Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve an integer version of Schittkowski's Test Problem 270. The source of this Test Problem 270 is L. W. Cornwell et al. of Argonne National Laboratory Technical Memorandum No. 320; see Schittkowski [14, p. 94]. Thus, the computer program below tries to minimize the following:
X(1)*X(2) *X(3) *X(4)-3* X(1)*X(2)*X(4) - 4*X(1)* X(2)*X(3)+12* X(1)*X(2) -X(2) *X( 3)*X(4)+3*X(2)*X(4) +4*X(2)*X(3)-12*X(2)-2*X(1)*X(3)*X(4)+6*X(1)*X(4)+8*X(1)*X(3)-24*X(1)
+ 2 *X(3) *X(4)-6* X(4) - 8*X(3) +24 + 1.5* X(5)^4 -5.75 *X( 5)^3+5.25*X(5)^2
subject to
34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2 >= 0
X(i) integer, i=1, 2, 3, 4, 5,
and the bounds are specified in line 171 through line 182.
Line 91 through line 95 are noteworthy.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
91 A(1)=1+RND*5000
92 A(2)=2+RND*5000
93 A(3)=3+RND*5000
94 A(4)=4+RND*5000
95 A(5)=-5000+RND*10000
128 FOR I=1 TO 3000
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5)
144 REM GOTO 167
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 5
174 IF X(J44)>5000 THEN X(J44)=A(J44)
177 NEXT J44
178 FOR J44=1 TO 4
179 IF X(J44)<J44 THEN X(J44)=A(J44)
182 NEXT J44
555 Y(6)= 34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2
565 IF Y(6)>0 THEN Y(6)=0
681 PDA=- X(1)*X(2) *X(3) *X(4)+3* X(1)*X(2)*X(4) + 4*X(1)* X(2)*X(3) - 12* X(1)*X(2) +X(2) *X( 3)*X(4)-3*X(2)*X(4) -4*X(2)*X(3)+12*X(2)+2*X(1)*X(3)*X(4)-6*X(1)*X(4)-8*X(1)*X(3)+24*X(1)
684 PDB=- 2 *X(3) *X(4)+6* X(4) + 8*X(3) -24 - 1.5* X(5)^4 +5.75 *X( 5)^3-5.25*X(5)^2 +500000!*Y(6)
695 PD1=PDA+PDB
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<.1 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
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=-31986 is shown below:
1 2 3 4 2
1 -31995
1 2 3 4 2
1 -31992
1 2 3 4 2
1 -31989
1 2 3 4 2
1 -31986
Above there is no rounding by hand.
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=-31986 was five seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
Thursday, August 14, 2014
A Unified Computer Program for an Integer Version of Schittkowski's Test Problem 270
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve an integer version of Schittkowski's Test Problem 270. The source of this Test Problem 270 is L. W. Cornwell et al. of Argonne National Laboratory Technical Memorandum No. 320; see Schittkowski [14, p. 94]. Thus, the computer program below tries to minimize the following:
X(1)*X(2) *X(3) *X(4)-3* X(1)*X(2)*X(4) - 4*X(1)* X(2)*X(3)+12* X(1)*X(2) -X(2) *X( 3)*X(4)+3*X(2)*X(4) +4*X(2)*X(3)-12*X(2)-2*X(1)*X(3)*X(4)+6*X(1)*X(4)+8*X(1)*X(3)-24*X(1)
+ 2 *X(3) *X(4)-6* X(4) - 8*X(3) +24 + 1.5* X(5)^4 -5.75 *X( 5)^3+5.25*X(5)^2
subject to
34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2 >= 0
X(i) integer, i=1, 2, 3, 4, 5.
and the bounds are specified in line 171 through line 182.
Line 91 through line 95 are noteworthy.
While line 1 of the preceding paper is 1 DEFINT J,K,B, here line 1 is 1 DEFINT J,K,B,X.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
91 A(1)=1+RND*300
92 A(2)=2+RND*300
93 A(3)=3+RND*300
94 A(4)=4+RND*300
95 A(5)=-300+RND*600
128 FOR I=1 TO 3000
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5)
144 REM GOTO 167
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 5
174 IF X(J44)>300 THEN X(J44)=A(J44)
177 NEXT J44
178 FOR J44=1 TO 4
179 IF X(J44)<J44 THEN X(J44)=A(J44)
182 NEXT J44
555 Y(6)= 34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2
565 IF Y(6)>0 THEN Y(6)=0
681 PDA=- X(1)*X(2) *X(3) *X(4)+3* X(1)*X(2)*X(4) + 4*X(1)* X(2)*X(3) - 12* X(1)*X(2) +X(2) *X( 3)*X(4)-3*X(2)*X(4) -4*X(2)*X(3)+12*X(2)+2*X(1)*X(3)*X(4)-6*X(1)*X(4)-8*X(1)*X(3)+24*X(1)
684 PDB=- 2 *X(3) *X(4)+6* X(4) + 8*X(3) -24 - 1.5* X(5)^4 +5.75 *X( 5)^3-5.25*X(5)^2 +500000!*Y(6)
695 PD1=PDA+PDB
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<.1 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
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=-31982 is shown below:
1 2 3 4 2
1 -31998
1 2 3 4 2
1 -31996
1 2 3 4 2
1 -31989
1 2 3 4 2
1 -31982
Above there is no rounding by hand.
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=-31982 was five seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
Similar to the computer programs of the preceding papers, the computer program below seeks to solve an integer version of Schittkowski's Test Problem 270. The source of this Test Problem 270 is L. W. Cornwell et al. of Argonne National Laboratory Technical Memorandum No. 320; see Schittkowski [14, p. 94]. Thus, the computer program below tries to minimize the following:
X(1)*X(2) *X(3) *X(4)-3* X(1)*X(2)*X(4) - 4*X(1)* X(2)*X(3)+12* X(1)*X(2) -X(2) *X( 3)*X(4)+3*X(2)*X(4) +4*X(2)*X(3)-12*X(2)-2*X(1)*X(3)*X(4)+6*X(1)*X(4)+8*X(1)*X(3)-24*X(1)
+ 2 *X(3) *X(4)-6* X(4) - 8*X(3) +24 + 1.5* X(5)^4 -5.75 *X( 5)^3+5.25*X(5)^2
subject to
34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2 >= 0
X(i) integer, i=1, 2, 3, 4, 5.
and the bounds are specified in line 171 through line 182.
Line 91 through line 95 are noteworthy.
While line 1 of the preceding paper is 1 DEFINT J,K,B, here line 1 is 1 DEFINT J,K,B,X.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
91 A(1)=1+RND*300
92 A(2)=2+RND*300
93 A(3)=3+RND*300
94 A(4)=4+RND*300
95 A(5)=-300+RND*600
128 FOR I=1 TO 3000
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5)
144 REM GOTO 167
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 5
174 IF X(J44)>300 THEN X(J44)=A(J44)
177 NEXT J44
178 FOR J44=1 TO 4
179 IF X(J44)<J44 THEN X(J44)=A(J44)
182 NEXT J44
555 Y(6)= 34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2
565 IF Y(6)>0 THEN Y(6)=0
681 PDA=- X(1)*X(2) *X(3) *X(4)+3* X(1)*X(2)*X(4) + 4*X(1)* X(2)*X(3) - 12* X(1)*X(2) +X(2) *X( 3)*X(4)-3*X(2)*X(4) -4*X(2)*X(3)+12*X(2)+2*X(1)*X(3)*X(4)-6*X(1)*X(4)-8*X(1)*X(3)+24*X(1)
684 PDB=- 2 *X(3) *X(4)+6* X(4) + 8*X(3) -24 - 1.5* X(5)^4 +5.75 *X( 5)^3-5.25*X(5)^2 +500000!*Y(6)
695 PD1=PDA+PDB
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<.1 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
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=-31982 is shown below:
1 2 3 4 2
1 -31998
1 2 3 4 2
1 -31996
1 2 3 4 2
1 -31989
1 2 3 4 2
1 -31982
Above there is no rounding by hand.
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=-31982 was five seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
A Unified Computer Program for Schittkowski's Test Problem 270
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 270. The source of this Test Problem 270 is L. W. Cornwell et al. of Argonne National Laboratory Technical Memorandum No. 320; see Schittkowski [14, p. 94]. Thus, the computer program below tries to minimize the following:
X(1)*X(2) *X(3) *X(4)-3* X(1)*X(2)*X(4) - 4*X(1)* X(2)*X(3)+12* X(1)*X(2) -X(2) *X( 3)*X(4)+3*X(2)*X(4) +4*X(2)*X(3)-12*X(2)-2*X(1)*X(3)*X(4)+6*X(1)*X(4)+8*X(1)*X(3)-24*X(1)
+ 2 *X(3) *X(4)-6* X(4) - 8*X(3) +24 + 1.5* X(5)^4 -5.75 *X( 5)^3+5.25*X(5)^2
subject to
34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2 >= 0
and the bounds are specified in line 171 through line 182.
Line 91 through line 95 are noteworthy.
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
91 A(1)=1+RND*300
92 A(2)=2+RND*300
93 A(3)=3+RND*300
94 A(4)=4+RND*300
95 A(5)=-300+RND*600
128 FOR I=1 TO 3000
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5)
144 REM GOTO 167
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 5
174 IF X(J44)>300 THEN X(J44)=A(J44)
177 NEXT J44
178 FOR J44=1 TO 4
179 IF X(J44)<J44 THEN X(J44)=A(J44)
182 NEXT J44
555 Y(6)= 34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2
565 IF Y(6)>0 THEN Y(6)=0
681 PDA=- X(1)*X(2) *X(3) *X(4)+3* X(1)*X(2)*X(4) + 4*X(1)* X(2)*X(3) - 12* X(1)*X(2) +X(2) *X( 3)*X(4)-3*X(2)*X(4) -4*X(2)*X(3)+12*X(2)+2*X(1)*X(3)*X(4)-6*X(1)*X(4)-8*X(1)*X(3)+24*X(1)
684 PDB=- 2 *X(3) *X(4)+6* X(4) + 8*X(3) -24 - 1.5* X(5)^4 +5.75 *X( 5)^3-5.25*X(5)^2 +500000!*Y(6)
695 PD1=PDA+PDB
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<.1 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
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=-31827 is shown below:
1.033385 2 3 4 1.982914
.9980621 -31909
1.001678 2.000072 3.000703 4.000962 1.996094
.9999046 -31842
1 2 3 4 1.999867
1.000002 -31835
1 2 3 4.000221 1.999494
1 -31827
Above there is no rounding by hand.
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=-31827 was 50 seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 270. The source of this Test Problem 270 is L. W. Cornwell et al. of Argonne National Laboratory Technical Memorandum No. 320; see Schittkowski [14, p. 94]. Thus, the computer program below tries to minimize the following:
X(1)*X(2) *X(3) *X(4)-3* X(1)*X(2)*X(4) - 4*X(1)* X(2)*X(3)+12* X(1)*X(2) -X(2) *X( 3)*X(4)+3*X(2)*X(4) +4*X(2)*X(3)-12*X(2)-2*X(1)*X(3)*X(4)+6*X(1)*X(4)+8*X(1)*X(3)-24*X(1)
+ 2 *X(3) *X(4)-6* X(4) - 8*X(3) +24 + 1.5* X(5)^4 -5.75 *X( 5)^3+5.25*X(5)^2
subject to
34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2 >= 0
and the bounds are specified in line 171 through line 182.
Line 91 through line 95 are noteworthy.
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
91 A(1)=1+RND*300
92 A(2)=2+RND*300
93 A(3)=3+RND*300
94 A(4)=4+RND*300
95 A(5)=-300+RND*600
128 FOR I=1 TO 3000
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5)
144 REM GOTO 167
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 5
174 IF X(J44)>300 THEN X(J44)=A(J44)
177 NEXT J44
178 FOR J44=1 TO 4
179 IF X(J44)<J44 THEN X(J44)=A(J44)
182 NEXT J44
555 Y(6)= 34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2
565 IF Y(6)>0 THEN Y(6)=0
681 PDA=- X(1)*X(2) *X(3) *X(4)+3* X(1)*X(2)*X(4) + 4*X(1)* X(2)*X(3) - 12* X(1)*X(2) +X(2) *X( 3)*X(4)-3*X(2)*X(4) -4*X(2)*X(3)+12*X(2)+2*X(1)*X(3)*X(4)-6*X(1)*X(4)-8*X(1)*X(3)+24*X(1)
684 PDB=- 2 *X(3) *X(4)+6* X(4) + 8*X(3) -24 - 1.5* X(5)^4 +5.75 *X( 5)^3-5.25*X(5)^2 +500000!*Y(6)
695 PD1=PDA+PDB
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<.1 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
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=-31827 is shown below:
1.033385 2 3 4 1.982914
.9980621 -31909
1.001678 2.000072 3.000703 4.000962 1.996094
.9999046 -31842
1 2 3 4 1.999867
1.000002 -31835
1 2 3 4.000221 1.999494
1 -31827
Above there is no rounding by hand.
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=-31827 was 50 seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
Wednesday, August 13, 2014
Errata: A Unified Computer Program for an Integer Version of Schittkowski's Test Problem 354
Jsun Yui Wong
The title of the paper preceding the last paper should read as follows:
A Unified Computer Program for Schittkowski's Test Problem 354.
The title of the paper preceding the last paper should read as follows:
A Unified Computer Program for Schittkowski's Test Problem 354.
A Unified Computer Program for Schittkowski's Test Problem 355, a Circuits Problem
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 355. The source of this Test Problem 355 is D. A. Pierre and M. J. Lowe/D. A. Pierre; see Schittkowski [14, p. 175]. Thus, the computer program below tries to minimize the following:
RONE^2+RTWO^2
subject to
RONE^2+RTWO^2-RTHR^2-RFOU^2 = 0
where
RONE=11-X(1)*X(4)-X(2)*X(4)+X(3)*X(4)
RTWO= X(1)+10*X(2)-X(3)+X(4)+X(2)*X(4)* (X(3)-X(1) )
RTHR=11-4*X(1)*X(4)-4*X(2)*X(4)+X(3)*X(4)
RFOU=2*X(1)+20*X(2)-.5*X(3)+2*X(4)+2*X(2)*X(4)* (X(3)-4*X(1) )
Lower bounds for the Xs are .1, .1, 0, 0, respectively. Upper bounds for the Xs are 2, 2, 2, 2, respectively. See line 171 through line 189.
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 2
112 A(J44)=.1+ RND*(2)
114 NEXT J44
116 FOR J44=3 TO 4
118 A(J44)= RND*(2)
119 NEXT J44
128 FOR I=1 TO 10000
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144 REM GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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 4
174 IF X(J44)>2 THEN X(J44)=A(J44)
177 NEXT J44
184 IF X(1)<.1 THEN X(1)=A(1)
185 IF X(2)<.1 THEN X(2)=A(2)
188 IF X(3)<0 THEN X(3)=A(3)
189 IF X(4)<0 THEN X(4)=A(4)
501 RONE=11-X(1)*X(4)-X(2)*X(4)+X(3)*X(4)
505 RTWO= X(1)+10*X(2)-X(3)+X(4)+X(2)*X(4)* (X(3)-X(1) )
507 RTHR=11-4*X(1)*X(4)-4*X(2)*X(4)+X(3)*X(4)
509 RFOU=2*X(1)+20*X(2)-.5*X(3)+2*X(4)+2*X(2)*X(4)* (X(3)-4*X(1) )
551 RFIV=-RONE^2-RTWO^2+RTHR^2+RFOU^2
561 IF ABS(RFIV)<.000001 THEN RFIV=0
671 PD1=-RONE^2-RTWO^2 -500000!*ABS(RFIV)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-70 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4)
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=-30329 is shown below:
1.977606 .1048642 1.574883E-03 1.918648
-69.7688 -30884
1.895019 .1001017 4.891302E-04 1.993295
-69.68565 -30790
1.970581 .1001387 1.877038E-02 1.92754
-69.91499 -30329
Above there is no rounding by hand.
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=-30329 was 24 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. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 355. The source of this Test Problem 355 is D. A. Pierre and M. J. Lowe/D. A. Pierre; see Schittkowski [14, p. 175]. Thus, the computer program below tries to minimize the following:
RONE^2+RTWO^2
subject to
RONE^2+RTWO^2-RTHR^2-RFOU^2 = 0
where
RONE=11-X(1)*X(4)-X(2)*X(4)+X(3)*X(4)
RTWO= X(1)+10*X(2)-X(3)+X(4)+X(2)*X(4)* (X(3)-X(1) )
RTHR=11-4*X(1)*X(4)-4*X(2)*X(4)+X(3)*X(4)
RFOU=2*X(1)+20*X(2)-.5*X(3)+2*X(4)+2*X(2)*X(4)* (X(3)-4*X(1) )
Lower bounds for the Xs are .1, .1, 0, 0, respectively. Upper bounds for the Xs are 2, 2, 2, 2, respectively. See line 171 through line 189.
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 2
112 A(J44)=.1+ RND*(2)
114 NEXT J44
116 FOR J44=3 TO 4
118 A(J44)= RND*(2)
119 NEXT J44
128 FOR I=1 TO 10000
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144 REM GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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 4
174 IF X(J44)>2 THEN X(J44)=A(J44)
177 NEXT J44
184 IF X(1)<.1 THEN X(1)=A(1)
185 IF X(2)<.1 THEN X(2)=A(2)
188 IF X(3)<0 THEN X(3)=A(3)
189 IF X(4)<0 THEN X(4)=A(4)
501 RONE=11-X(1)*X(4)-X(2)*X(4)+X(3)*X(4)
505 RTWO= X(1)+10*X(2)-X(3)+X(4)+X(2)*X(4)* (X(3)-X(1) )
507 RTHR=11-4*X(1)*X(4)-4*X(2)*X(4)+X(3)*X(4)
509 RFOU=2*X(1)+20*X(2)-.5*X(3)+2*X(4)+2*X(2)*X(4)* (X(3)-4*X(1) )
551 RFIV=-RONE^2-RTWO^2+RTHR^2+RFOU^2
561 IF ABS(RFIV)<.000001 THEN RFIV=0
671 PD1=-RONE^2-RTWO^2 -500000!*ABS(RFIV)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-70 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4)
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=-30329 is shown below:
1.977606 .1048642 1.574883E-03 1.918648
-69.7688 -30884
1.895019 .1001017 4.891302E-04 1.993295
-69.68565 -30790
1.970581 .1001387 1.877038E-02 1.92754
-69.91499 -30329
Above there is no rounding by hand.
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=-30329 was 24 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. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
A Unified Computer Program for an Integer Version of Schittkowski's Test Problem 354
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 354. The source of this Test Problem 354 S. Walukiewicz; see Schittkowski [14]. Thus, the computer program below tries to minimize the following:
( X(1)+10*X(2) )^2+5*( X(3)-X(4) )^2 +( X(2)-2*X(3) )^4+ 10*( X(1)-X(4) )^4
subject to
X(1)+X(2)+X(3)+X(4)-1 >=0
Lower bounds are -5, -5, -5, -5, respectively.
Upper bounds are 20, 20, 20, 20, respectively.
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 4
112 A(J44)=-5+ RND*(25)
114 NEXT J44
128 FOR I=1 TO 10000
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144 REM GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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 4
174 IF X(J44)>20 THEN X(J44)=A(J44)
177 NEXT J44
178 FOR J44=1 TO 4
179 IF X(J44)<-5 THEN X(J44)=A(J44)
180 NEXT J44
555 X(5)=X(1)+X(2)+X(3)+X(4)-1
565 IF X(5)>0 THEN X(5)=0
681 PD1=-( X(1)+10*X(2) )^2-5*( X(3)-X(4) )^2 -( X(2)-2*X(3) )^4- 10*( X(1)-X(4) )^4 +500000!*X(5)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-9999 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4)
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:
.5027103 -4.520061E-02 .2363648 .3061256
-.1137975 -32000
.5035425 -4.552651E-02 .2360212 .3059629
-.1137877 -31999
.5022817 -4.535842E-02 .2362913 .3067854
-.1137903 -31998
Above there is no rounding by hand.
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 six seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 354. The source of this Test Problem 354 S. Walukiewicz; see Schittkowski [14]. Thus, the computer program below tries to minimize the following:
( X(1)+10*X(2) )^2+5*( X(3)-X(4) )^2 +( X(2)-2*X(3) )^4+ 10*( X(1)-X(4) )^4
subject to
X(1)+X(2)+X(3)+X(4)-1 >=0
Lower bounds are -5, -5, -5, -5, respectively.
Upper bounds are 20, 20, 20, 20, respectively.
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 4
112 A(J44)=-5+ RND*(25)
114 NEXT J44
128 FOR I=1 TO 10000
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144 REM GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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 4
174 IF X(J44)>20 THEN X(J44)=A(J44)
177 NEXT J44
178 FOR J44=1 TO 4
179 IF X(J44)<-5 THEN X(J44)=A(J44)
180 NEXT J44
555 X(5)=X(1)+X(2)+X(3)+X(4)-1
565 IF X(5)>0 THEN X(5)=0
681 PD1=-( X(1)+10*X(2) )^2-5*( X(3)-X(4) )^2 -( X(2)-2*X(3) )^4- 10*( X(1)-X(4) )^4 +500000!*X(5)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-9999 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4)
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:
.5027103 -4.520061E-02 .2363648 .3061256
-.1137975 -32000
.5035425 -4.552651E-02 .2360212 .3059629
-.1137877 -31999
.5022817 -4.535842E-02 .2362913 .3067854
-.1137903 -31998
Above there is no rounding by hand.
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 six seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
Friday, August 8, 2014
A Unified Computer Program for an Integer Version of Schittkowski's Test Problem 281 but with 9000 General Integer Variables
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve an integer version of Schittkowski's Test Problem 281 [14, p. 105] but with 9000 unknowns instead 10 unknowns. The source of this Test Problem 281 is S. Walukiewicz; see Schittkowski [14]. Thus, the computer program below tries to minimize the following:
9000
[ SIGMA i^3*(X(i)-1)^2 ]^(1/3)
i=1
subject to
-5<= X(i) <=5, X(i) integer, i=1, 2, 3,..., 9000.
One notes line 214 and line 217, which are 214 IF X(J44)<-5 THEN 1670 and 217 IF X(J44)>5 THEN 1670, respectively,
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(9000),X(9000)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 9000
112 A(J44)=-5+ RND*(10)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*9000)
144 GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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
211 FOR J44=1 TO 9000
214 IF X(J44)<-5 THEN 1670
217 IF X(J44)>5 THEN 1670
219 NEXT J44
400 SONE=0
401 FOR J44=1 TO 9000
403 SONE=SONE+J44^3*(X(J44)-1)^2
405 NEXT J44
666 PD1= -SONE^(1/3)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-.5 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10)
1924 PRINT A(11),A(12),A(13),A(14),A(15),A(16),A(17),A(18),A(19),A(20)
1925 PRINT A(4991),A(4992),A(4993),A(4994),A(4995),A(4996),A(4997),A(4998),A(4999),A(5000)
1927 PRINT A(5991),A(5992),A(5993),A(5994),A(5995),A(5996),A(5997),A(5998),A(5999),A(6000)
1928 PRINT A(7991),A(7992),A(7993),A(7994),A(7995),A(7996),A(7997),A(7998),A(7999),A(8000)
1929 PRINT A(8991),A(8992),A(8993),A(8994),A(8995),A(8996),A(8997),A(8998),A(8999),A(9000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-32000 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000
M=0 is optimal; see Schittkowski [14, p.105].
Above there is no rounding by hand.
Of the 9000 A's, only the 60 A's of line 1923, line 1924, line 1925, line 1927, line 1928, and line 1929 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was eight hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
Similar to the computer programs of the preceding papers, the computer program below seeks to solve an integer version of Schittkowski's Test Problem 281 [14, p. 105] but with 9000 unknowns instead 10 unknowns. The source of this Test Problem 281 is S. Walukiewicz; see Schittkowski [14]. Thus, the computer program below tries to minimize the following:
9000
[ SIGMA i^3*(X(i)-1)^2 ]^(1/3)
i=1
subject to
-5<= X(i) <=5, X(i) integer, i=1, 2, 3,..., 9000.
One notes line 214 and line 217, which are 214 IF X(J44)<-5 THEN 1670 and 217 IF X(J44)>5 THEN 1670, respectively,
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(9000),X(9000)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 9000
112 A(J44)=-5+ RND*(10)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*9000)
144 GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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
211 FOR J44=1 TO 9000
214 IF X(J44)<-5 THEN 1670
217 IF X(J44)>5 THEN 1670
219 NEXT J44
400 SONE=0
401 FOR J44=1 TO 9000
403 SONE=SONE+J44^3*(X(J44)-1)^2
405 NEXT J44
666 PD1= -SONE^(1/3)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-.5 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10)
1924 PRINT A(11),A(12),A(13),A(14),A(15),A(16),A(17),A(18),A(19),A(20)
1925 PRINT A(4991),A(4992),A(4993),A(4994),A(4995),A(4996),A(4997),A(4998),A(4999),A(5000)
1927 PRINT A(5991),A(5992),A(5993),A(5994),A(5995),A(5996),A(5997),A(5998),A(5999),A(6000)
1928 PRINT A(7991),A(7992),A(7993),A(7994),A(7995),A(7996),A(7997),A(7998),A(7999),A(8000)
1929 PRINT A(8991),A(8992),A(8993),A(8994),A(8995),A(8996),A(8997),A(8998),A(8999),A(9000)
1939 PRINT M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the computer program's complete output through JJJJ=-32000 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000
M=0 is optimal; see Schittkowski [14, p.105].
Above there is no rounding by hand.
Of the 9000 A's, only the 60 A's of line 1923, line 1924, line 1925, line 1927, line 1928, and line 1929 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-32000 was eight hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.
[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.
[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[18] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
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