Similar to the computer programs of the preceding papers, the computer program below seeks to solve an integer version of Schittkowski's Test Problem 272. The source of this Test Problem 272 is S. Walukiewicz; see Schittkowski [14, p. 96]. Thus, the computer program below tries to minimize the following:
13
SIGMA ( X(4)*EXP(-X(1)*T(i) ) -X(5)*EXP(-X(2)*T(i) )+X(6)*EXP(-X(3)*T(i) ) -Y(i) ) ^2
i=1
T(i)=i/10
Y(i)=EXP(-T(i) ) -5*EXP(-10*T(i) )+3*EXP(-4*T(i) )
0<= X(i) <= 200, X(i) integer, i=1, 2, 3, 4, 5, 6.
One notes line 111 through line 117, which are
111 FOR J44=1 TO 6
114 A(J44)=RND*200
117 NEXT J44
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(90),X(90),T(90),Y(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
91 GOTO 111
92 A(2)=2+RND*4000
93 A(3)=3+RND*4000
94 A(4)=4+RND*4000
95 A(5)=-4000+RND*8000
111 FOR J44=1 TO 6
114 A(J44)=RND*200
117 NEXT J44
128 FOR I=1 TO 20000
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*6)
144 REM GOTO 167
145 IF RND<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
170 REM GOTO 601
171 FOR J44=1 TO 6
174 IF X(J44)<0 THEN X(J44)=A(J44)
175 IF X(J44)>200 THEN X(J44)=A(J44)
177 NEXT J44
555 REM Y(6)= 34 -X(1)^2-X(2)^2-X(3)^2-X(4)^2-X(5)^2
565 REM IF Y(6)>0 THEN Y(6)=0
591 GOTO 601
592 X(2)=10
593 X(3)=4
594 X(4)=1
595 X(5)=5
596 X(6)=3
601 SUMM=0
602 FOR J44=1 TO 13
604 T(J44)=J44/10
607 Y(J44)=EXP(-T(J44) ) -5*EXP(-10*T(J44) )+3*EXP(-4*T(J44) )
611 SUMM=SUMM+ ( X(4)*EXP(-X(1)*T(J44) ) -X(5)*EXP(-X(2)*T(J44) )+X(6)*EXP(-X(3)*T(J44) ) -Y(J44) ) ^2
621 NEXT J44
622 GOTO 699
681 PDA=- X(1)*X(2) *X(3) *X(4)+3* X(1)*X(2)*X(4) + 4*X(1)* X(2)*X(3) - 12* X(1)*X(2) +X(2) *X( 3)*X(4)-3*X(2)*X(4) -4*X(2)*X(3)+12*X(2)+2*X(1)*X(3)*X(4)-6*X(1)*X(4)-8*X(1)*X(3)+24*X(1)
684 PDB=- 2 *X(3) *X(4)+6* X(4) + 8*X(3) -24 - 1.5* X(5)^4 +5.75 *X( 5)^3-5.25*X(5)^2 +500000!*Y(6)
695 PD1=PDA+PDB
699 PD1=-SUMM
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 6
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.01 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5)
1939 PRINT A(6),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, this computer program's complete output through JJJJ=-30685 is shown below:
4 10 1 3 5
1 -8.881784E-16 -31760
4 10 1 3 5
1 -8.881784E-16 -31460
1 10 4 1 5
3 0 -30685
Above there is no rounding by hand.
M=0 is optimal; see Schittkowski [14, p. 96].
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-30685 was five hours and twenty minutes.
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] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.
[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[8] 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.
[9] 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.
[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[11] 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.
[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html
[16] 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/
[17] 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/
[18] 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
