Jsun Yui Wong
Case One: Four General Integer Variables
Similar to the computer programs of the preceding papers, the following computer program seeks to solve the C. F. Wood/Westinghouse Research Laboratory problem on page 403 of Himmelblau [5] plus two new restrictions of the lower bounds and the upper bounds of -100's and +100's instead
of -10's and +10's and of the variables here are general integer variables instead of continuous variables. Thus, the problem is to minimize the following:
100*(X(2)-X(1)^2 )^2+(1-X(1) )^2+90*( X(4)-X(3)^2 )^2+(1-X(3) )^2+10.1* ( (X(2)-1)^2 +(X(4)-1)^2 )+19.8*( X(2)-1 )*( X(4)-1 )
subject to
-100<=X(i)<=100, X(i) integer, i=1, 2, 3, 4. See line 112, line 213, and line 214 below.
One notes that 201*201*201*201 equals 1,632,240,801.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(1001),X(1001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 4
112 A(1)=-100+ RND*200
114 NEXT J44
128 FOR I=1 TO 32000
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<.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
212 FOR J44=1 TO 4
213 IF X(J44)<-100 THEN X(J44)=A(J44)
214 IF X(J44)>100 THEN X(J44)=A(J44)
215 NEXT J44
449 PD1=-100*(X(2)-X(1)^2 )^2-(1-X(1) )^2-90*( X(4)-X(3)^2 )^2-(1-X(3) )^2-10.1* ( (X(2)-1)^2 +(X(4)-1)^2 )-19.8*( X(2)-1 )*( X(4)-1 )
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4
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),A(4)
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:
-1 1 1 1
-4 -32000
1 1 -1 1
-4 -31999
1 1 1 1
0 -31998
-1 1 1 1
-4 -31997
1 1 -1 1
-4 -31996
-1 1 -1 1
-8 -31995
1 1 1 1
0 -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 twelve seconds.
Case Two: Seven Thousand Binary 0-1 Integer Variables
Similar to the computer program above, the computer program below seeks to solve Schittkowski's Test Problem 305 [14, p. 129] but with 7000 unknowns instead of 100 unknowns and with the modification that the 7000 unknowns are 0-1 variables. Thus, the problem is to minimize the following:
7000 7000 7000
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,..., 7000. 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(7001),X(7001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 7000
112 A(J44)=FIX( RND*2)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 7000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*7000)
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 7000
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 7000
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 7000
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 7000
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 7000
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)
1923 PRINT A(16),A(17),A(18),A(19),A(20)
1924 PRINT A(21),A(22),A(23),A(24),A(25)
1929 PRINT A(6976),A(6977),A(6978),A(6979),A(6980)
1930 PRINT A(6981),A(6982),A(6983),A(6984),A(6985)
1931 PRINT A(6986),A(6987),A(6988),A(6989),A(6990)
1932 PRINT A(6991),A(6992),A(6993),A(6994),A(6995)
1933 PRINT A(6996),A(6997),A(6998),A(6999),A(7000)
1937 PRINT M,JJJJ
1977 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 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -32000
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 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-1.037206E+11 -31999
-1.037206E+11 -31999
Immediately above there is no rounding by hand. One notes that M=0 at JJJJ=-32000 and
M=-1.037206E+11 at JJJJ=-31999 and that only 50 A's of 7000 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 three hours.
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/
Monday, June 16, 2014
Sunday, June 15, 2014
Unified Computer Programs for Nonlinear Integer/Continuous/Discrete Programming Problems Including One Here with Four Thousand Binary 0-1 Integer Variables
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/
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/
Saturday, June 14, 2014
Unified Computer Programs for Nonlinear Integer/Continuous/Discrete Programming Problems
Jsun Yui Wong
Case One: Two Hundred Continuous Variables
Similar to the computer programs of the preceding papers, the first computer program below seeks to solve Schittkowski's Test Problem 305 [14, p. 129] but with 200 unknowns instead of 100 unknowns. Thus, the problem is to minimize the following:
200 200 200
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 -5<= X(j)<=10, j=1, 2, 3,..., 200. See Schittkowski [14, p. 129]. These lower bounds of -5's and these upper bounds of 10's are usually used in the literature--see the Zakharov function [15].
One notes line 144, which is 144 REM GOTO 167.
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 200
112 A(J44)=-5+FIX( RND*16)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 200
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*200)
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
212 FOR J44=1 TO 200
213 IF X(J44)<-5 THEN X(J44)=A(J44)
214 IF X(J44)>10 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 200
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 200
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 200
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 200
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1551 PRINT A(1),A(2),A(3),A(4),A(5)
1552 PRINT A(196),A(197),A(198),A(199),A(200)
1553 PRINT M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1904 GOTO 1922
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1908 PRINT A(21),A(22),A(23),A(24),A(25)
1909 PRINT A(26),A(27),A(28),A(29),A(30)
1910 PRINT A(31),A(32),A(33),A(34),A(35)
1911 PRINT A(36),A(37),A(38),A(39),A(40)
1912 PRINT A(41),A(42),A(43),A(44),A(45)
1913 PRINT A(46),A(47),A(48),A(49),A(50)
1914 PRINT A(51),A(52),A(53),A(54),A(55)
1915 PRINT A(56),A(57),A(58),A(59),A(60)
1916 PRINT A(61),A(62),A(63),A(64),A(65)
1917 PRINT A(66),A(67),A(68),A(69),A(70)
1918 PRINT A(71),A(72),A(73),A(74),A(75)
1919 PRINT A(76),A(77),A(78),A(79),A(80)
1920 PRINT A(81),A(82),A(83),A(84),A(85)
1921 PRINT A(86),A(87),A(88),A(89),A(90)
1922 PRINT A(1),A(2),A(3),A(4),A(5)
1923 PRINT A(196),A(197),A(198),A(199),A(200)
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 BASIC manual [11]. Copied by hand from the screen, the output partly through
JJJJ=-32000 is as follows:
-5.987215E-19 8.747792E-09 -4.113391E-15
-5.474714E-07 -9.680436E-07
9.264764E-14 -9.924814E-05 3.112297E-04
3.539233E-04 1.256368E-17
-9.993773E-07 32000
Immediately above there is no rounding by hand. One notes M=-9.993773E-07 and that only ten of the 200 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 shown above was 22 hours.
Case Two: One General Integer Variable and Two Continuous Variables
Similar to the computer program above, the following computer program seeks to solve Hock and Schittkowski's Problem 25 [6, p. 48] plus the modification that the second variable is an integer instead of continuous. Thus, the problem 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
0<=X(2)<=25.6 and X(2) is an 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
110 REM FOR J44=1 TO 3
112 A(1)=.1+ RND*99.9
121 A(2)= CINT( RND*25.6 )
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
170 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 BASIC manual [11]. Copied by hand from the screen, the complete output through
JJJJ=-31997 is as follows:
49.99962 25 1.499997
-1.14382E-11 -32000
85.51085 24 1.657802
-3.974902E-03 -31999
50.00013 25 1.500001
-2.172113E-12 -31998
50.00013 25 1.500001
-2.172113E-12 -31997
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=-31997 was 33 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/
Case One: Two Hundred Continuous Variables
Similar to the computer programs of the preceding papers, the first computer program below seeks to solve Schittkowski's Test Problem 305 [14, p. 129] but with 200 unknowns instead of 100 unknowns. Thus, the problem is to minimize the following:
200 200 200
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 -5<= X(j)<=10, j=1, 2, 3,..., 200. See Schittkowski [14, p. 129]. These lower bounds of -5's and these upper bounds of 10's are usually used in the literature--see the Zakharov function [15].
One notes line 144, which is 144 REM GOTO 167.
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 200
112 A(J44)=-5+FIX( RND*16)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 200
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*200)
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
212 FOR J44=1 TO 200
213 IF X(J44)<-5 THEN X(J44)=A(J44)
214 IF X(J44)>10 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 200
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 200
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 200
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 200
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1551 PRINT A(1),A(2),A(3),A(4),A(5)
1552 PRINT A(196),A(197),A(198),A(199),A(200)
1553 PRINT M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1904 GOTO 1922
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1908 PRINT A(21),A(22),A(23),A(24),A(25)
1909 PRINT A(26),A(27),A(28),A(29),A(30)
1910 PRINT A(31),A(32),A(33),A(34),A(35)
1911 PRINT A(36),A(37),A(38),A(39),A(40)
1912 PRINT A(41),A(42),A(43),A(44),A(45)
1913 PRINT A(46),A(47),A(48),A(49),A(50)
1914 PRINT A(51),A(52),A(53),A(54),A(55)
1915 PRINT A(56),A(57),A(58),A(59),A(60)
1916 PRINT A(61),A(62),A(63),A(64),A(65)
1917 PRINT A(66),A(67),A(68),A(69),A(70)
1918 PRINT A(71),A(72),A(73),A(74),A(75)
1919 PRINT A(76),A(77),A(78),A(79),A(80)
1920 PRINT A(81),A(82),A(83),A(84),A(85)
1921 PRINT A(86),A(87),A(88),A(89),A(90)
1922 PRINT A(1),A(2),A(3),A(4),A(5)
1923 PRINT A(196),A(197),A(198),A(199),A(200)
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 BASIC manual [11]. Copied by hand from the screen, the output partly through
JJJJ=-32000 is as follows:
-5.987215E-19 8.747792E-09 -4.113391E-15
-5.474714E-07 -9.680436E-07
9.264764E-14 -9.924814E-05 3.112297E-04
3.539233E-04 1.256368E-17
-9.993773E-07 32000
Immediately above there is no rounding by hand. One notes M=-9.993773E-07 and that only ten of the 200 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 shown above was 22 hours.
Case Two: One General Integer Variable and Two Continuous Variables
Similar to the computer program above, the following computer program seeks to solve Hock and Schittkowski's Problem 25 [6, p. 48] plus the modification that the second variable is an integer instead of continuous. Thus, the problem 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
0<=X(2)<=25.6 and X(2) is an 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
110 REM FOR J44=1 TO 3
112 A(1)=.1+ RND*99.9
121 A(2)= CINT( RND*25.6 )
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
170 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 BASIC manual [11]. Copied by hand from the screen, the complete output through
JJJJ=-31997 is as follows:
49.99962 25 1.499997
-1.14382E-11 -32000
85.51085 24 1.657802
-3.974902E-03 -31999
50.00013 25 1.500001
-2.172113E-12 -31998
50.00013 25 1.500001
-2.172113E-12 -31997
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=-31997 was 33 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/
Tuesday, June 10, 2014
Unified Computer Programs for Nonlinear Integer/Continuous/Discrete Programming Problems
Jsun Yui Wong
Case One: One Hundred Continuous Variables
Similar to the computer programs of the preceding papers, the first computer program below seeks to solve Schittkowski's Problem 305 [14, p. 129]. This problem is to minimize the following:
100 100 100
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 -5<= X(j)<=10, j=1, 2, 3,..., 100. See Schittkowski [14, p. 129]. The lower bounds
-5's and the upper bounds 10's are usually used in the literature; see Zakharov function [15].
Noteworthy is line 163, which is 163 IF RND<.5 THEN X(B)=(A(B)-.00005) ELSE X(B)=(A(B) +.00005 ). One notes line 1, which is 1 DEFINT J,K,B.
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 100
112 A(J44)=-5+FIX( RND*16)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*100)
144 REM GOTO 167
145 IF RND<.5 THEN 163 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
163 IF RND<.5 THEN X(B)=(A(B)-.00005) ELSE X(B)=(A(B) +.00005 )
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
212 FOR J44=1 TO 100
213 IF X(J44)<-5 THEN X(J44)=A(J44)
214 IF X(J44)>10 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 100
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 100
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 100
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 100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1499 REM PRINT M,JJJJ,A(1),A(50),A(100)
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1904 GOTO 1923
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1908 PRINT A(21),A(22),A(23),A(24),A(25)
1909 PRINT A(26),A(27),A(28),A(29),A(30)
1910 PRINT A(31),A(32),A(33),A(34),A(35)
1911 PRINT A(36),A(37),A(38),A(39),A(40)
1912 PRINT A(41),A(42),A(43),A(44),A(45)
1913 PRINT A(46),A(47),A(48),A(49),A(50)
1914 PRINT A(51),A(52),A(53),A(54),A(55)
1915 PRINT A(56),A(57),A(58),A(59),A(60)
1916 PRINT A(61),A(62),A(63),A(64),A(65)
1917 PRINT A(66),A(67),A(68),A(69),A(70)
1918 PRINT A(71),A(72),A(73),A(74),A(75)
1919 PRINT A(76),A(77),A(78),A(79),A(80)
1920 PRINT A(81),A(82),A(83),A(84),A(85)
1921 PRINT A(86),A(87),A(88),A(89),A(90)
1922 PRINT A(91),A(92),A(93),A(94),A(95)
1923 PRINT A(96),A(97),A(98),A(99),A(100)
1929 PRINT M,JJJJ,A(1),A(2)
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See [11]. Copied by hand from the screen, the complete output through JJJJ=-31999 is as follows:
-2.908819E-06 -1.460285E-08 -5.40603E-09
1.229637E-09 -2.659363E-08
-7.57602E-08 -32000 -4.06726E-09 -1.040462E-09
5.002695E-05 1.646549E-08 5.002997E-05
1.000198E-04 1.029982E-04
-7.82044E-08 -31999 8.432835E-09 3.38332E-09
Immediately above there is no rounding by hand. One notes M=-7.57602E-08 at JJJJ=-32000 and
M=-7.82044E-08 at JJJJ=-31999 and that only seven of the 100 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 four hours and forty minutes.
Case Two: Fifty General Integer Variables
Similar to the computer program above, the following computer program below seeks to solve a problem based on Schittkowski's Problem 304 [14, p. 128]. For this case only integer solutions are of interest. Thus, the problem is to minimize the following:
50 50 50
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 -5000<= X(j)<=5000, X(j) integer, j=1, 2, 3,..., 50. See Schittkowski [14, p. 128].
One notes line 1, which is 1 DEFINT J,K,B,X.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
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)=-5000+FIX( RND*10001!)
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 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)
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
212 FOR J44=1 TO 50
213 IF X(J44)<-5000 THEN X(J44)=A(J44)
214 IF X(J44)>5000 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 50
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 50
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 50
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 50
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1908 PRINT A(21),A(22),A(23),A(24),A(25)
1909 PRINT A(26),A(27),A(28),A(29),A(30)
1910 PRINT A(31),A(32),A(33),A(34),A(35)
1911 PRINT A(36),A(37),A(38),A(39),A(40)
1912 PRINT A(41),A(42),A(43),A(44),A(45)
1913 PRINT A(46),A(47),A(48),A(49),A(50)
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 [11]. Copied by hand from the screen, the complete output through JJJJ=-31995 is as follows:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 -1
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 0 -1
0 0 0 0 0
0 0 0 0 0
-4 -32000
0 0 1 0 0
0 0 0 -1 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 1
0 0 -1 0 -1
-7 -31999
0 0 0 0 0
0 0 0 1 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 1
0 0 0 0 0
0 0 0 -1 0
-3 -31998
0 0 0 0 -1
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 1 0
0 0 0 -1 0
-1 0 0 0 0
-6 -31997
0 0 0 0 0
0 0 0 0 0
-1 -1 0 0 0
0 0 0 0 0
0 0 1 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
-3 -31996
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 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -31995
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=-31995 was one hour and a half.
Case Three: All Continuous Variables
.
Similar to the computer program above, the following computer program seeks to solve Hock and Schittkowski's Problem 110 [6, p. 110]; the source of this problem is Himmelblau [5, Problem 17, p. 416]. The problem is to minimize the following:
10 10
SIGMA [ ( LOG ( X(i)-2) )^2 +( LOG (10-X(i) ) )^2 ] -[ PI X(i) ]^0.2
i=1 i=1
subject to 2.001 <= X(i) <= 9.999, i=1,..., 10. See Himmelblau [5, p. 416] and/or Hock and Schittkowski [6, p. 110].
One notes that while line 1 of Case Two is 1 DEFINT J,K,B,X, line 1 below is
1 DEFINT J,K,B.
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 10
111 A(J44)=2.001+( RND*7.998)
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<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
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
202 FOR J44=1 TO 10
213 IF X(J44)<2.001 THEN X(J44)=A(J44)
214 IF X(J44)>9.998999 THEN X(J44)=A(J44)
215 NEXT J44
217 GOTO 301
220 SUMM=0
222 FOR J44=1 TO 5
225 SUMM=SUMM+100*(X(J44)^2+X(J44+5) ) ^2 + ( X(J44)- 1 )^2 +90* ( X(J44+10)^2+X(J44+15) )^2 + (X(J44+10)-1 )^2+10.1*((X(J44+5)-1)^2+(X(J44+15)-1)^2)+19.8*(X(J44+5)-1)*(X(J44+15)-1)
226 NEXT J44
301 SONE=0
303 FOR J44=1 TO 10
306 IF ( X(J44)-2)<.0001 THEN 1670
307 IF (10-X(J44) ) <.0001 THEN 1670
308 SONE=SONE+ ( LOG ( X(J44)-2) )^2 +( LOG (10-X(J44) ) )^2
309 NEXT J44
371 PROD=1
373 FOR J44=1 TO 10
375 PROD=PROD*X(J44)
379 NEXT J44
381 PRODPROD=PROD^.2
444 REM PD1=-SUMM
447 PD1=-SONE+PRODPROD
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 REM IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1927 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 [11]. Copied by hand from the screen, the complete output through JJJJ=-31998 is as follows:
9.350314 9.351601 9.351381 9.350377 9.350853
9.348362 9.350105 9.350344 9.351095 9.350406
45.77849 -32000
9.349132 9.350902 9.352669 9.348278 9.348939
9.348385 9.35071 9.352462 9.351089 9.349948
45.77846 -31999
9.35129 9.351302 9.35021 9.349292 9.350794
9.350446 9.348862 9.351012 9.348884 9.350422
45.7785 -31998
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=-31998 was 32 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] Ssurjano, 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/
Case One: One Hundred Continuous Variables
Similar to the computer programs of the preceding papers, the first computer program below seeks to solve Schittkowski's Problem 305 [14, p. 129]. This problem is to minimize the following:
100 100 100
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 -5<= X(j)<=10, j=1, 2, 3,..., 100. See Schittkowski [14, p. 129]. The lower bounds
-5's and the upper bounds 10's are usually used in the literature; see Zakharov function [15].
Noteworthy is line 163, which is 163 IF RND<.5 THEN X(B)=(A(B)-.00005) ELSE X(B)=(A(B) +.00005 ). One notes line 1, which is 1 DEFINT J,K,B.
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 100
112 A(J44)=-5+FIX( RND*16)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*100)
144 REM GOTO 167
145 IF RND<.5 THEN 163 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
163 IF RND<.5 THEN X(B)=(A(B)-.00005) ELSE X(B)=(A(B) +.00005 )
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
212 FOR J44=1 TO 100
213 IF X(J44)<-5 THEN X(J44)=A(J44)
214 IF X(J44)>10 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 100
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 100
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 100
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 100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1499 REM PRINT M,JJJJ,A(1),A(50),A(100)
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1904 GOTO 1923
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1908 PRINT A(21),A(22),A(23),A(24),A(25)
1909 PRINT A(26),A(27),A(28),A(29),A(30)
1910 PRINT A(31),A(32),A(33),A(34),A(35)
1911 PRINT A(36),A(37),A(38),A(39),A(40)
1912 PRINT A(41),A(42),A(43),A(44),A(45)
1913 PRINT A(46),A(47),A(48),A(49),A(50)
1914 PRINT A(51),A(52),A(53),A(54),A(55)
1915 PRINT A(56),A(57),A(58),A(59),A(60)
1916 PRINT A(61),A(62),A(63),A(64),A(65)
1917 PRINT A(66),A(67),A(68),A(69),A(70)
1918 PRINT A(71),A(72),A(73),A(74),A(75)
1919 PRINT A(76),A(77),A(78),A(79),A(80)
1920 PRINT A(81),A(82),A(83),A(84),A(85)
1921 PRINT A(86),A(87),A(88),A(89),A(90)
1922 PRINT A(91),A(92),A(93),A(94),A(95)
1923 PRINT A(96),A(97),A(98),A(99),A(100)
1929 PRINT M,JJJJ,A(1),A(2)
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See [11]. Copied by hand from the screen, the complete output through JJJJ=-31999 is as follows:
-2.908819E-06 -1.460285E-08 -5.40603E-09
1.229637E-09 -2.659363E-08
-7.57602E-08 -32000 -4.06726E-09 -1.040462E-09
5.002695E-05 1.646549E-08 5.002997E-05
1.000198E-04 1.029982E-04
-7.82044E-08 -31999 8.432835E-09 3.38332E-09
Immediately above there is no rounding by hand. One notes M=-7.57602E-08 at JJJJ=-32000 and
M=-7.82044E-08 at JJJJ=-31999 and that only seven of the 100 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 four hours and forty minutes.
Case Two: Fifty General Integer Variables
Similar to the computer program above, the following computer program below seeks to solve a problem based on Schittkowski's Problem 304 [14, p. 128]. For this case only integer solutions are of interest. Thus, the problem is to minimize the following:
50 50 50
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 -5000<= X(j)<=5000, X(j) integer, j=1, 2, 3,..., 50. See Schittkowski [14, p. 128].
One notes line 1, which is 1 DEFINT J,K,B,X.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
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)=-5000+FIX( RND*10001!)
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 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)
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
212 FOR J44=1 TO 50
213 IF X(J44)<-5000 THEN X(J44)=A(J44)
214 IF X(J44)>5000 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 50
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 50
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 50
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 50
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1908 PRINT A(21),A(22),A(23),A(24),A(25)
1909 PRINT A(26),A(27),A(28),A(29),A(30)
1910 PRINT A(31),A(32),A(33),A(34),A(35)
1911 PRINT A(36),A(37),A(38),A(39),A(40)
1912 PRINT A(41),A(42),A(43),A(44),A(45)
1913 PRINT A(46),A(47),A(48),A(49),A(50)
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 [11]. Copied by hand from the screen, the complete output through JJJJ=-31995 is as follows:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 -1
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 0 -1
0 0 0 0 0
0 0 0 0 0
-4 -32000
0 0 1 0 0
0 0 0 -1 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 1
0 0 -1 0 -1
-7 -31999
0 0 0 0 0
0 0 0 1 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 1
0 0 0 0 0
0 0 0 -1 0
-3 -31998
0 0 0 0 -1
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 1 0
0 0 0 -1 0
-1 0 0 0 0
-6 -31997
0 0 0 0 0
0 0 0 0 0
-1 -1 0 0 0
0 0 0 0 0
0 0 1 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
-3 -31996
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 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -31995
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=-31995 was one hour and a half.
Case Three: All Continuous Variables
.
Similar to the computer program above, the following computer program seeks to solve Hock and Schittkowski's Problem 110 [6, p. 110]; the source of this problem is Himmelblau [5, Problem 17, p. 416]. The problem is to minimize the following:
10 10
SIGMA [ ( LOG ( X(i)-2) )^2 +( LOG (10-X(i) ) )^2 ] -[ PI X(i) ]^0.2
i=1 i=1
subject to 2.001 <= X(i) <= 9.999, i=1,..., 10. See Himmelblau [5, p. 416] and/or Hock and Schittkowski [6, p. 110].
One notes that while line 1 of Case Two is 1 DEFINT J,K,B,X, line 1 below is
1 DEFINT J,K,B.
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 10
111 A(J44)=2.001+( RND*7.998)
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<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
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
202 FOR J44=1 TO 10
213 IF X(J44)<2.001 THEN X(J44)=A(J44)
214 IF X(J44)>9.998999 THEN X(J44)=A(J44)
215 NEXT J44
217 GOTO 301
220 SUMM=0
222 FOR J44=1 TO 5
225 SUMM=SUMM+100*(X(J44)^2+X(J44+5) ) ^2 + ( X(J44)- 1 )^2 +90* ( X(J44+10)^2+X(J44+15) )^2 + (X(J44+10)-1 )^2+10.1*((X(J44+5)-1)^2+(X(J44+15)-1)^2)+19.8*(X(J44+5)-1)*(X(J44+15)-1)
226 NEXT J44
301 SONE=0
303 FOR J44=1 TO 10
306 IF ( X(J44)-2)<.0001 THEN 1670
307 IF (10-X(J44) ) <.0001 THEN 1670
308 SONE=SONE+ ( LOG ( X(J44)-2) )^2 +( LOG (10-X(J44) ) )^2
309 NEXT J44
371 PROD=1
373 FOR J44=1 TO 10
375 PROD=PROD*X(J44)
379 NEXT J44
381 PRODPROD=PROD^.2
444 REM PD1=-SUMM
447 PD1=-SONE+PRODPROD
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 REM IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1927 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 [11]. Copied by hand from the screen, the complete output through JJJJ=-31998 is as follows:
9.350314 9.351601 9.351381 9.350377 9.350853
9.348362 9.350105 9.350344 9.351095 9.350406
45.77849 -32000
9.349132 9.350902 9.352669 9.348278 9.348939
9.348385 9.35071 9.352462 9.351089 9.349948
45.77846 -31999
9.35129 9.351302 9.35021 9.349292 9.350794
9.350446 9.348862 9.351012 9.348884 9.350422
45.7785 -31998
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=-31998 was 32 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] Ssurjano, 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/
Saturday, June 7, 2014
A Unifying Algorithm for Nonlinear Integer/Continuous/Discrete Programming
Jsun Yui Wong
Case One: All Integer Variables
Similar to the computer program of the preceding paper, the first of the two computer programs below seeks to solve a problem based on Schittkowski's Problem 303 [13, p. 127]. Here only integer solutions are of interest. Thus, in the present paper the first problem is to minimize the following:
20 20 20
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 -5000<= X(j)<=5000, X(j) integer, j=1, 2, 3,..., 20. See Schittkowski [13, p. 127].
One notes line 1, which is 1 DEFINT J,K,B,X.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(1001),X(1001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 20
112 A(J44)=-5000+FIX( RND*10001!)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 20
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*20)
144 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)
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
212 FOR J44=1 TO 20
213 IF X(J44)<-5000 THEN X(J44)=A(J44)
214 IF X(J44)>5000 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 20
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 20
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 20
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 20
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1927 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 [10]. Copied by hand from the screen, the complete output through JJJJ=-31998 is as follows:
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 -31999
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -31998
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=-31998 was two minutes and ten seconds.
Case Two: All Continuous Variables
.
Similar to the computer program above, the following computer program seeks to solve Hock and Schittkowski's Problem 110 [5, p. 110]; the source of this problem is Himmelblau [4, Problem 17, p. 416]. The problem is to minimize the following:
10 10
SIGMA [ ( LOG ( X(i)-2) )^2 +( LOG (10-X(i) ) )^2 ] -[ PI X(i) ]^.2
i=1 i=1
subject to 2.001 <= X(i) <= 9.999, i=1,..., 10. See Himmelblau [4, p. 416] and Hock and Schittkowski [5, p. 110].
One notes that while line 1 above is 1 DEFINT J,K,B,X, line 1 below is
1 DEFINT J,K,B.
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 10
111 A(J44)=2.001+( RND*7.998)
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<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
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
202 FOR J44=1 TO 10
213 IF X(J44)<2.001 THEN X(J44)=A(J44)
214 IF X(J44)>9.998999 THEN X(J44)=A(J44)
215 NEXT J44
217 GOTO 301
220 SUMM=0
222 FOR J44=1 TO 5
225 SUMM=SUMM+100*(X(J44)^2+X(J44+5) ) ^2 + ( X(J44)- 1 )^2 +90* ( X(J44+10)^2+X(J44+15) )^2 + (X(J44+10)-1 )^2+10.1*((X(J44+5)-1)^2+(X(J44+15)-1)^2)+19.8*(X(J44+5)-1)*(X(J44+15)-1)
226 NEXT J44
301 SONE=0
303 FOR J44=1 TO 10
306 IF ( X(J44)-2)<.0001 THEN 1670
307 IF (10-X(J44) ) <.0001 THEN 1670
308 SONE=SONE+ ( LOG ( X(J44)-2) )^2 +( LOG (10-X(J44) ) )^2
309 NEXT J44
371 PROD=1
373 FOR J44=1 TO 10
375 PROD=PROD*X(J44)
379 NEXT J44
381 PRODPROD=PROD^.2
444 REM PD1=-SUMM
447 PD1=-SONE+PRODPROD
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 REM IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1927 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 [10]. Copied by hand from the screen, the complete output through JJJJ=-31998 is as follows:
9.350314 9.351601 9.351381 9.350377 9.350853
9.348362 9.350105 9.350344 9.351095 9.350406
45.77849 -32000
9.349132 9.350902 9.352669 9.348278 9.348939
9.348385 9.35071 9.352462 9.351089 9.349948
45.77846 -31999
9.35129 9.351302 9.35021 9.349292 9.350794
9.350446 9.348862 9.351012 9.348884 9.350422
45.7785 -31998
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=-31998 was 32 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 MonthlyThe American , Volume 18 #2, pp. 29-32.
[4] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[5] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[6] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[7] 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.
[8] 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.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[10] 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.
[11] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[12] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[13] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[14] 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/
[15] 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/
Case One: All Integer Variables
Similar to the computer program of the preceding paper, the first of the two computer programs below seeks to solve a problem based on Schittkowski's Problem 303 [13, p. 127]. Here only integer solutions are of interest. Thus, in the present paper the first problem is to minimize the following:
20 20 20
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 -5000<= X(j)<=5000, X(j) integer, j=1, 2, 3,..., 20. See Schittkowski [13, p. 127].
One notes line 1, which is 1 DEFINT J,K,B,X.
0 REM DEFDBL A-Z
1 DEFINT J,K,B,X
2 DIM A(1001),X(1001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 20
112 A(J44)=-5000+FIX( RND*10001!)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 20
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*20)
144 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)
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
212 FOR J44=1 TO 20
213 IF X(J44)<-5000 THEN X(J44)=A(J44)
214 IF X(J44)>5000 THEN X(J44)=A(J44)
215 NEXT J44
301 SONE=0
303 FOR J44=1 TO 20
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 20
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 20
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 20
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1927 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 [10]. Copied by hand from the screen, the complete output through JJJJ=-31998 is as follows:
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 -31999
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -31998
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=-31998 was two minutes and ten seconds.
Case Two: All Continuous Variables
.
Similar to the computer program above, the following computer program seeks to solve Hock and Schittkowski's Problem 110 [5, p. 110]; the source of this problem is Himmelblau [4, Problem 17, p. 416]. The problem is to minimize the following:
10 10
SIGMA [ ( LOG ( X(i)-2) )^2 +( LOG (10-X(i) ) )^2 ] -[ PI X(i) ]^.2
i=1 i=1
subject to 2.001 <= X(i) <= 9.999, i=1,..., 10. See Himmelblau [4, p. 416] and Hock and Schittkowski [5, p. 110].
One notes that while line 1 above is 1 DEFINT J,K,B,X, line 1 below is
1 DEFINT J,K,B.
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 10
111 A(J44)=2.001+( RND*7.998)
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<.5 THEN 150 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
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
202 FOR J44=1 TO 10
213 IF X(J44)<2.001 THEN X(J44)=A(J44)
214 IF X(J44)>9.998999 THEN X(J44)=A(J44)
215 NEXT J44
217 GOTO 301
220 SUMM=0
222 FOR J44=1 TO 5
225 SUMM=SUMM+100*(X(J44)^2+X(J44+5) ) ^2 + ( X(J44)- 1 )^2 +90* ( X(J44+10)^2+X(J44+15) )^2 + (X(J44+10)-1 )^2+10.1*((X(J44+5)-1)^2+(X(J44+15)-1)^2)+19.8*(X(J44+5)-1)*(X(J44+15)-1)
226 NEXT J44
301 SONE=0
303 FOR J44=1 TO 10
306 IF ( X(J44)-2)<.0001 THEN 1670
307 IF (10-X(J44) ) <.0001 THEN 1670
308 SONE=SONE+ ( LOG ( X(J44)-2) )^2 +( LOG (10-X(J44) ) )^2
309 NEXT J44
371 PROD=1
373 FOR J44=1 TO 10
375 PROD=PROD*X(J44)
379 NEXT J44
381 PRODPROD=PROD^.2
444 REM PD1=-SUMM
447 PD1=-SONE+PRODPROD
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 REM IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1927 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 [10]. Copied by hand from the screen, the complete output through JJJJ=-31998 is as follows:
9.350314 9.351601 9.351381 9.350377 9.350853
9.348362 9.350105 9.350344 9.351095 9.350406
45.77849 -32000
9.349132 9.350902 9.352669 9.348278 9.348939
9.348385 9.35071 9.352462 9.351089 9.349948
45.77846 -31999
9.35129 9.351302 9.35021 9.349292 9.350794
9.350446 9.348862 9.351012 9.348884 9.350422
45.7785 -31998
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=-31998 was 32 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 MonthlyThe American , Volume 18 #2, pp. 29-32.
[4] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[5] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[6] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[7] 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.
[8] 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.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[10] 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.
[11] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[12] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[13] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[14] 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/
[15] 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/
Tuesday, June 3, 2014
Testing the Nonlinear Integer Programming Solver with a Problem Based on Schittkowski's Test Problem 287 with Lower Bounds of -32000's and Upper Bounds of 32000's for the Integer Variables
Jsun Yui Wong
Similar to the computer program of the preceding paper, the following computer program seeks to solve a problem based on Schittkowski's Problem 287 [11, p. 111]; only integer solutions are of interest in the present paper. Thus, in the present paper the problem is to minimize the following:
5
SIGMA [ 100*(X(i)^2+X(i+5) ) ^2 + ( X(i)- 1 )^2 +90* ( X(i+10)^2
i=1
+X(i+15) )^2 + (X(i+10)-1 )^2+10.1*((X(i+5)-1)^2+(X(i+15)-1)^2)+19.8*(X(i+5)-1)*(X
(i+15)-1) ]
subject to -32000<= X(j)<=32000, X(j) integer, j=1, 2, 3,..., 20. See line 225 and Schittkowski [11, p. 111].
One notes line 112, line 213, and line 214, which are 112 A(J44)=-32000+FIX( RND*64001), 213 IF X(J44)<-32000 THEN X(J44)=A(J44), and 214 IF X(J44)>32000 THEN X(J44)=A(J44), respectively.
0 REM DEFDBL A-Z
1 DEFINT J,K,X,B
2 DIM A(1001),X(1001),T(100)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 20
112 A(J44)=-32000+FIX( RND*64001!)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 20
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*20)
143 IF RND<.5 THEN 150 ELSE 167
144 REM GOTO 167
150 R=(1-RND*2)*A(B)
155 T(B)=(A(B) +RND^3*R)
158 IF ABS(T(B))>32000 THEN 1670
159 X(B)=T(B)
160 REM X(B)=(A(B) +RND^3*R)
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
212 FOR J44=1 TO 20
213 IF X(J44)<-32000 THEN X(J44)=A(J44)
214 IF X(J44)>32000 THEN X(J44)=A(J44)
215 NEXT J44
220 SUMM=0
222 FOR J44=1 TO 5
225 SUMM=SUMM+100*(X(J44)^2+X(J44+5) ) ^2 + ( X(J44)- 1 )^2 +90* ( X(J44+10)^2+X(J44+15) )^2 + (X(J44+10)-1 )^2+10.1*((X(J44+5)-1)^2+(X(J44+15)-1)^2)+19.8*(X(J44+5)-1)*(X(J44+15)-1)
226 NEXT J44
231 REM SUMN=0
233 REM FOR J45=1 TO 5
236 REM SUMN =SUMN+10.1*( (X(J45+5)-1)^2 +(X(J45+15)-1)^2 )+ 19.8*( X(J45+5) -1 ) *(X(J45+15) -1)
239 REM NEXT J45
244 REM SUMS=SUMM
444 PD1=-SUMM
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 20
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 1557
1544 IF M<-8 THEN 1557
1546 PRINT I,A(30),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1927 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 [8]. Copied by hand from the screen, the complete output through JJJJ=-31990 is as follows:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -32000
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31999
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31998
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31997
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31996
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31995
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31994
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31993
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31992
-2 0 0 0 0
-4 0 0 0 0
0 0 0 0 0
1 0 0 0 0
-520.5 -31991
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31990
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=-31990 was two minutes and a half.
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] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[5] 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.
[6] 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.
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[8] 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.
[9] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[10] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[11] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[12] 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/
[13] 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/
Similar to the computer program of the preceding paper, the following computer program seeks to solve a problem based on Schittkowski's Problem 287 [11, p. 111]; only integer solutions are of interest in the present paper. Thus, in the present paper the problem is to minimize the following:
5
SIGMA [ 100*(X(i)^2+X(i+5) ) ^2 + ( X(i)- 1 )^2 +90* ( X(i+10)^2
i=1
+X(i+15) )^2 + (X(i+10)-1 )^2+10.1*((X(i+5)-1)^2+(X(i+15)-1)^2)+19.8*(X(i+5)-1)*(X
(i+15)-1) ]
subject to -32000<= X(j)<=32000, X(j) integer, j=1, 2, 3,..., 20. See line 225 and Schittkowski [11, p. 111].
One notes line 112, line 213, and line 214, which are 112 A(J44)=-32000+FIX( RND*64001), 213 IF X(J44)<-32000 THEN X(J44)=A(J44), and 214 IF X(J44)>32000 THEN X(J44)=A(J44), respectively.
0 REM DEFDBL A-Z
1 DEFINT J,K,X,B
2 DIM A(1001),X(1001),T(100)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 20
112 A(J44)=-32000+FIX( RND*64001!)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 20
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*20)
143 IF RND<.5 THEN 150 ELSE 167
144 REM GOTO 167
150 R=(1-RND*2)*A(B)
155 T(B)=(A(B) +RND^3*R)
158 IF ABS(T(B))>32000 THEN 1670
159 X(B)=T(B)
160 REM X(B)=(A(B) +RND^3*R)
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
212 FOR J44=1 TO 20
213 IF X(J44)<-32000 THEN X(J44)=A(J44)
214 IF X(J44)>32000 THEN X(J44)=A(J44)
215 NEXT J44
220 SUMM=0
222 FOR J44=1 TO 5
225 SUMM=SUMM+100*(X(J44)^2+X(J44+5) ) ^2 + ( X(J44)- 1 )^2 +90* ( X(J44+10)^2+X(J44+15) )^2 + (X(J44+10)-1 )^2+10.1*((X(J44+5)-1)^2+(X(J44+15)-1)^2)+19.8*(X(J44+5)-1)*(X(J44+15)-1)
226 NEXT J44
231 REM SUMN=0
233 REM FOR J45=1 TO 5
236 REM SUMN =SUMN+10.1*( (X(J45+5)-1)^2 +(X(J45+15)-1)^2 )+ 19.8*( X(J45+5) -1 ) *(X(J45+15) -1)
239 REM NEXT J45
244 REM SUMS=SUMM
444 PD1=-SUMM
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 20
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 1557
1544 IF M<-8 THEN 1557
1546 PRINT I,A(30),M,JJJJ
1557 GOTO 128
1670 NEXT I
1889 IF M<-999 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1927 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 [8]. Copied by hand from the screen, the complete output through JJJJ=-31990 is as follows:
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -32000
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31999
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31998
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31997
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31996
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31995
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31994
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31993
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31992
-2 0 0 0 0
-4 0 0 0 0
0 0 0 0 0
1 0 0 0 0
-520.5 -31991
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-210 -31990
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=-31990 was two minutes and a half.
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] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING
[5] 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.
[6] 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.
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951
[8] 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.
[9] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).
[10] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).
[11] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[12] 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/
[13] 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/
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