Substance: February 2015

General Mixed Integer Nonlinear Programming (MINLP) Solver with a Divide-and-Conquer Strategy To Solve Li and Sun's Problem 14.3 but with n=15110 General Integer Variables

Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but with 15110 unknowns instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

15110-1
(X(1)-1)^2 + ( X(15110)-1)^2 + 15110* SIGMA (15110-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15110.

One notes that the problem above is equivalent to minimize

15110-1
(X(1)-1)^2 + ( X(15110)-1)^2 + 15110* SIGMA (15110-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15110)^7 >= 15110

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15110

and to minimize

15110-1
(X(1)-1)^2 + ( X(15110)-1)^2 + 15110* SIGMA (15110-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15110)^7 <= 15110

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15110.

Then one takes the best produced.

Generally speaking, while directly dealing with an initial problem is advantageous if the initial problem is "small," dealing with the initial problem plus an additional constraint--twice--is advantageous if the initial problem is "large" because each of these two problems has a smaller penalty-free region than that of the initial problem; see line 251, line 257, and line 492 of each of the two computer programs below.

(1) The Additional Constraint Used Immediately Below Is X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15110)^7 >= 15110

The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15113),X(15113)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15110
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15110
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15113)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15110
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15110
228 SFE=SFE+X(J44)^7
233 NEXT J44
251 TSL=-15110+SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 15109
405 SUMNEWZ=SUMNEWZ+ (15110-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15110)-1)^2 -15110* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15110
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15108),A(15109),A(15110),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31997 is shown below:

1 1 1 1 1
0 -32000

1 1 1 1 1
0 -31999

1 1 1 1 0
-4.911188E+11 -31998

1 1 1 1 1
0 -31997

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

M=0 is optimal. See Li and Sun [12, pp. 414-415].

Of the 15110 A's, only the 5 A's of line 1778 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=-31997 was 20 hours.

(2) The Additional Constraint Used Below Is X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15110)^7 <= 15110

While line 251 above is 251 TSL= -15110+SFE, line 251 below is 251 TSL= 15110-SFE.

The following computer program also uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15113),X(15113)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15110
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15110
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15113)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15110
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15110
228 SFE=SFE+X(J44)^7
233 NEXT J44
251 TSL= 15110-SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 15109
405 SUMNEWZ=SUMNEWZ+ (15110-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15110)-1)^2 -15110* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15110
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15108),A(15109),A(15110),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31997 is shown below:

0 0 0 0 1
-4.223938E+11 -32000

0 0 0 0 1
-4.543545E+11 -31999

0 0 0 0 0
-2 -31998

0 0 0 0 0
-2 -31997

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

Of the 15110 A's, only the 5 A's of of line 1778 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=-31997 was 17 hours.

(3) The realized solution with M=0 at JJJJ=-32000, at JJJJ=-31999, and at JJJJ=-31997 and produced through the additional constraint X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15110)^7 >= 15110 is the best produced.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

Acknowledgment

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

References

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

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

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

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

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

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

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

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

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

[10] 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.

[11] 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.

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

[13] 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.

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

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

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

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

[18] 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/

[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[20] 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/

[21] 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

Posted at 03:08 AM | Permalink | Comments (0)

General Mixed Integer Nonlinear Programming (MINLP) Solver with the Help of an Additional Constraint Solving Li and Sun's Problem 14.3 but with n=15100 General Integer Variables

Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but with 15100 unknowns instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

15100-1
(X(1)-1)^2 + ( X(15100)-1)^2 + 15100* SIGMA (15100-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15100.

One notes that the problem above is equivalent to minimize

15100-1
(X(1)-1)^2 + ( X(15100)-1)^2 + 15100* SIGMA (15100-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15100)^7 >= 15100

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15100

and to minimize

15100-1
(X(1)-1)^2 + ( X(15100)-1)^2 + 15100* SIGMA (15100-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15100)^7 <= 15100

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15100.

Then one takes the best produced.

Generally speaking, while dealing with a given problem directly is advantageous if the given problem is "small," dealing with a given problem with an additional constraint--twice--is advantageous if the given problem is "large" because the two problems each with a smaller search region are easier.

(1) The Additional Constraint Used Immediately Below Is X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15100)^7 >= 15100

The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15103),X(15103)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15100
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15100
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15100
228 SFE=SFE+X(J44)^7
233 NEXT J44
251 TSL=-15100+SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 15099
405 SUMNEWZ=SUMNEWZ+ (15100-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15100)-1)^2 -15100* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15098),A(15099),A(15100),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31998 is shown below:

1 1 1 1 1
-1.028267E+10 -32000

1 1 0 0 1
-4.944112E+11 -31999

1 1 1 1 1
0 -31998

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

M=0 is optimal. See Li and Sun [12, pp. 414-415].

Of the 15100 A's, only the 5 A's of of 1778 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through
JJJJ=-31998 was 13 hours.

(2) The Additional Constraint Used Below Is X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15100)^7 <= 15100

While line 251 above is 251 TSL=-15100+SFE, line 251 below is 251 TSL= 15100-SFE.

The following computer program also uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15103),X(15103)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15100
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15103)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15100
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15100
228 SFE=SFE+X(J44)^7
233 NEXT J44
251 TSL= 15100-SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 15099
405 SUMNEWZ=SUMNEWZ+ (15100-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15100)-1)^2 -15100* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15098),A(15099),A(15100),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 [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31996 is shown below:

0 0 1 1 0
-4.28105E+11 -32000

0 0 0 0 0
-2 -31999

-1 1 0 0 1
-4.853535E+11 -31998

0 -1 -1 2 3
-3.763683E+11 -31997

0 -1 0 0 -1
-5.508909E+11 -31996

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

M=0 is optimal. See Li and Sun [12, pp. 414-415].

Of the 15100 A's, only the 5 A's of of 1778 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=-31996 was 23 hours.

(3) The realized solution with M=0 at JJJJ=-31998 and produced through the additional constraint X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15100)^7 >= 15100 is the best produced.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

Acknowledgment

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

References

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

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

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

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

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

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

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

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

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

[10] 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.

[11] 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.

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

[13] 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.

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

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

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

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

[18] 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/

[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[20] 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/

[21] 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

Posted at 08:04 AM | Permalink | Comments (0)

General Mixed Integer Nonlinear Programming (MINLP) Solver Solving Li and Sun's Problem 14.3 but of n=15000 General Integer Variables Subject to an Additional Constraint, X(1) + X(2) + X(3) + ... + X(15000) >= 15000

Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but of n=15000 general integer variables subject to an additional constraint; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is as follows:

Minimize

15000-1
(X(1)-1)^2 + ( X(15000)-1)^2 + 15000* SIGMA (15000-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1) + X(2) + X(3) + ... + X(15000) >= 15000

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15000.

One notes the starting solution vectors of line 111 through line 117, which are as follows:
111 FOR J44=1 TO 15000
114 A(J44)=-3+FIX(RND*6.98)
117 NEXT J44

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.

0 DEFINT J,K,B,X,A
2 DIM A(15003),X(15003)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15000
114 A(J44)=-3+FIX(RND*6.98)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15003)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
170 FOR J44=1 TO 15000
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15000
228 SFE=SFE+X(J44)
233 NEXT J44
251 TSL=-15000+SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 14999
405 SUMNEWZ=SUMNEWZ+ (15000-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15000)-1)^2 -15000* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(2),A(3),A(4),A(5)
1778 PRINT A(14997),A(14998),A(14999),A(15000),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Ver
sion 1.00. See the BASIC manual [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31995 is shown below:

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

1 1 1 2 1
2 2 2 0 -2.798091E+12
-31999

0 0 -1 1 1
2 2 3 3 -2.847327E+12
-31998

0 0 0 0 0
0 0 0 0 -7.5E+13
-31997

0 0 0 0 0
0 0 0 0 -7.5E+13
-31996

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

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

M=0 is optimal. See Li and Sun [12, pp. 414-415].

Of the 15000 A's, only the 9 A's of of line 1773 and line 1778 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=-31995 was 24 hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.

Acknowledgment

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

References

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

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

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

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

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

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

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

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

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

[10] 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.

[11] 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.