Jsun Yui Wong
Case One: Four Thousand Binary 0-1 Integer Variables
Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's Test Problem 305 [14, p. 129] but with 4000 unknowns instead of 100 unknowns and with the modification that the 4000 unknowns are 0-1 variables. Thus, the problem is to minimize the following:
4000 4000 4000
SIGMA X(i)^2 + [ SIGMA (1/2) * i *X(i)^2 ] ^2 + [ SIGMA (1/2) * i *X(i)^2 ] ^4
i=1 i=1 i=1
subject to 0<= X(j) <=1, X(j) integer, j=1, 2, 3,..., 4000. See Schittkowski [14, p. 129].
One notes line 144, which is 144 GOTO 168.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(4001),X(4001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 4000
112 A(J44)=FIX( RND*2)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 4000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4000)
144 GOTO 168
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 IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
211 GOTO 301
212 FOR J44=1 TO 4000
213 IF X(J44)<0 THEN X(J44)=A(J44)
214 IF X(J44)>1 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 4000
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 4000
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 4000
345 STHR=STHR+(1/2)*J44*X(J44)
349 NEXT J44
351 STHRTHR= ( STHR)^4
447 PD1=-SONEONE-STWOTWO-STHRTHR
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1551 REM PRINT A(1),A(2),A(3),A(4),A(5)
1552 REM PRINT A(196),A(197),A(198),A(199),A(200)
1553 REM PRINT M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1920 PRINT A(1),A(2),A(3),A(4),A(5)
1921 PRINT A(6),A(7),A(8),A(9),A(10)
1922 PRINT A(11),A(12),A(13),A(14),A(15)
1931 PRINT A(3986),A(3987),A(3988),A(3989),A(3990)
1932 PRINT A(3991),A(3992),A(3993),A(3994),A(3995)
1933 PRINT A(3996),A(3997),A(3998),A(3999),A(4000)
1937 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 as follows:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -32000
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -31999
Immediately above there is no rounding by hand. One notes that M=0 and that only 30 A's of 4000 A's 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 through JJJJ=-31999 was 70 minutes.
Case Two: Two General Integer Variables and One Continuous Variable
Similar to the computer program above, the following computer program seeks to solve Hock and Schittkowski's Problem 25 [6, p. 48] with the modification that the first two variables are general integer variables. Thus, the problem here is to minimize the following:
99
SIGMA (-.01*i+ EXP( -(1/X(1))*( U(i) -X(2))^X(3) ) )^2
i=1
where U(i)=25+( -50*LOG(.01*i ) )^(1/1.5), i=1, 2, 3,..., 99
subject to
.1<=X(1)<=100, X(1) integer
0<=X(2)<=25.6, X(2) integer
0<=X(3)<=5.
See Hock and Schittkowski [6, p. 48] and Himmelblau [5, p. 422].
0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(1001),X(1001),U(111)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
112 A(1)=1+ FIX( RND*100 )
121 A(2)= FIX( RND*26)
123 A(3)= RND*5
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 3
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*2)
140 B=1+FIX(RND*3)
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
171 X(1)=CINT(X(1))
172 X(2)=CINT(X(2))
181 IF X(1)<.1 THEN X(1)=A(1)
182 IF X(1)>100 THEN X(1)=A(1)
183 IF X(2)<0 THEN X(2)=A(2)
184 IF X(2)>25.6 THEN X(2)=A(2)
185 IF X(3)<0 THEN X(3)=A(3)
186 IF X(3)>5 THEN X(3)=A(3)
231 FOR J44=1 TO 99
233 IF -50*LOG(.01*J44 ) <1E-09 THEN 1670
234 U(J44)=25+( -50*LOG(.01*J44 ) )^(1/1.5)
237 NEXT J44
301 SONE=0
303 FOR J44=1 TO 99
307 SONE=SONE+(-.01*J44+ EXP( -(1/X(1))*( U(J44) -X(2))^X(3) ) )^2
309 NEXT J44
447 PD1=-SONE
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-.0001 THEN 1999
1922 PRINT A(1),A(2),A(3)
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=-31994 is as follows:
50 25 1.5
-7.681148E-13 -32000
50 25 1.5
-7.681148E-13 -31999
50 25 1.5
-7.681148E-13 -31998
50 25 1.5
-7.681148E-13 -31997
50 25 1.5
-7.681148E-13 -31996
86 24 1.659983
-3.978813E-03 -31995
50 25 1.5
-7.681148E-13 -31994
Immediately 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=-31994 was 15 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/
