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/
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