0 -1006 -31999
1 1 0 -805 -31999
1 1 0 -604 -31999
1 1 0 -603 -31999
1 1 0 -402 -31999
1 1 0 -201 -31999
1 1 0 0 -31999

Above there is no rounding by hand.

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

Of the 10000 A’s, only the two A’s–A(1) and A(10000)–of line 1459 are shown above.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

by

A Unified Computer Program Solving Li and Sun’s Problem 14.4 Plus a Less-Than-or-Equal-to Constraint and with 8000 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 a less-than-or-equal-to constraint and with 8000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

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

subject to

X(1)+X(2)+X(3)+…+X(8000) <= 8000

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

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

One also notes line 255, which is 255 X(8001)=8000- SUMNEW.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 8000
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 8000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*8000)
143 GOTO 167
144 REM GOTO 168
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
171 FOR J44=1 TO 8000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
200 SUMNEW=0
203 FOR J44=1 TO 8000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(8001)=8000- SUMNEW
303 IF X(8001)<0 THEN X(8001)=X(8001) ELSE X(8001)=0
401 SONE=0
402 FOR J44=1 TO 7999
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(8001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 8001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-10000 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(7996),A(7997),A(7998),A(7999),A(8000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

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

-1      1      1      1      1
1      1      1      1      1
-4      -32000

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

Above there is no rounding by hand.

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

Of the 8000 A’s, only the 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=-31999 was 18 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

by

A Unified Computer Program Solving Li and Sun’s Problem 14.4 Plus an Additional Constraint and with 4000 Unknowns instead of 100 Unknowns

Jsun Yui Wong

Case One: The Additional Constraint is a Less-Than-or-Equal-to Constraint

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Li and Sun’s Problem 14.4 plus a less-than-or-equal-to constraint and with 4000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

4000-1
SIGMA [ 100* ( X(i+1) – X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

X(1)+X(2)+X(3)+…+X(4000) <= 4000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,…, 4000.

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

One also notes line 255, which is 255 X(4001)=4000- SUMNEW.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 4000
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 4000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4000)
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 4000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
200 SUMNEW=0
203 FOR J44=1 TO 4000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
255 X(4001)=4000- SUMNEW
303 IF X(4001)<0 THEN X(4001)=X(4001) ELSE X(4001)=0
401 SONE=0
402 FOR J44=1 TO 3999
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE +5000000!*X(4001)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4001
1455 A(KLX)=X(KLX)
1456 NEXT KLX
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(3996),A(3997),A(3998),A(3999),A(4000)
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=-31997 is shown below:

1 1 1 1 1
1 1 1 1 1
-201 -32000

-1 1 1 1 1
1 1 1 1 1
-4 -31999

-1 1 1 1 1
1 1 0 -2 4
-514 -31998

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

Above there is no rounding by hand.

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

Of the 4000 A’s, only the 10 A’s of lin 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=-31997 was 7 hours and 40 minutes.

Case Two: The Additional Constraint is an Equality Constraint

The computer program below seeks to solve Li and Sun’s Problem 14.4 plus an equality constraint and with 4000 unknowns instead of 100 unknowns; see Li and Sun [10, p. 415]. Specifically the computer program below tries to minimize the following:

4000-1
SIGMA [ 100* ( X(i+1) – X(i)^2 )^2 + ( 1-X(i) )^2 ]
i=1

subject to

X(1)+X(2)+X(3)+…+X(4000) = 4000

-5<=X(i)<=5, X(i) integer, i=1, 2, 3,…, 4000.

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

One also notes line 211, which is 211 X(1)=4000-SUMNEW.

0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(10000),X(10000)
81 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 4000
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000 STEP .5
129 FOR KKQQ=1 TO 4000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4000)
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=2 TO 4000
174 IF X(J44)>5 THEN X(J44 )=A(J44 )
175 IF X(J44)<-5 THEN X(J44 )=A(J44 )
177 NEXT J44
200 SUMNEW=0
203 FOR J44=2 TO 4000
205 SUMNEW=SUMNEW+X(J44)
207 NEXT J44
211 X(1)=4000-SUMNEW
222 IF X(1)>5 THEN X(1 )=A(1 )
225 IF X(1)<-5 THEN X(1 )=A(1 )
401 SONE=0
402 FOR J44=1 TO 3999
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-4444 THEN 1999
1931 PRINT A(1),A(2),A(3),A(4),A(5)
1936 PRINT A(3996),A(3997),A(3998),A(3999),A(4000)
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=-31998 is shown below:

1 1 1 1 1
1 1 1 1 1
-405 -32000

0 1 1 1 1
1 1 1 1 1
-708 -31999

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

Above there is no rounding by hand.

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

Of the 4000 A’s, only the 10 A’s of lin 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=-31998 was 8 hours.

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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