Jsun Yui Wong
The problem here is Li and Sun's Problem 14.3 but with 15120 unknowns instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize
15120-1
(X(1)-1)^2 + ( X(15120)-1)^2 + 15120* SIGMA (15120-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15120.
One notes that the problem above is equivalent to minimize
15120-1
(X(1)-1)^2 + ( X(15120)-1)^2 + 15120* SIGMA (15120-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^3 + X(2)^3 + X(3)^3 + ... + X(15120)^3 >= 15120
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15120
and to minimize
15120-1
(X(1)-1)^2 + ( X(15120)-1)^2 + 15120* SIGMA (15120-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^3 + X(2)^3 + X(3)^3 + ... + X(15120)^3 <= 15120
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 15120.
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 and line 257 and line 492 of each of the two computer programs below.
(1) The Additional Constraint Used Immediately Below Is X(1)^3 + X(2)^3 + X(3)^3 + ... + X(15120)^3 >= 15120
The following computer program uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.
0 DEFINT J,K,B,X,A
2 DIM A(15123),X(15123)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15120
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15123)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15120
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15120
228 SFE=SFE+X(J44)^3
233 NEXT J44
251 TSL=-15120+SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SZ=0
403 FOR J44=1 TO 15119
405 SZ=SZ+ (15120-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15120)-1)^2 -15120* SZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15120
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15118),A(15119),A(15120),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31990 is shown below:
1 1 1 1 1
-5.063338E+09 -32000
0 0 0 0 0
-7.56E+13 -31999
0 0 0 0 0
-7.560546E+13 -31998
0 0 1 1 -1
-2.161696E+12 -31997
0 -1 0 -1 4
-1.745691E+12 -31996
0 0 2 1 0
-6.621315E+11 -31995
1 1 1 1 1
-1.037247E+10 -31994
1 1 1 1 1
0 -31993
0 0 0 0 0
-7.560509E+13 -31992
1 1 2 2 0
-2.411565E+12 -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 15120 A's, only the 5 A's of line 1778 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31990 was 42 hours.
(2) The Additional Constraint Used Below Is X(1)^3 + X(2)^3 + X(3)^3 + ... + X(15120)^3 <= 15120
While line 251 above is 251 TSL= -15120+SFE, line 251 below is 251 TSL= 15120-SFE.
The following computer program also uses the IBM Personal BASIC Compiler--through A:\>bascom and A:\>link--Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft.
0 DEFINT J,K,B,X,A
2 DIM A(15123),X(15123)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 15120
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 15120
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*15123)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 15120
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 15120
228 SFE=SFE+X(J44)^3
233 NEXT J44
251 TSL= 15120-SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SZ=0
403 FOR J44=1 TO 15119
405 SZ=SZ+ (15120-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(15120)-1)^2 -15120* SZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15120
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(15118),A(15119),A(15120),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with the IBM Personal Computer BASIC Compiler, Version 1.00. See the BASIC manual [13, page iii, Preface]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31996 is shown below:
0 0 0 0 -1
-4.232866E+11 -32000
0 0 0 1 0
-5.537017E+11 -31999
1 1 0 1 1
-5.232461E+11 -31998
0 0 -1 1 0
-4.482497E+11 -31997
1 1 0 -1 2
-4.507229E+11 -31996
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 15120 A's, only the 5 A's of of line 1778 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM Personal Computer BASIC Compiler, Copyright IBM Corp 1982 Version 1.00/Copyright Microsoft, Inc. 1982, the wall-clock time for obtaining the output through JJJJ=-31996 was 22 hours.
(3) Produced through the additional constraint X(1)^3 + X(2)^3 + X(3)^3 + ... + X(15120)^3 >= 15120, the realized solution with M=0 at JJJJ=-31993 and at JJJJ=-31990 is the best produced.
For a computer program involving a mix of continuous variables and integer variables, see Wong [19], for instance.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey--Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0
[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[10] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.
[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.
[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[13] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.
[14] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[15] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[18] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[19] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[20] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[21] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
