Substance

Solving A Large Nonlinear Integer Programming Test Problem of 32760 General Integer Variables with an Assist from Two Artificial Constraints

Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but with 32760 unknowns instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski Test Problem 282 [17, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize

                                                                             32760-1
(X(1)-1)^2 + ( X(32760)-1)^2 + 32760* SIGMA (32760-i)* ( X(i)^2-X(i+1) )^2
                                                                                    i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32760.

One notes that the problem above is equivalent to minimize

                                                                               32760-1
(X(1)-1)^2 + ( X(32760)-1)^2 + 32760* SIGMA (32760-i)* ( X(i)^2-X(i+1) )^2
                                                                                    i=1

subject to

X(1) + X(2) + X(3) + ... + X(32760) >= 32750

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32760

and to minimize

                                                                                 32760-1
(X(1)-1)^2 + ( X(32760)-1)^2 + 32760* SIGMA (32760-i)* ( X(i)^2-X(i+1) )^2
                                                                                        i=1

subject to

X(1) + X(2) + X(3) + ... + X(32760) <= 32750

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32760.

And then one takes the better of the two.

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 257, which are
221 SFE = 0
225 FOR J44 = 1 TO 32760
228 SFE = SFE + X(J44)
233 NEXT J44
251 TSL = -32750 + SFE
257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0.

(1) The Additional Constraint Used Immediately Below Is X(1) + X(2) + X(3) + ... + X(32760) >= 32750

The following computer program uses QB64 [19, 20].

0 DEFINT J, K, B, X, A
2 DIM A(32763), X(32763)

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 32760

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

129 FOR KKQQ = 1 TO 32760

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32760

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 32760

 

228 SFE = SFE + X(J44)

233 NEXT J44
251 TSL = -32750 + SFE

 

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32759

405 SUMNEWZ = SUMNEWZ + (32760 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32760) - 1) ^ 2 - 32760 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32760

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(3), A(32509), A(32760), M, JJJJ

1788 PRINT A(1111), A(11111), A(32757), A(32758), A(32759)

1999 NEXT JJJJ

Modelled after the computer programs in Wong [25], this BASIC computer program was run with QB64 [19, 20]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31992 is shown below:

-1      1      1      2      3
-1.705676E+13      -32000
0      1      0      2      2

0 0 0 1 2
-1.309933E+13      -31999
1 1 2 2 2

1 1 2 2 1
-2.028592E+13      -31998
1 1 0 2 3

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

0 0 1 2 0
-2.072362E+13      -31996
1 2 1 4 2

1 1 1 1 1
-1.790793E+09      -31995
1 1 1 1 1

1 1 1 0 0
-2.038182E+13      -31994
2 1 2 1 2

1 1 1 1 0
-1.426115E+13      -31993
1 1 2 2 2

-1 1 1 2 1
-1.493485E+13      -31992
1 1 0 2 2

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

At JJJJ=-31997, M=0 is optimal. See Li and Sun [12, pp. 414-415].

Of the 32760 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 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31992 was 21 hours.

(2) The Additional Constraint Used Immediately Below Is X(1) + X(2) + X(3) + ... + X(32760) <= 32750

One notes that line 251 above is 251 TSL=-32750+SFE and that line 251 below is 251 TSL=32750-SFE.

The following computer program uses QB64 [19, 20].

0 DEFINT J, K, B, X, A
2 DIM A(32763), X(32763)

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 32760

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

129 FOR KKQQ = 1 TO 32760

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32760

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 32760

 

228 SFE = SFE + X(J44)

 

233 NEXT J44
251 TSL = 32750 - SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32759

405 SUMNEWZ = SUMNEWZ + (32760 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32760) - 1) ^ 2 - 32760 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32760

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(3), A(32509), A(32760), M, JJJJ

1788 PRINT A(1111), A(11111), A(32757), A(32758), A(32759)

1999 NEXT JJJJ

Modelled after the computer programs in Wong [25], this BASIC computer program was run with QB64 [19, 20]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31996 is shown below:

-1      1      1      1      3
-4.967824E+12      -32000
0      1      -1      1      2

0      0      0      0      0
-2      -31999
0      0      0      0      0

0 0 0 0 -4
-4.725988E+12      -31998
1 1 -1 1 0

-1 1 1 0 3
-4.616374E+12      -31997
1 1 0 -1 -1

0 -1 1 1 1
-5.744561E+12      -31996
1 0 -1 1 1

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

Although not optimal, the solution at JJJJ=-31999 with M=-2 is noteworthy.

Of the 32760 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=-31996 was nine hours.

(3) The realized solution with M=0 at JJJJ=-31997, which was produced through the artificial constraint X(1) + X(2) + X(3) + ... + X(32760) >= 32750
of the first computer program, is the best produced.

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. 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] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operatons Research, Volume 14 (1988), Issue 1, pp.. 291-304.

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

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

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

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

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

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

[21] 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/

[22] 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/

[23] 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/

[24] 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

[25] 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/

 

Solving a Large Nonlinear Integer Programming Test Problem of 32760 General Integer Variables, Fourth Edition

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 32760 unknowns instead of their 100 unknowns [12, pp. 415-417]. The function is based on the widely known Rosenbrock function–-see, for example, Li and Sun [12, p. 415] and Schittkowski [17, Test Problems 294-299]. Specifically, the test example here is as follows:

Minimize

32760-1
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,…, 32760.

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

111 FOR J44=1 TO 32760
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44.

While line 128 of the third edition is 128 FOR I = 1 TO 100000, here line 128 is 128 FOR I = 1 TO 90000.

The following computer program uses QB64 [19, 20].

0 DEFINT J, K, B, X, A

2 DIM A(32763), X(32763)

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 32760

116 A(J44) = -5 + FIX(RND * 11)
118 NEXT J44
128 FOR I = 1 TO 90000

129 FOR KQ = 1 TO 32760

130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
171 FOR J9 = 1 TO 32760

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 32759

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 32760

1455 A(KX) = X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1), A(32167), A(32168), A(32709), A(32760), M, JJJJ
1783 PRINT A(111), A(30167), A(31168), A(32758), A(32759)

1999 NEXT JJJJ

Modelled after the computer program in Wong [25], this BASIC computer program was run with QB64 [19, 20]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31991 is shown below:

-1      1      1      1      3
-205      -32000
1      1      1      1      2

-1      1      1      1      1
--205      -31999
1      1      1      1      1

-1      1      1      1      1
-4      -31998
1      1      1      1      1

0      0      0      0      0
-32759      -31997
0      0      0      0      0

0      0      0      0      0
-32759      -31996
0      0      0      0      0

-1      1      1      1      1
-4    -31995
1      1      1      1      1

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

-1      1      1      1      1
-4      -31993
1      1      1      1      1

0      0      0      0      0
-32759      -31992
0      0      0      0      0

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

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

At JJJJ=-31994 and JJJJ=-31991, M=0, which is optimal. See Li and Sun [12, p. 415].

Of the 32760 A's, only the 10 A's of line 1773 and line 1783 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 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31991 was 24 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. 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] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operatons Research, Volume 14 (1988), Issue 1, pp.. 291-304.

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

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

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

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

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

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

[21] 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/

[22] 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/

[23] 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/

[24] 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

[25] 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/

Solving a Large Nonlinear Integer Programming Problem of 32760 General Integer Variables, Fourth Edition

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun's Problem 14.5 but with 32760 unknowns instead of their 100 unknowns [12, pp. 416-417]. Specifically, the test example here is as follows:

Minimize

32760                     32760
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,..., 32760.

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

111 FOR J44=1 TO 32760
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

While line 128 of the third edition is 128 FOR I = 1 TO 90000, here line 128 is 128 FOR I = 1 TO 150000.

The following computer program uses QB64 [19, 20].

0 DEFINT J, K, B, X, A
2 DIM A(32763), X(32763)

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 32760

114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 150000

129 FOR KQ = 1 TO 32760

130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32760

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 32760

485 SY = SY + X(J44) ^ 4
487 NEXT J44
488 SZ = 0
489 FOR J44 = 1 TO 32760

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 32760

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I

1781 PRINT A(1), A(2), A(3), A(222), A(32760), M, JJJJ

1788 PRINT A(1111), A(11111), A(32757), A(32758), A(32759)

1999 NEXT JJJJ

Modelled after the computer program in Wong [25], this BASIC computer program was run with QB64 [19, 20]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31990 is shown below:

-1 1 0 1 0
-17730      -32000
0 1 -1 -1 0

0 0 0 0 0
0      -31999
0 0 0 0 0

1 0 -1 1 0
-19210      -31998
0-1010

1 1 -1 1 -1
-23528      -31997
-1 -1 -1 1 1

0 0 0 0 0
0      -31996
0 0 0 0 0

0 1 0 -1 1
-16988      -31995
-1 0 -1 -1 0

0 1 1 1 1
-21630      -31994
1 1 1 -1 1

0 -1 -1 -1 1
-23636      -31993
1 0 -1 1 1

0 0 0 0 0
0      -31992
0 0 0 0 0

0 0 0 0 0
0      -31991
0 0 0 0 0

1 -1 0 1 0
-17772      -31990
0 0 1 -1 -1

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

At JJJJ=-31999, JJJJ=-31996, JJJJ=-31992, and JJJJ=-31991, M=0, which is optimal. See Li and Sun [12, p. 416].

Of the 32760 A's, only the 10 A's of line 1781 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 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31990 was 14 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. 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] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operatons Research, Volume 14 (1988), Issue 1, pp.. 291-304.

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

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

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

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

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

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

[21] 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/

[22] 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/

[23] 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/

[24] 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

[25] Jsun Yui Wong (2015, February 11). General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun's Problem 14.5 but Involving 15120 General Integer Variables instead of Their 100 General Integer Variables. http://myblogsubstance.typepad.com/substance/2015/02/

Solving a Large Nonlinear Integer Programming Problem of 32760 General Integer Variables, Third Edition

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun's Problem 14.5 but with 32760 unknowns instead of their 100 unknowns [12, pp. 416-417]. Specifically, the test example here is as follows:

Minimize

32760                      32760
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,..., 32760.

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

111 FOR J44=1 TO 32760
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

While line 128 of the second edition is 128 FOR I = 1 TO 100000, here line 128 is 128 FOR I = 1 TO 90000.

The following computer program uses QB64 [19, 20].

0 DEFINT J, K, B, X, A
2 DIM A(32763), X(32763)

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 32760

114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 90000

129 FOR KQ = 1 TO 32760

130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32760

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 32760

485 SY = SY + X(J44) ^ 4
487 NEXT J44
488 SZ = 0
489 FOR J44 = 1 TO 32760

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 32760

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I

1781 PRINT A(1), A(2), A(3), A(222), A(32760), M, JJJJ

1788 PRINT A(1111), A(11111), A(32757), A(32758), A(32759)

1999 NEXT JJJJ

Modelled after the computer program in Wong [25], this BASIC computer program was run with QB64 [19, 20]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31988 is shown below:

-1 1 0 1 0
-17730      -32000
0 1 -1 -1 0

-1 1 0 -1 0
-19422      -31999
0 1 -1 -1 -1

0 0 0 0 0
-17808      -31998
0 1 -1 1 1

0 0 1 -1 0
-17552      -31997
0 1 0 0 0

-1 0 1 0 -1
-19444      -31996
-1 1 -1 -1 -1

1 1 0 -1 1
-29658      -31995
-1 1 1 1 -1

0 -1 0 -1 1
-17750      -31994
1 0 -1 0 0

1 0 0 -1 1
-23528      -31993
0 -1 1 1 0

0 1 0 0 1
-21584      -31992
-1 -1 -1 -1 1

1 -1 0 0 0
-18690      -31991
1 0 0 -1 -1

0 0 0 0 0
-6      -31990
0 0 0 0 0

0 0 0 0 0
0      -31989
0 0 0 0 0

0 0 0 0 0
-2      -31988
0 0 0 0 0

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

At JJJJ=-31989, M=0, which is optimal. See Li and Sun [12, p. 416].

Of the 32760 A's, only the 10 A's of line 1781 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 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31988 was 24 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. 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] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operatons Research, Volume 14 (1988), Issue 1, pp.. 291-304.

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

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

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

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

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

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

[21] 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/

[22] 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/

[23] 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/

[24] 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

[25] Jsun Yui Wong (2015, February 11). General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun's Problem 14.5 but Involving 15120 General Integer Variables instead of Their 100 General Integer Variables. http://myblogsubstance.typepad.com/substance/2015/02/

 

Solving a Large Nonlinear Integer Programming Problem of 32760 General Integer Variables, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun's Problem 14.5 but with 32760 unknowns instead of their 100 unknowns [12, pp. 416-417]. Specifically, the test example here is as follows:

Minimize

32760                     32760
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,..., 32760.

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

111 FOR J44=1 TO 32760
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

The following computer program uses QB64 [19, 20].

0 DEFINT J, K, B, X, A
2 DIM A(32763), X(32763)

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 32760

114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KQ = 1 TO 32760

130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32760

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 32760

485 SY = SY + X(J44) ^ 4
487 NEXT J44
488 SZ = 0
489 FOR J44 = 1 TO 32760

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 32760

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I

1781 PRINT A(1), A(2), A(3), A(222), A(32760), M, JJJJ

1788 PRINT A(1111), A(11111), A(32757), A(32758), A(32759)

1999 NEXT JJJJ

Modelled after the computer program in Wong [25], this BASIC computer program was run with QB64 [19, 20]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31995 is shown below:

-1      1      0      1      0
-17730      -32000
0      1      -1      -1      0

0      0      -1      0      -1
-18706      -31999
0      1      0      -1      0

-1      1      -1      -1      1
-18738      -31998
0      0      0      0      0

1      -1      0      1      0
-16668      -31997
0      0      0      -1      -1

0      0      0      0      0
0      -31996
0      0      0      0      0

0      0      0      0      0
0      -31995
0      0      0      0      0

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

At JJJJ=-31996 and at JJJJ=-31995, M=0, which is optimal. See Li and Sun [12, p. 416].

Of the 32760 A's, only the 10 A's of line 1781 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 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31995 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. 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] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operatons Research, Volume 14 (1988), Issue 1, pp.. 291-304.

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

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

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

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

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

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

[21] 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/

[22] 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/

[23] 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/

[24] 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

[25] Jsun Yui Wong (2015, February 11). General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun's Problem 14.5 but Involving 15120 General Integer Variables instead of Their 100 General Integer Variables. http://myblogsubstance.typepad.com/substance/2015/02/

Solving a Large Nonlinear Integer Programming Test Problem of 32760 General Integer Variables, Third Edition

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 32760 unknowns instead of their 100 unknowns [12, pp. 415-417]. The function is based on the widely known Rosenbrock function–-see, for example, Li and Sun [12, p. 415] and Schittkowski [17, Test Problems 294-299]. Specifically, the test example here is as follows:

Minimize

32760-1
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,…, 32760.

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

111 FOR J44=1 TO 32760
116 A(J44)=-5+FIX(RND*11)
118 NEXT J44.

While line 128 of the Second Edition is 128 FOR I = 1 TO 125000, here line 128 is 128 FOR I = 1 TO 100000.

The following computer program uses QB64 [19, 20].

0 DEFINT J, K, B, X, A

2 DIM A(32763), X(32763)

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 32760

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

129 FOR KQ = 1 TO 32760

130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
171 FOR J9 = 1 TO 32760

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 32759

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 32760

1455 A(KX) = X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1), A(32167), A(32168), A(32709), A(32760), M, JJJJ
1783 PRINT A(111), A(30167), A(31168), A(32758), A(32759)

1999 NEXT JJJJ

Based on the computer program in Wong [25], this BASIC computer program was run with QB64 [19, 20]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31994 is shown below:

-1      1      1      1      3
--205      -32000
1      1      1      1      2

-1      1      1      1      1
-205      -31999
1      1      1      1      1

1      1      1      1      4
-101      -31998
1      1      1      1      2

-1      1      1      1      4
-105      -31997
1      1      1      1      2

0      0      0      0      0
-33165      -31996
0      0      0      0      0

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

0      0      0      0      0
-32759      -31994
0      0      0      0      0

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

At JJJJ=-31995, M=0, which is optimal. See Li and Sun [12, p. 415].

Of the 32760 A's, only the 10 A's of line 1773 and line 1783 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 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31994 was 17 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. 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] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Programmming Problem on a Vector Processing Computer, Annals of Operations Research, Volume 14 (1988), Issue 1, pp.. 291-304.

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

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

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

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

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

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

[21] 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/

[22] 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/

[23] 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/

[24] 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

[25] 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/

Solving a Large Nonlinear Integer Programming Test Problem of 32760 General Integer Variables, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.4 but with 32760 unknowns instead of their 100 unknowns [12, pp. 415-417]. 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

32760-1
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,…, 32760.

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

111 FOR J44=1 TO 32760
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(32763), X(32763)

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 32760

116 A(J44) = -5 + FIX(RND * 11)
118 NEXT J44
128 FOR I = 1 TO 125000

129 FOR KQ = 1 TO 32760

130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
171 FOR J9 = 1 TO 32760

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 32759

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 32760

1455 A(KX) = X(KX)
1456 NEXT KX
1559 GOTO 128
1670 NEXT I
1773 PRINT A(1), A(32167), A(32168), A(32709), A(32760), M, JJJJ
1783 PRINT A(111), A(30167), A(31168), A(32758), A(32759)

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=-31996 is shown below:

-1      1      1      1      3
-205      -32000
1      1      1      1      2

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

-1      1      1      1      1
-4      -31998
1      1      1      1      1

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

1      1      1      1      1
0      -31996
1      1      1      1      1

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

At JJJJ=-31997 and JJJJ=-31996, M=0, which is optimal. See Li and Sun [12, p. 415].

Of the 32760 A's, only the 10 A's of line 1773 and line 1783 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=-31996 was 25 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. 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] 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/

Solving a Test Example with 32510 General Integer Variables with an Assist from Another Pair of Artificial Constraints

Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but with 32510 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

                                                                    32510-1
(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2
                                                                              i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510.

One notes that the problem above is equivalent to minimize

                                                                        32510-1
(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2
                                                                            i=1

subject to

X(1) + X(2) + X(3) + ... + X(32510) >= 32400

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510

and to minimize

                                                                    32510-1
(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2
                                                                       i=1

subject to

X(1) + X(2) + X(3) + ... + X(32510) <= 32400

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510.

One notes that here the artificial constraint is X(1) + X(2) + X(3) + ... + X(32510) <= 32400, while the corresponding artificial constraint of the preceding paper is
X(1) + X(2) + X(3) + ... + X(32510) <= 32500. The former is easier to be estimated for a brand-new problem.

And then one takes the better of the two.

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

(1) The Additional Constraint Used Immediately Below Is X(1) + X(2) + X(3) + ... + X(32510) >= 32400

One notes line 221 through line 257 of the following computer program, which are
221 SFE = 0
225 FOR J44 = 1 TO 32510
228 SFE = SFE + X(J44) 
233 NEXT J44
251 TSL = -32400 + SFE
257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0.

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

0 DEFINT J, K, B, X, A
2 DIM A(32522), X(32522)

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 32510

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

129 FOR KKQQ = 1 TO 32510

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32513)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32510

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 32510

228 SFE = SFE + X(J44)

233 NEXT J44
251 TSL = -32400 + SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32509

405 SUMNEWZ = SUMNEWZ + (32510 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32510) - 1) ^ 2 - 32510 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32510

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(32108), A(32509), A(32510), 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=-31997 is shown below:

-1 1 1 3 3
-1.677766E+13      -32000
1 1 1 1 2

1 1 1 1 1
-4.487713E+09      -31999
1 1 1 1 1

2 2 2 3 4
-1.979362E+13      -31998
0 1 0 1 1

1 1 1 1 1
0      -31997
1 1 1 1 1

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

At JJJJ=-31997, M=0 is optimal. See Li and Sun [12, pp. 414-415].

Of the 32510 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=-31997 was 41 hours.

(2) The Additional Constraint Used Immediately Below Is X(1) + X(2) + X(3) + ... + X(32510) <= 32400

One notes that line 251 above is 251 TSL=-32400+SFE and that line 251 below is 251 TSL=32400-SFE.

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

0 DEFINT J, K, B, X, A
2 DIM A(32522), X(32522)

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 32510

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

129 FOR KKQQ = 1 TO 32510

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32513)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32510

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 32510

228 SFE = SFE + X(J44)

233 NEXT J44
251 TSL = 32400 - SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32509

405 SUMNEWZ = SUMNEWZ + (32510 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32510) - 1) ^ 2 - 32510 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32510

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(32108), A(32509), A(32510), 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=-31997 is shown below:

-1 1 1 2 3
-4.878518E+12      -32000
0 1 1 1 1

0 0 0 0 0
-2      -31999
0 0 0 0 0

0 0 0 0 0
-2      -31998
0 0 0 0 0

0 0 0 0 0
-2      -31997
0 0 0 0 0

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

Of the 32510 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=-31997 was six hours.

(3) The realized solution with M=0 at JJJJ=-31997, which was produced through the artificial constraint X(1) + X(2) + X(3) + ... + X(32510) >= 32400
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. 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] 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/

Solving a Test Example with 32510 General Integer Variables with an Assist from two Artificial Constraints

Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but with 32510 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

                                                                              32510-1
(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2
                                                                                 i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510.

One notes that the problem above is equivalent to minimize

                                                                              32510-1
(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2
                                                                                  i=1

subject to

X(1) + X(2) + X(3) + ... + X(32510) >= 32500

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510

and to minimize

                                                                              32510-1
(X(1)-1)^2 + ( X(32510)-1)^2 + 32510* SIGMA (32510-i)* ( X(i)^2-X(i+1) )^2
                                                                                  i=1

subject to

X(1) + X(2) + X(3) + ... + X(32510) <= 32500

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32510.

One notes that here the artificial constraint is X(1) + X(2) + X(3) + ... + X(32510) <= 32500 while the corresponding artificial constraint of the preceding paper is
X(1)^5 + X(2)^5 + X(3)^5 + ... + X(32710)^5 <= 32710. The former is simpler to use.

And then one takes the better of the two.

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 257 of the first computer program below, which are
221 SFE = 0
225 FOR J44 = 1 TO 32510
228 SFE = SFE + X(J44)
233 NEXT J44
251 TSL = -32500 + SFE
257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0

(1) The Additional Constraint Used Immediately Below Is X(1) + X(2) + X(3) + ... + X(32510) >= 32500

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

0 DEFINT J, K, B, X, A
2 DIM A(32522), X(32522)

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 32510

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

129 FOR KKQQ = 1 TO 32510

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32513)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32510

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 32510

228 SFE = SFE + X(J44)

233 NEXT J44
251 TSL = -32500 + SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32509

405 SUMNEWZ = SUMNEWZ + (32510 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32510) - 1) ^ 2 - 32510 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32510

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(32108), A(32509), A(32510), 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=-31997 is shown below:

-1 1 1 3 3
-1.668491E+13      -32000
1 1 1 1 2

1 1 1 1 1
-1.84491E+09      -31999
1 1 1 1 1

0 0 0 0 0
-1.625E+14      -31998
0 0 0 0 0

1 1 1 1 1
0      -31997
1 1 1 1 1

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

At JJJJ=-31997, M=0 is optimal. See Li and Sun [12, pp. 414-415].

Of the 32510 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=-31997 was nine hours.

(2) The Additional Constraint Used Immediately Below Is X(1) + X(2) + X(3) + ... + X(32510) <= 32500

One notes that line 251 above is 251 TSL=-32500+SFE and that line 251 below is 251 TSL=32500-SFE.

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

0 DEFINT J, K, B, X, A
2 DIM A(32522), X(32522)

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 32510

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

129 FOR KKQQ = 1 TO 32510

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32513)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32510

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 32510

228 SFE = SFE + X(J44)

233 NEXT J44
251 TSL = 32500 - SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32509

405 SUMNEWZ = SUMNEWZ + (32510 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32510) - 1) ^ 2 - 32510 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32510

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(32108), A(32509), A(32510), 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=-31997 is shown below:

-1 1 1 2 3
-4.678518      -32000
0 1 1 1 1

0 0 0 0 0
-2      -31999
0 0 0 0 0

0 0 0 0 0
-2      -31998
0 0 0 0 0

0 0 0 0 0
-2     -31997
0 0 0 0 0

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

Of the 32510 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=-31997 was six hours.

(3) The realized solution with M=0 at JJJJ=-31997, which was produced through the artificial constraint X(1) + X(2) + X(3) + ... + X(32510) >= 32500
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. 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] 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/

 

Solving Another Test Example of 32710 General Integer Variables

Jsun Yui Wong

The problem here is Li and Sun's Problem 14.3 but with 32710 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

                                                                                     32710-1
(X(1)-1)^2 + ( X(32710)-1)^2 + 32710* SIGMA (32710-i)* ( X(i)^2-X(i+1) )^2
                                                                                      i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32710.

One notes that the problem above is equivalent to minimize

                                                                                      32710-1
(X(1)-1)^2 + ( X(32710)-1)^2 + 32710* SIGMA (32710-i)* ( X(i)^2-X(i+1) )^2
                                                                                          i=1

subject to

X(1)^5 + X(2)^5 + X(3)^5 + ... + X(32710)^5 >= 32710

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32710

and to minimize

                                                                                 32710-1
(X(1)-1)^2 + ( X(32710)-1)^2 + 32710* SIGMA (32710-i)* ( X(i)^2-X(i+1) )^2
                                                                                  i=1

subject to

X(1)^5 + X(2)^5 + X(3)^5 + ... + X(32710)^5 <= 32710

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32710.

And then one takes the better of the two.

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 32710
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(32710)^5 >= 32710

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

0 DEFINT J, K, B, X, A
2 DIM A(32717), X(32717)

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 32710

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

129 FOR KKQQ = 1 TO 32710

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32713)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32710

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 32710

228 SFE = SFE + X(J44) ^ 5
233 NEXT J44
251 TSL = -32710 + SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32709

405 SUMNEWZ = SUMNEWZ + (32710 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32710) - 1) ^ 2 - 32710 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32710

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(32108), A(32709), A(32710), 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=-31997 is shown below:

0      0      1      2      -2
-6.141141E+12      -32000
0      1      1      1      1

0      0      0      0      0
-1.635562E+14      -31999
0      0      0      0      0

0      0      0      2      5
-6.106011E+12      -31998
1      1      1      1      1

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

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

At JJJJ=-31997, M=0, which is optimal. See Li and Sun [12, pp. 414-415].

Of the 32710 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=-31997 was 12 hours.

(2) The Additional Constraint Used Immediately Below Is X(1)^5 + X(2)^5 + X(3)^5 + ... + X(32710)^5 <= 32710

One notes that line 251 above is 251 TSL=-32710+SFE and that line 251 below is 251 TSL=32710-SFE.

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

0 DEFINT J, K, B, X, A
2 DIM A(32717), X(32717)

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 32710

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

129 FOR KKQQ = 1 TO 32710

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32713)

167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32710

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 32710

228 SFE = SFE + X(J44) ^ 5
233 NEXT J44
251 TSL = 32710 - SFE

257 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32709

405 SUMNEWZ = SUMNEWZ + (32710 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2

407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32710) - 1) ^ 2 - 32710 * SUMNEWZ

492 PD1 = SONE + 5000000000# * TSL
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32710

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(32108), A(32709), A(32710), 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=-31997 is shown below:

0      0      1      0      -2
-4.97328E+12      -32000
0      1      1      1      1

0      0      0      0      -2
-5.030801E+12      -31999
0      0      0      1      0

0      0      0      -2      2
-5.278709E+12      -31998
0      0      -1      0      1

0      0      0      -1      3
-4.737936E+12      -31997
1      -1      0      1      -1

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

Of the 32710 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=-31997 was 13 hours.

(3) The realized solution with M=0 at JJJJ=-31997, which was produced with the artificial constraint X(1)^5 + X(2)^5 + X(3)^5 + ... + X(32710)^5 >= 32710 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. 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] 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/