Jsun Yui Wong
The problem here is Li and Sun’s Problem 14.3 but with 32110 unknowns instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski Test Problem 282 [16, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize
32110-1
(X(1)-1)^2 + ( X(32110)-1)^2 + 32110* SIGMA (32110-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32110.
One notes that the problem above is equivalent to minimize
32110-1
(X(1)-1)^2 + ( X(32110)-1)^2 + 32110* SIGMA (32110-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 >= 32110
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32110
and to minimize
32110-1
(X(1)-1)^2 + ( X(32110)-1)^2 + 32110* SIGMA (32110-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 <= 32110
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32110.
Then one takes the best produced.
Generally speaking, while directly dealing with an initial problem is advantageous if the initial problem is “small,” dealing with the initial problem plus an additional constraint–twice–is advantageous if the initial problem is “large” because each of these two problems has a smaller penalty-free region than that of the initial problem; see line 251, line 257, and line 492 of each of the two computer programs below.
One notes line 221 through line 233, which are
221 SFE=0
225 FOR J44=1 TO 32110
228 SFE=SFE+X(J44)^5
233 NEXT J44.
(1) The Additional Constraint Used Immediately Below Is X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 >= 32110
The following computer program uses QB64 [18, 19].
0 DEFINT J,K,B,X,A
2 DIM A(32222),X(32222)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 32110
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 100000
129 FOR KKQQ=1 TO 32110
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*32113)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 32110
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 32110
228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL=-32110+SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 32109
405 SUMNEWZ=SUMNEWZ+ (32110-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= – (X(1)-1)^2 – ( X(32110)-1)^2 -32110* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 32110
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(32108),A(32109),A(32110),M,JJJJ
1788 PRINT A(1111),A(11111),A(23333),A(27777),A(28888)
1999 NEXT JJJJ
Based on the computer programs in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31998 is shown below:
0 0 1 2 2
-5.593623E+12 -32000
0 1 1 0 1
1 1 1 -1 4
-6.64949E+12 -31999
0 0 1 1 0
1 1 1 1 1
0 -31998
1 1 1 1 1
Above there is no rounding by hand; it is just straight copying by hand from the screen.
M=0 is optimal. See Li and Sun [12, pp. 414-415].
Of the 32110 A’s, only the ten A’s of line 1778 and line 1788 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31998 was eight hours.
(2) The Additional Constraint Used Immediately Below Is X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 <= 32110
One notes that line 251 above is 251 TSL=-32110+SFE and that line 251 below is 251 TSL=32110-SFE.
.
The following computer program uses QB64 [18, 19].
0 DEFINT J,K,B,X,A
2 DIM A(32222),X(32222)
81 FOR JJJJ=-32000 TO 32000
85 LB=-FIX(RND*6)
86 UB=FIX(RND*6)
89 RANDOMIZE JJJJ
90 M=-1.5D+38
111 FOR J44=1 TO 32110
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 100000
129 FOR KKQQ=1 TO 32110
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*32113)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 32110
171 IF X(J44)<LB THEN X(J44 )=LB
172 IF X(J44)>UB THEN X(J44 )=UB
173 NEXT J44
221 SFE=0
225 FOR J44=1 TO 32110
228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL=32110-SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 32109
405 SUMNEWZ=SUMNEWZ+ (32110-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= – (X(1)-1)^2 – ( X(32110)-1)^2 -32110* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 32110
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(32108),A(32109),A(32110),M,JJJJ
1788 PRINT A(1111),A(11111),A(23333),A(27777),A(28888)
1999 NEXT JJJJ
Based on the computer programs in Wong [24], this BASIC computer program was run with QB64 [18, 19]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31998 is shown below:
0 0 0 1 1
-4.680451E+12 -32000
0 1 1 0 1
0 0 1 1 -1
-5.337537E+12 -31999
0 1 1 -1 1
-1 1 0 0 0
-4.636656E+12 -31998
1 -1 0 0 0
Above there is no rounding by hand; it is just straight copying by hand from the screen.
M=0 is optimal. See Li and Sun [12, pp. 414-415].
Of the 32110 A’s, only the ten A’s of line 1778 and line 1788 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31998 was nine hours.
(3) The realized solution with M=0 at JJJJ=-31998, which was produced through the artificial constraint X(1)^5 + X(2)^5 + X(3)^5 + … + X(32110)^5 >= 32110 of the first computer program, is the best produced.
For a computer program involving a mix of continuous variables and integer variables, see Wong [21], for instance.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING
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[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).
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[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html
[18] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview
[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[20] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[21] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[22] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/
[23] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski’s Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
[24] Jsun Yui Wong (2015, March 08). Nonconvex Mixed Integer Nonlinear Programming (MINLP) Computer Programs with a Divide-and-Conquer Strategy To Solve Li and Sun’s Problem 14.3 but with n=15110 General Integer Variables. http://myblogsubstance.typepad.com/substance/2015/03/
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