Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's last test problem [14, p. 213, Test Problem 395]. The problem is to minimize the following:
50
SIGMA i*(X(i)^2+X(i)^4 )
i=1
subjec to
50
SIGMA X(i)^2 =1.
i=1
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(1001),X(1001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 50
112 A(J44)=-.1+RND*.2
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 50
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*50)
144 REM GOTO 167
145 IF RND<.33 THEN 150 ELSE IF RND<.5 THEN 163 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 GOTO 400
396 FOR J44=1 TO 50
397 IF X(J44)<-1 THEN X(J44)=A(J44)
398 IF X(J44)>1 THEN X(J44)=A(J44)
399 NEXT J44
400 SONE=0
401 FOR J44=2 TO 50
403 SONE=SONE-X(J44)^2
404 NEXT J44
405 IF ( 1+SONE )<.0000001 THEN 1670
406 X(1)=( 1+SONE )^(1/2)
410 STWO=0
411 FOR J44=1 TO 50
413 STWO=STWO+J44*(X(J44)^2+X(J44)^4 )
415 NEXT J44
457 PD1= - STWO
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 50
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-999999999# THEN 1999
1911 GOTO 1945
1922 LPRINT A(1),A(2),A(3),A(4),A(5)
1923 LPRINT A(6),A(7),A(8),A(9),A(10)
1924 LPRINT A(11),A(12),A(13),A(14),A(15)
1925 LPRINT A(16),A(17),A(18),A(19),A(20)
1926 LPRINT A(21),A(22),A(23),A(24),A(25)
1927 LPRINT A(26),A(27),A(28),A(29),A(30)
1928 LPRINT A(31),A(32),A(33),A(34),A(35)
1929 LPRINT A(36),A(37),A(38),A(39),A(40)
1930 LPRINT A(41),A(42),A(43),A(44),A(45)
1931 LPRINT A(46),A(47),A(48),A(49),A(50)
1939 LPRINT M,JJJJ
1945 PRINT A(1),A(2),A(3),A(4),A(5)
1947 PRINT A(46),A(47),A(48),A(49),A(50)
1949 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 output through
JJJJ=-31999 is summarized below:
.9128926 -.4081998 8.617702E-05 -7.757013E-05
-7.474125E-05
2.186212E-05 -3.765578E-06 2.219921E-05
2.747354E-05 4.096901E-06
-1.916666 -32000
.9128328 -.4083324 -8.775965E-04 2.634784E-04
-3.219565E-05
-1.469409E-05 2.279203E-05 1.603028E-05
-1.14861E-05 -3.286808E-05
-1.916667 -31999
Of the fifty A's, only the ten A's of line 1945 and line 1947 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 basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output shown above was 30 seconds.
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/
