Substance

A Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Solver Using Microsoft GW-BASIC Interpreter and Applied to a Problem with 15120 General Integer Variables and with X(1)^3 + X(2)^3 + X(3)^3 + ... + X(15120)^3 = 15120

Jsun Yui Wong

The problem here is based on Li and Sun's Problem 14.3, which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

                                                                   15120-1
(X(1)-1)^2 + ( X(15120)-1)^2 + 15120* SIGMA (15120-i)* ( X(i)^2-X(i+1) )^2
                                                                          i=1

subject to

X(1)^3 + X(2)^3 + X(3)^3 + ... + X(15120)^3 = 15120

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15120.

One notes line 111 through line 117, which are
111 FOR J44=1 TO 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

One also notes line 225 through line 233, which are

225 FOR J44=1 TO 15120
228 SFE=SFE+X(J44)^3
233 NEXT J44.

The following computer program uses Microsoft's GW-BASIC 3.11 interpreter.

0 DEFINT J,K,B,X,A
2 DIM A(15123),X(15123)
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 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 15120
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15123)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15120
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 15120
228 SFE=SFE+X(J44)^3
233 NEXT J44
251 TSL=-15120+SFE
267 IF TSL=0 THEN TSL=TSL ELSE TSL=-ABS(TSL)
400 SZ=0
403 FOR J44=1 TO 15119
405 SZ=SZ+ (15120-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15120)-1)^2 -15120* SZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15120
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15118),A(15119),A(15120),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with basica/D of Microsoft's GW-BASIC 3.11 interpreter for Dos. See the BASIC manual [13]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31996 is shown below:

0      0      0      2      0
-4.486499E+12      -32000

0      1      0      0      2
-5.400013E+12      -31999

0      0      0      0      0
-7.56E+13      -31998

1      3      3      1      3
-1.236227E+14      -31997

1      1      1      1      1
0      -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 15120 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 basica/D interpreter, version GW-BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31996 was 34 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.

[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

Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Computer Program Applied to a Problem with 15120 General Integer Variables and with X(1)^9 + X(2)^9 + X(3)^9 + ... + X(15120)^9 = 15120

Jsun Yui Wong

The problem here is based on Li and Sun's Problem 14.3, which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

                                                              15120-1
(X(1)-1)^2 + ( X(15120)-1)^2 + 15120* SIGMA (15120-i)* ( X(i)^2-X(i+1) )^2
                                                                  i=1

subject to

X(1)^9 + X(2)^9 + X(3)^9 + ... + X(15120)^9 = 15120

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15120.

One notes line 221 through line 267, which are
111 FOR J44=1 TO 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

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(15123),X(15123)
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 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15120
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15123)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15120
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 15120
228 SFE=SFE+X(J44)^9
233 NEXT J44
251 TSL=-15120+SFE
257 REM IF TSL<0 THEN TSL=TSL ELSE TSL=0
267 IF TSL=0 THEN TSL=TSL ELSE TSL=-ABS(TSL)
400 SZ=0
403 FOR J44=1 TO 15119
405 SZ=SZ+ (15120-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15120)-1)^2 -15120* SZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15120
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15118),A(15119),A(15120),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Verson 1.00. See the BASIC manual [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31988 is shown below:

1      1      1      1      1
-5.063338E+09      -32000

0      0      0      0      0
-7.56E+13      -31999

0      0      0      0      0
-7.560546E+13      -31998

0      1      -1      -1      -2
-3.199152E+13     -31997

1     1     1     1     1
-1.045414E+10      -31996

1      -2      2      -2      2
-2.603691E+13      -31995

0      0      0      0      0
-1.269369E+11      -31994

0      0      0      1      0
-1.91615E+11      -31993

4      4      4      -4      3
-2.045462E+14      -31992

-1      -1      -1      0      -2
-6.993109E+13      -31991

1      1      1      1      1
-1.056246E+10      -31990

-1      -1      -1      -1      -1
-5.560215E+12      -31989

1      1      1      1      1
0      -31988

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 15120 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=-31988 was 39 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.

[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

Nonconvex Mixed Integer Nonlinear Programming (MINLP) Computer Programs 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)^5 + X(2)^5 + X(3)^5 + ... + X(15110)^5 >= 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)^5 + X(2)^5 + X(3)^5 + ... + X(15110)^5 <= 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.

One notes line 221 through line 233, which are
221 SFE=0
225 FOR J44=1 TO 15110
228 SFE=SFE+X(J44)^5
233 NEXT J44.

(1) The Additional Constraint Used Immediately Below Is X(1)^5 + X(2)^5 + X(3)^5 + ... + X(15110)^5 >= 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)^5
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
-7.29404E+11      -31998

1      1      0      -1      0
-7.191392E+11      -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 17 hours.

(2) The Additional Constraint Used Below Is X(1)^5 + X(2)^5 + X(3)^5 + ... + X(15110)^5 <= 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)^5
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=-31999 is shown below:

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

0      0      0      0      1
-4.56575E+11       -31999

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=-31999 was 13 hours.

(3) The realized solution with M=0 at JJJJ=-32000 and at JJJJ=-31999 and produced through the additional constraint X(1)^5 + X(2)^5 + X(3)^5 + ... + X(15110)^5 >= 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 MonthlyThe , 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

Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Solver Applied to a Problwm with 15120 General Integer Variables and with X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15120)^7 = 15120

Jsun Yui Wong

The problem here is based on Li and Sun's Problem 14.3, which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

                                                                       15120-1
(X(1)-1)^2 + ( X(15120)-1)^2 + 15120* SIGMA (15120-i)* ( X(i)^2-X(i+1) )^2
                                                                             i=1

subject to

X(1)^7 + X(2)^7 + X(3)^7 + ... + X(15120)^7 = 15120

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15120.

One notes line 111 through line 117, which are
111 FOR J44=1 TO 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

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(15123),X(15123)
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 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15120
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15123)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15120
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 15120
228 SFE=SFE+X(J44)^7
233 NEXT J44
251 TSL=-15120+SFE
257 REM IF TSL<0 THEN TSL=TSL ELSE TSL=0
267 IF TSL=0 THEN TSL=TSL ELSE TSL=-ABS(TSL)
400 SZ=0
403 FOR J44=1 TO 15119
405 SZ=SZ+ (15120-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15120)-1)^2 -15120* SZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15120
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15118),A(15119),A(15120),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=-31987 is shown below:

1      1      1      1      1
-5.063338E+09      -32000

0      0      0      0      0
-7.56E+13      -31999

0      0      0      0      0
-7.560546E+13      -31998

0      -2      0      0      -2
-4.624016E+13      -31997

4      -4      -2      -1      1
-2.052905E+14      -31996

0      0      0      0      0
-3.566939E+11      -31995

1      1      1      1      1
-1.551204E+10      -31994

0      0      0      0      0
-1.340931E+11      -31993

0      0      0      0      0
-7.56E+13      -31992

-3      -4      -1      4      -1
-2.06143E+14      -31991

-2      -2      -2      -2      -1
-4.624902E+13      -31990

0      -3      -1      3      -3
-1.095236E+14      -31989

1      1      1      1      1
-2.076547E+10      -31988

1      1      1      1      1
0      -31987

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 15120 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=-31987 was 37 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.

[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

Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Solver Applied to a Problwm with 15120 General Integer Variables and with X(1)^11 + X(2)^11 + X(3)^11 + ... + X(15120)^11 = 15120

Jsun Yui Wong

The problem here is based on Li and Sun's Problem 14.3, which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

                                                                            15120-1
(X(1)-1)^2 + ( X(15120)-1)^2 + 15120* SIGMA (15120-i)* ( X(i)^2-X(i+1) )^2
                                                                                  i=1

subject to

X(1)^11 + X(2)^11 + X(3)^11 + ... + X(15120)^11 = 15120

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15120.

One notes line 111 through line 117, which are
111 FOR J44=1 TO 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

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(15123),X(15123)
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 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15120
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15123)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15120
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 15120
228 SFE=SFE+X(J44)^11
233 NEXT J44
251 TSL=-15120+SFE
257 REM IF TSL<0 THEN TSL=TSL ELSE TSL=0
267 IF TSL=0 THEN TSL=TSL ELSE TSL=-ABS(TSL)
400 SZ=0
403 FOR J44=1 TO 15119
405 SZ=SZ+ (15120-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15120)-1)^2 -15120* SZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15120
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15118),A(15119),A(15120),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=-31995 is shown below:

1      1      1      1      1
-5.063338E+09      -32000

0      0      0      0      0
-7.56E+13      -31999

0      0      0      0      0
-7.560546E+13      -31998

0      0      -1      -1      -2
-2.551876E+13      -31997

0      0      0      0      0
-1.814342E+11      -31996

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 15120 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=-31995 was 21 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.

[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

Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Solver Applied to a Problwm with 15110 General Integer Variables

Jsun Yui Wong

The problem here is based on Li and Sun's Problem 14.3, which 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

X(1)^11 + X(2)^11 + X(3)^11 + ... + X(15110)^11 >= 15110

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

One notes line 221 through line 233, which are
221 SFE=0
225 FOR J44=1 TO 15110
228 SFE=SFE+X(J44)^11
233 NEXT J44.

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)^11
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
-4.726992E+11      -31998

0      0      0      0      1
-4.358813E+11      -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 19 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

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[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