Friday, June 27, 2014

A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition

Jsun Yui Wong
 
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 377 [14, p. 196].  The source of this Test Problem 377 is S. Walukiewicz; see Schittkowski [14].  The problem is to minimize the following:

10
SIGMA        X(i) *  ( C(i)  +LOG(  X(i)/SONE       )   )
i=1

            10
where SONE= SIGMA   X(j)
            j=1

C(1) through C(10) are given in line 11 and line 13

subject to

X(1)     -  2*X(2) +  2* X(3)    +   X(6)    + X(10)    -    2   =0

X(4) -  2* X(5)    +   X(6)    +     X(7)   -    1               =0

X(3)   +   X(7)    +   X(8) +2* X(9)   +     X(10)   -    1     =0

0.1E-04<= X(i)<=10, i=1 , 2 , 3,..., 10.

The computer program's arrangement of line 380, line 381, and line 383 is to induce domino effect.

While line 163 of the preceding paper is 163 IF RND<.5 THEN X(B)=(A(B)-.001)   ELSE X(B)=(A(B)     +.001   ), here line 163 is 163 IF RND<.5 THEN X(B)=(A(B)-.0001)   ELSE X(B)=(A(B)     +.0001   ).

0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(1001),X(1001)
11 C(1)=-6.089 :C(2)=- 17.164 :C(3)=- 34.054:C(4)=- 5.914 :C(5)=- 24.721
13 C(6)=-14.986 :C(7)=- 24.1 :C(8)=- 10.708:C(9)=-26.662 :C(10)=- 22.179
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 10
113 A(J44)=  RND*(.1)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10)
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)-.0001)   ELSE X(B)=(A(B)     +.0001   )
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
370 FOR J44=1 TO 10
371 IF X(J44)<.00001 THEN 1670
372 IF X(J44)>10 THEN 1670
375 NEXT J44
380     X(7)=  -X(4)   +2* X(5)    -   X(6)       +    1
381     X(10)=  -X(3)   -   X(7)    -   X(8)    -2* X(9)    +    1
383     X(1)=2*X(2)   -2* X(3)    -   X(6)    - X(10)    +    2
396 FOR J44=1 TO 10
397 IF X(J44)<.00001 THEN 1670
398 IF X(J44)>10 THEN 1670
399 NEXT J44
400 SONE=0
401 FOR J44=1 TO 10
403 SONE=SONE+X(J44)
405 NEXT J44
410 STWO=0
411 FOR J44=1 TO 10
412 IF   (  X(J44)/SONE       )   <    1E-11 THEN 1670
413 STWO=STWO+X(J44) *  ( C(J44)  +LOG(  X(J44)/SONE       )   )
415 NEXT J44
455 PD1= -   STWO
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-999999! THEN 1999
1923   PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10)
1929   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 complete output through
JJJJ=-31996 is shown below:

9.999998   9.981577   .9815478   9.999987   9.50916
9.999988   1.834583E-02   1.3025E-05   1.106528E-05
7.116795E-03
794.8063   -32000

10   9.983363   .9833495   10   9.508292
9.999997   1.658726E-02   1.141463E-05
1.10852E-05   2.962351E-05
794.8277   -31999

10   9.988516   .9884736   10   9.505688
9.999984   1.139164E-02   1.299235E-05
1.094947E-05   9.983778E-05
794.885   -31998

10   9.987944   .987878   9.999989   9.505974
9.999999   1.195908E-02   1.069538E-05
1.000319E-05   1.322031E-04
794.8788   -31997

9.999996   9.989264   .9892115   9.999985   9.505311
9.999993   1.064396E-02   1.05749E-05   1.007373E-05
1.138449E-04
794.8931   -31996

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=-31996 was thirty-one 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/