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=32110 General Integer Variables

Jsun Yui Wong

The problem here is Li and Sun’s Problem 14.3 but with 32110 unknowns instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski Test Problem 282 [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

32110-1
(X(1)-1)^2 + ( X(32110)-1)^2 + 32110* SIGMA (32110-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32110.

One notes that the problem above is equivalent to minimize

32110-1
(X(1)-1)^2 + ( X(32110)-1)^2 + 32110* SIGMA (32110-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 >= 32110

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32110

and to minimize

32110-1
(X(1)-1)^2 + ( X(32110)-1)^2 + 32110* SIGMA (32110-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 <= 32110

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32110.

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 32110
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(32110)^5 >= 32110

The following computer program uses QB64 [18, 19].

0 DEFINT J,K,B,X,A
2 DIM A(32222),X(32222)

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 32110

114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 100000

129 FOR KKQQ=1 TO 32110

130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*32113)

167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 32110

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 32110

228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL=-32110+SFE

257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 32109

405 SUMNEWZ=SUMNEWZ+ (32110-J44)* ( X(J44)^2-X(J44+1) )^2

407 NEXT J44
411 SONE= – (X(1)-1)^2 – ( X(32110)-1)^2 -32110* SUMNEWZ

492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 32110

1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(32108),A(32109),A(32110),M,JJJJ

1788 PRINT A(1111),A(11111),A(23333),A(27777),A(28888)

1999 NEXT JJJJ

Based on the computer programs in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31998 is shown below:

0 0 1 2 2
-5.593623E+12 -32000
0 1 1 0 1

1 1 1 -1 4
-6.64949E+12 -31999
0 0 1 1 0

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

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 32110 A’s, only the ten A’s of line 1778 and line 1788 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31998 was eight hours.

(2) The Additional Constraint Used Immediately Below Is X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 <= 32110

One notes that line 251 above is 251 TSL=-32110+SFE and that line 251 below is 251 TSL=32110-SFE.
.
The following computer program uses QB64 [18, 19].

0 DEFINT J,K,B,X,A
2 DIM A(32222),X(32222)

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 32110

114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 100000

129 FOR KKQQ=1 TO 32110

130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*32113)

167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 32110

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 32110

228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL=32110-SFE

257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 32109

405 SUMNEWZ=SUMNEWZ+ (32110-J44)* ( X(J44)^2-X(J44+1) )^2

407 NEXT J44
411 SONE= – (X(1)-1)^2 – ( X(32110)-1)^2 -32110* SUMNEWZ

492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 32110

1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(32108),A(32109),A(32110),M,JJJJ

1788 PRINT A(1111),A(11111),A(23333),A(27777),A(28888)

1999 NEXT JJJJ

Based on the computer programs in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31998 is shown below:

0 0 0 1 1
-4.680451E+12 -32000
0 1 1 0 1

0 0 1 1 -1
-5.337537E+12 -31999
0 1 1 -1 1

-1 1 0 0 0
-4.636656E+12 -31998
1 -1 0 0 0

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 32110 A’s, only the ten A’s of line 1778 and line 1788 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31998 was nine hours.

(3) The realized solution with M=0 at JJJJ=-31998, which was produced through the artificial constraint X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 >= 32110 of the first computer program, is the best produced.

For a computer program involving a mix of continuous variables and integer variables, see Wong [21], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, March 08). 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. http://myblogsubstance.typepad.com/substance/2015/03/

General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 32170 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 32170 unknowns instead of their 100 unknowns [12, p. 415]. The function is based on the widely known Rosenbrock function–-see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:

Minimize

32170-1
SIGMA 100* ( X(i+1) – X(i)^2 )^2 + ( 1-X(i) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32170.

One notes the starting solution vectors of line 111 through line 118, which are

111 FOR J44=1 TO 32170
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44

The following computer program uses QB64 [18, 19].

0 DEFINT J,K,B,X,A
2 DIM A(32177),X(32177)
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 32170

116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 64000

129 FOR KQ=1 TO 32170

130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*32173)

167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 32170

173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
401 SONE=0
402 FOR J44=1 TO 32169

411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 32170

1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(32167),A(32168),A(32169),A(32170),M,JJJJ

1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31997 is shown below:

-1      0      -1      2      3
-611      -32000

0      0      0      0      0
-32169      -31999

1      1      1      1      1
0      -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, p. 415].

Of the 32170 A’s, only the 5 A’s of line 1773 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31997 was twelve hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [21], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, February 2). Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 15170 General Integer Variables. https://computerprogramsandresults.wordpress.com/2015/02/02/

General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.5 but Involving 30170 General Integer Variables instead of Their 100 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.5 but with 30170 unknowns instead of their 100 unknowns [12, p. 416]. Specifically, the test example here is as follows:

Minimize

30170                 30170
SIGMA X(i) ^4 + [ SIGMA X(i) ] ^2
i=1                       i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 30170.

One notes the starting solutions vectors of line 111 through line 117, which are as follows:

111 FOR J44=1 TO 30170
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

The following computer program uses QB64 [18, 19].

0 DEFINT J, K, B, X, A
2 DIM A(35173), X(35173)

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 30170

114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 64000
129 FOR KQ = 1 TO 30170

130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 30173)

167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 30170

171 IF X(J44) < LB THEN X(J44) = LB
172 IF X(J44) > UB THEN X(J44) = UB
173 NEXT J44
482 SY = 0
483 FOR J44 = 1 TO 30170

485 SY = SY + X(J44) ^ 4
487 NEXT J44
488 SZ = 0
489 FOR J44 = 1 TO 30170

490 SZ = SZ + X(J44)
491 NEXT J44
492 PD1 = -SY – SZ ^ 2
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 30170

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1779 PRINT A(1), A(30170), M, JJJJ

1788 PRINT A(1111), A(11111), A(22222), A(23333), A(30111)
1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31997 is shown below:

-1      -1      -16430      -32000
0      1      1      0      -1

0      0      -2      -31999
0      0      0      0      0

-1      -1      -15414      -31998
0      0      0      0      -1

0      0      0      -31997
0      0      0      0      0

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, p. 416].

Of the 30170 A’s, only the 7 A’s of line 1779 and line 1788 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31997 was eight hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [21], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, February 9). General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.5 but Involving 15170 General Integer Variables instead of Their 100 General Integer Variables.
http://nonlinearintegerprogrammingsolver.blogspot.ca/2015/02/

Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Computer Program Applied to a Problem with 22120 General Integer Variables and with X(1)^3 + X(2)^3 + X(3)^3 + … + X(22120)^3 = 22120

Jsun Yui Wong

The problem here is based on Li and Sun’s Problem 14.3 [12, 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

22120-1
(X(1)-1)^2 + ( X(22120)-1)^2 + 22120* SIGMA (22120-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^3 + X(2)^3 + X(3)^3 + … + X(22120)^3 = 22120

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 22120.

The equality constraint above is presently added.

One notes the starting solution vectors of line 111 through line 117, which are as follows:
111 FOR J44 = 1 TO 22120
114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44.

The following computer program uses QB64 [18, 19].

0 DEFINT J, K, B, X, A
2 DIM A(33123), X(33123)
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 22120
114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 32000
129 FOR KKQQ = 1 TO 22120
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 22123)
167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 22120
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 22120
228 SFE = SFE + X(J44) ^ 3
233 NEXT J44
251 TSL = -22120 + 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 22119
405 SZ = SZ + (22120 – J44) * (X(J44) ^ 2 – X(J44 + 1)) ^ 2
407 NEXT J44
411 SONE = -(X(1) – 1) ^ 2 – (X(22120) – 1) ^ 2 – 22120 * SZ
492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 22120
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(22118), A(22119), A(22120), M, JJJJ
1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31967 is shown below:

-2 3 0 -1 -3
-3.270111E+14 -32000

0 0 0 0 0
-1.106998E+14 -31999

-1 2 2 0 2
-1.145444E+14 -31998

2 1 2 -4 2
-9.9405E+13 -31997

0 0 0 0 0
-1.106053E+14 -31996

0 0 0 0 0
-1.106484E+14 -31995

0 0 0 0 0
-1.106E+14 -31994

-2 -5 -2 0 3
-3.834372E+14 -31993

0 0 0 0 0
-1.106055E+14 -31992

-3 -2 -2 -2 -1
-4.581762E+14 -31991

0 0 0 0 0
-1.106E+14 -31990

-1 -1 0 0 -1
-3.29308E+13 -31989

0 0 0 0 0
-1.106E+14 -31988

-2 5 4 -1 -2
-1.844149E+14 -31987

2 -1 2 -2 0
-9.990742E+13 -31986

1 1 1 1 1
-1.655205E+10 -31985

1 1 1 1 1
-1.017886E+10 -31984

0 0 0 0 0
-1.106165E+14 -31983

0 0 0 0 0
-1.106E+14 -31982

2 2 0 2 -3
-8.414546E+13 -31981

1 1 1 1 1
-5.645041E+09 -31980

1 1 1 1 1
-5.033556E+09 -31979

0 -3 -3 1 -3
-2.652683E+14 -31978

-1 -1 -1 1 -1
-3.282038E+13 -31977

0 0 0 0 0
-1.106054E+14 -31976

1 1 1 1 1
-5.687512E+09 -31975

0 0 1 1 0
-1.266092E+13 -31974

-4 -4 1 3 3
-3.729486E+14 -31973

2 2 1 -1 0
-6.559005E+13 -31972

2 1 -2 2 2
-1.006949E+14 -31971

0 -1 -1 -3 4
-7.753536E+14 -31970

1 3 3 -1 2
-3.800816E+14 -31969

-2 -2 -1 2 2
-1.001469E+14 -31968

1 1 1 1 1
0 -31967

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 22120 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 with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31967 was 26 hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [21], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, March 11). Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Computer Program Applied to a Problwm with 15120 General Integer Variables and with X(1)^9 + X(2)^9 + X(3)^9 + … + X(15120)^9 = 15120. http://myblogsubstance.typepad.com/substance/2015/03/

General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.5 but Involving 21170 General Integer Variables instead of Their 100 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.5 but with 21170 unknowns instead of their 100 unknowns [12, p. 416]. Specifically, the test example here is as follows:

Minimize

21170                  21170
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1                        i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 21170.

One notes the starting solutions vectors of line 111 through line 117, which are as follows:

111 FOR J44=1 TO 21170
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44

The following computer program uses QB64 [18, 19].

0 DEFINT J,K,B,X,A
2 DIM A(35173),X(35173)

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 21170
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 21170
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*21173)

167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 21170
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
482 SY=0
483 FOR J44=1 TO 21170
485 SY=SY+X(J44)^4
487 NEXT J44
488 SZ=0
489 FOR J44=1 TO 21170
490 SZ=SZ+ X(J44)
491 NEXT J44
492 PD1=-SY-SZ^2
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 21170
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1779 PRINT A(1),A(21170),M,JJJJ
1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31994 is shown below:

-1    -1    -11568    -32000

1    -1    -9980    -31999

0    -1    -13730    -31998

1    -1    -15140    -31997

0    0    -2    -31996

0    0    0    -31995

0    -1    -13776    -31994

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, p. 416].

Of the 21170 A’s, only the 2 A’s of line 1779 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31994 was three hours and a half.

For a computer program involving a mix of continuous variables and integer variables, see Wong [21], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, February 9). General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.5 but Involving 15170 General Integer Variables instead of Their 100 General Integer Variables.
http://nonlinearintegerprogrammingsolver.blogspot.ca/2015/02/

General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 22170 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 22170 general integer variables instead of their 100 general integer variables [12, p. 415]. The function is based on the widely known Rosenbrock function–-see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:

Minimize

22170-1
SIGMA 100* ( X(i+1) – X(i)^2 )^2 + ( 1-X(i) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 22170.

One notes line 111 through line 118, which are
111 FOR J44=1 TO 22170
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44.

The following computer program uses QB64 [18, 19].

0 DEFINT J,K,B,X,A
2 DIM A(33173),X(33173)
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 22170

116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 22170

130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*22173)

167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 22170

173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
401 SONE=0
402 FOR J44=1 TO 22169

411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 22170

1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(22167),A(22168),A(22169),A(22170),M,JJJJ

1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31995 is shown below:

-1 1 1 2 3
-608      -32000

1 1 1 2 4
-302      -31999

0 0 0 0 0
-22778      -31998

0 0 0 0 0
-22169      -31997

1 1 1 1 1
0      -31996

1 1 1 2 3
-405      -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, p. 415].

Of the 22170 A’s, only the 5 A’s of line 1773 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31995 was seven hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [21], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, February 2). Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 15170 General Integer Variables. https://computerprogramsandresults.wordpress.com/2015/02/02/

A General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 21170 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 21170 general integer variables instead of their 100 general integer variables [12, p. 415]. The function is based on the widely known Rosenbrock function–-see, for example, Li and Sun [12, p. 415] and Schittkowski [16, Test Problems 294-299]. Specifically, the test example here is as follows:

Minimize

21170-1
SIGMA 100* ( X(i+1) – X(i)^2 )^2 + ( 1-X(i) )^2
i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 21170.

One notes line 111 through line 118, which are
111 FOR J44=1 TO 21170
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44.

The following computer program uses QB64 [18, 19].

0 DEFINT J,K,B,X,A
2 DIM A(33173),X(33173)
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 21170
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44
128 FOR I=1 TO 32000
129 FOR KQ=1 TO 21170
130 X(KQ)=A(KQ)
131 NEXT KQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*21173)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
171 FOR J9=1 TO 21170
173 IF X(J9)<LB THEN X(J9)=LB
175 IF X(J9)>UB THEN X(J9)=UB
177 NEXT J9
401 SONE=0
402 FOR J44=1 TO 21169
411 SONE=SONE+ 100* ( X(J44+1) – X(J44)^2 )^2 + ( 1-X(J44) )^2
421 NEXT J44
689 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KX=1 TO 21170
1455 A(KX)=X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1),A(21167),A(21168),A(21169),A(21170),M,JJJJ
1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31994 is shown below:

1 1 1 1 1
-205      -32000

0 0 0 0 0
-21169      -31999

-1 1 1 1 1
-4      -31998

-1 1 1 1 1
-4      -31997

1 1 1 1 1
0      -31996

1 1 1 1 1
0      -31995

1 1 1 1 1
0      -31994

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, p. 415].

Of the 21170 A’s, only the 5 A’s of line 1773 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31994 was eight hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [22], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2014, February 2). Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve a Nonlinear Integer Programming Problem Involving 15170 General Integer Variables. https://computerprogramsandresults.wordpress.com/2015/02/02/

Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Computer Program Applied to a Problem with 21120 General Integer Variables and with X(1)^3 + X(2)^3 + X(3)^3 + … + X(21120)^3 = 21120

Jsun Yui Wong

The problem here is based on Li and Sun’s Problem 14.3 [12, 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

21120-1
(X(1)-1)^2 + ( X(21120)-1)^2 + 21120* SIGMA (21120-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^3 + X(2)^3 + X(3)^3 + … + X(21120)^3 = 21120

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 21120.

The equality constraint above is presently added.

One notes the starting solution vectors of line 111 through line 117, which are as follows:
111 FOR J44 = 1 TO 21120
114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44.

The following computer program uses QB64 [18, 19].

0 DEFINT J, K, B, X, A
2 DIM A(33333), X(33333)
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 21120

114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 32000
129 FOR KKQQ = 1 TO 21120
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 21123)

167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 21120

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 21120

228 SFE = SFE + X(J44) ^ 3
233 NEXT J44
251 TSL = -21120 + 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 21119

405 SZ = SZ + (21120 – J44) * (X(J44) ^ 2 – X(J44 + 1)) ^ 2
407 NEXT J44
411 SONE = -(X(1) – 1) ^ 2 – (X(21120) – 1) ^ 2 – 21120 * SZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 21120
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(21118), A(21119), A(21120), M, JJJJ

1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31991 is shown below:

-1 3 2 -1 3
-2.873852E+14      -32000

1 1 1 1 1
-5.598013E+09      -31999

1 0 0 1 0
-9.48158E+12      -31998

1 2 -2 2 2
-8.695591E+13      -31997

0 0 0 0 0
-1.056E+14      -31996

1 1 1 1 1
-5.626483E+09      -31995

0 0 0 0 0
-1.056051E+14      -31994

-3 2 2 -4 2
-9.973437E+13      -31993

2 -4 3 -1 1
-2.902891E+14      -31992

1 1 1 1 1
0      -31991

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 21120 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 with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31991 was eight hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [22], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, March 11). Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Computer Program Applied to a Problwm with 15120 General Integer Variables and with X(1)^9 + X(2)^9 + X(3)^9 + … + X(15120)^9 = 15120. http://myblogsubstance.typepad.com/substance/2015/03/

Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Computer Program Applied to a Problem with 20120 General Integer Variables and with X(1)^3 + X(2)^3 + X(3)^3 + … + X(20120)^3 = 20120

Jsun Yui Wong

The problem here is based on Li and Sun’s Problem 14.3 [12, 414-415], which is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

20120-1
(X(1)-1)^2 + ( X(20120)-1)^2 + 20120* SIGMA (20120-i)* ( X(i)^2-X(i+1) )^2
i=1

subject to

X(1)^3 + X(2)^3 + X(3)^3 + … + X(20120)^3 = 20120

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 20120.

The following computer program uses QB64 [18, 19].

0 DEFINT J, K, B, X, A
2 DIM A(33333), X(33333)

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 20120

114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 32000
129 FOR KKQQ = 1 TO 20120

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 20123)

167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 20120

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 20120

228 SFE = SFE + X(J44) ^ 3
233 NEXT J44
251 TSL = -20120 + 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 20119

405 SZ = SZ + (20120 – J44) * (X(J44) ^ 2 – X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) – 1) ^ 2 – (X(20120) – 1) ^ 2 – 20120 * SZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 20120

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(20118), A(20119), A(20120), M, JJJJ

1999 NEXT JJJJ

Based on the computer program in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31990 is shown below:

2 3 3 -1 0
-2.478626E+14 -32000

0 2 0 -2 2
-5.006566E+13 -31999

0 0 0 0 0
-1.006054E+14 -31998

-5 2 0 -2 0
-7.411186E+14 -31997

-3 -3 2 0 -2
-3.459166E+14 -31996

0 0 0 0 0
-1.006E+14 -31995

5 -4 3 -1 5
-5.799339E+14 -31994

0 0 0 0 0
-1.006057E+14 -31993

-1 -1 -1 0 1
-2.005778E+13 -31992

-3 3 2 -1 -3
-2.925154E+14 -31991

1 1 1 1 1
0 -31990

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 20120 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 with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31990 was six hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [22], 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. http://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. http://www.sfu.ca/~ssurjano/zakharov.html

[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”.

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[20] 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/

[21] 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/

[22] 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/

[23] 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

[24] Jsun Yui Wong (2015, March 11). Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Computer Program Applied to a Problwm with 15120 General Integer Variables and with X(1)^9 + X(2)^9 + X(3)^9 + … + X(15120)^9 = 15120. http://myblogsubstance.typepad.com/substance/2015/03/