Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 354. The source of this Test Problem 354 S. Walukiewicz; see Schittkowski [14]. Thus, the computer program below tries to minimize the following:
( X(1)+10*X(2) )^2+5*( X(3)-X(4) )^2 +( X(2)-2*X(3) )^4+ 10*( X(1)-X(4) )^4
subject to
X(1)+X(2)+X(3)+X(4)-1 >=0
Lower bounds are -5, -5, -5, -5, respectively.
Upper bounds are 20, 20, 20, 20, respectively.
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(90),X(90)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 4
112 A(J44)=-5+ RND*(25)
114 NEXT J44
128 FOR I=1 TO 10000
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144 REM GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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
171 FOR J44=1 TO 4
174 IF X(J44)>20 THEN X(J44)=A(J44)
177 NEXT J44
178 FOR J44=1 TO 4
179 IF X(J44)<-5 THEN X(J44)=A(J44)
180 NEXT J44
555 X(5)=X(1)+X(2)+X(3)+X(4)-1
565 IF X(5)>0 THEN X(5)=0
681 PD1=-( X(1)+10*X(2) )^2-5*( X(3)-X(4) )^2 -( X(2)-2*X(3) )^4- 10*( X(1)-X(4) )^4 +500000!*X(5)
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-9999 THEN 1999
1923 PRINT A(1),A(2),A(3),A(4)
1939 PRINT 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, the computer program's complete output through JJJJ=-31998 is shown below:
.5027103 -4.520061E-02 .2363648 .3061256
-.1137975 -32000
.5035425 -4.552651E-02 .2360212 .3059629
-.1137877 -31999
.5022817 -4.535842E-02 .2362913 .3067854
-.1137903 -31998
Above there is no rounding by hand.
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=-31998 was six seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.
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[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
