The problem here is Li and Sun's Problem 14.3 but with 20110 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
20110-1
(X(1)-1)^2 + ( X(20110)-1)^2 + 20110* SIGMA (20110-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 20110.
One notes that the problem above is equivalent to minimize
20110-1
(X(1)-1)^2 + ( X(20110)-1)^2 + 20110* SIGMA (20110-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^5 + X(2)^5 + X(3)^5 + ... + X(20110)^5 >= 20110
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 20110
and to minimize
20110-1
(X(1)-1)^2 + ( X(20110)-1)^2 + 20110* SIGMA (20110-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
X(1)^5 + X(2)^5 + X(3)^5 + ... + X(20110)^5 <= 20110
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 20110.
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 20110
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(20110)^5 >= 20110
The following computer program uses QB64 [18, 19].
0 DEFINT J,K,B,X,A
2 DIM A(35113),X(35113)
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 20110
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 20110
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*20113)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 20110
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 20110
228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL=-20110+SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 20109
405 SUMNEWZ=SUMNEWZ+ (20110-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(20110)-1)^2 -20110* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 20110
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(20108),A(20109),A(20110),M,JJJJ
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=-31995 is shown below:
0 0 1 0 1
-1.461463E+12 -32000
0 0 0 0 0
-1.0055E+14 -31999
0 0 1 2 3
-1.196455E+12 -31998
0 0 -1 0 -1
-1.668631E+12 -31997
0 0 -1 0 -2
-1.551935E+12 -31996
1 1 1 1 1
0 -31995
Above there is no rounding by hand; it is just straight copying by hand from the screen.
M=0 is optimal. See Li and Sun [12, pp. 414-415].
Of the 20110 A's, only the 5 A's of line 1778 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31995 was five hours.
(2) The Additional Constraint Used Below Is X(1)^5 + X(2)^5 + X(3)^5 + ... + X(20110)^5 <= 20110
While line 251 above is 251 TSL= -20110+SFE, line 251 below is 251 TSL= 20110-SFE.
The following computer program uses QB64 [18, 19].
0 DEFINT J,K,B,X,A
2 DIM A(35113),X(35113)
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 20110
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 20110
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*.3)
140 B=1+FIX(RND*20113)
167 IF RND<.5 THEN X(B)=(A(B)-1) ELSE X(B)=(A(B) +1 )
169 NEXT IPP
170 FOR J44=1 TO 20110
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 20110
228 SFE=SFE+X(J44)^5
233 NEXT J44
251 TSL= 20110-SFE
257 IF TSL<0 THEN TSL=TSL ELSE TSL=0
400 SUMNEWZ=0
403 FOR J44=1 TO 20109
405 SUMNEWZ=SUMNEWZ+ (20110-J44)* ( X(J44)^2-X(J44+1) )^2
407 NEXT J44
411 SONE= - (X(1)-1)^2 - ( X(20110)-1)^2 -20110* SUMNEWZ
492 PD1=SONE +5000000000#*TSL
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 20110
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1),A(2),A(20108),A(20109),A(20110),M,JJJJ
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=-31995 is shown below:
0 0 1 1 1
-1.137198E+12 -32000
0 -1 0 0 0
-1.26094E+12 -31999
0 0 0 0 0
-1.23041E+09 -31998
0 0 0 0 0
-5.918373E+07 -31997
1 1 1 0 -1
-1.154406E+12 -31996
0 0 1 0 -1
-1.165531E+12 -31995
Above there is no rounding by hand; it is just straight copying by hand from the screen.
M=0 is optimal. See Li and Sun [12, pp. 414-415].
Of the 20110 A's, only the 5 A's of line 1778 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and with QB64 [18, 19], the wall-clock time for obtaining the output through JJJJ=-31995 was six hours.
(3) The realized solution with M=0 at JJJJ=-31995, which was produced through the additional constraint X(1)^5 + X(2)^5 + X(3)^5 + ... + X(15110)^5 >= 20110, 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/
No comments:
Post a Comment