Nonlinear Integer/Continuous/Discrete Programming Solver: June 2014

Friday, June 27, 2014

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

Jsun Yui Wong

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

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

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

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

subject to

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

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

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

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

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

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

0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(1001),X(1001)
11 C(1)=-6.089 :C(2)=- 17.164 :C(3)=- 34.054:C(4)=- 5.914 :C(5)=- 24.721
13 C(6)=-14.986 :C(7)=- 24.1 :C(8)=- 10.708:C(9)=-26.662 :C(10)=- 22.179
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 10
113 A(J44)= RND*(.1)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10)
144 REM GOTO 167
145 IF RND<.33 THEN 150 ELSE IF RND<.5 THEN 163 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.0001) ELSE X(B)=(A(B) +.0001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
370 FOR J44=1 TO 10
371 IF X(J44)<.00001 THEN 1670
372 IF X(J44)>10 THEN 1670
375 NEXT J44
380 X(7)= -X(4) +2* X(5) - X(6) + 1
381 X(10)= -X(3) - X(7) - X(8) -2* X(9) + 1
383 X(1)=2*X(2) -2* X(3) - X(6) - X(10) + 2
396 FOR J44=1 TO 10
397 IF X(J44)<.00001 THEN 1670
398 IF X(J44)>10 THEN 1670
399 NEXT J44
400 SONE=0
401 FOR J44=1 TO 10
403 SONE=SONE+X(J44)
405 NEXT J44
410 STWO=0
411 FOR J44=1 TO 10
412 IF ( X(J44)/SONE ) < 1E-11 THEN 1670
413 STWO=STWO+X(J44) * ( C(J44) +LOG( X(J44)/SONE ) )
415 NEXT J44
455 PD1= - STWO
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-999999! THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10)
1929 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the complete output through
JJJJ=-31996 is shown below:

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

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

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

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

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

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31996 was thirty-one minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

A Unified Computer Program for Schittkowski's Test Problem 377

Jsun Yui Wong

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

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

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

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

subject to

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

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

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

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

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

0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(1001),X(1001)
11 C(1)=-6.089 :C(2)=- 17.164 :C(3)=- 34.054:C(4)=- 5.914 :C(5)=- 24.721
13 C(6)=-14.986 :C(7)=- 24.1 :C(8)=- 10.708:C(9)=-26.662 :C(10)=- 22.179
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1.5D+38
110 FOR J44=1 TO 10
113 A(J44)= RND*(.1)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10)
144 REM GOTO 167
145 IF RND<.33 THEN 150 ELSE IF RND<.5 THEN 163 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.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
370 FOR J44=1 TO 10
371 IF X(J44)<.00001 THEN 1670
372 IF X(J44)>10 THEN 1670
375 NEXT J44
380 X(7)= -X(4) +2* X(5) - X(6) + 1
381 X(10)= -X(3) - X(7) - X(8) -2* X(9) + 1
383 X(1)=2*X(2) -2* X(3) - X(6) - X(10) + 2
396 FOR J44=1 TO 10
397 IF X(J44)<.00001 THEN 1670
398 IF X(J44)>10 THEN 1670
399 NEXT J44
400 SONE=0
401 FOR J44=1 TO 10
403 SONE=SONE+X(J44)
405 NEXT J44
410 STWO=0
411 FOR J44=1 TO 10
412 IF ( X(J44)/SONE ) < 1E-11 THEN 1670
413 STWO=STWO+X(J44) * ( C(J44) +LOG( X(J44)/SONE ) )
415 NEXT J44
455 PD1= - STWO
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-999999! THEN 1999
1923 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10)
1929 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the complete output through
JJJJ=-31996 is shown below:

9.999997 9.921316 .9206928 9.999953 9.538964
10 7.797528E-02 5.814566E-05
1.181062E-05 1.250207E-03
794.0433 -32000

9.999997 9.952159 .9517721 9.999995 9.523699
9.999996 .0474062 1.866661E-05 1.0937E-05
7.811189E-04
794.4446 -31999

10 9.9450008 .9445719 9.999955 9.527228
9.999991 5.451107E-02 1.329805E-05
1.000197E-05 8.836985E-04
794.3527 -31998

10 9.954234 .9541342 9.999999 9.522805
9.999994 4.561711E-02 2.054263E-05
1.107116E-05 2.059937E-04
794.4748 -31997

9.999995 9.976694 .9764906 10 9.511514
9.999997 2.303219E-02 4.105517E-05
1.103431E-05 4.140735E-04
794.7468 -31996

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31996 was seven minutes and ten seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

Wednesday, June 25, 2014

A Unified Computer Program for Schittkowski's Test Problem 395 but with 3000 Unknowns instead of 50 Unknowns

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's last test problem [14, p. 213, Test Problem 395] but with 3000 unknowns instead of 50 unknowns. The source of this Test Problem 395 is given in Schittkowski [14]. Thus, the problem is to minimize the following:

3000
SIGMA i*(X(i)^2+X(i)^4 )
i=1

subjec to

3000
SIGMA X(i)^2 =1.
i=1

0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(3000),X(3000)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 3000
112 A(J44)=-.01+RND*.02
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 3000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*3000)
144 REM GOTO 167
145 IF RND<.33 THEN 150 ELSE IF RND<.5 THEN 163 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
170 GOTO 400
396 FOR J44=1 TO 50
397 REM IF X(J44)<-1 THEN X(J44)=A(J44)
398 REM IF X(J44)>1 THEN X(J44)=A(J44)
399 NEXT J44
400 SONE=0
401 FOR J44=2 TO 3000
403 SONE=SONE-X(J44)^2
404 NEXT J44
405 IF ( 1+SONE )<.0000001 THEN 1670
406 X(1)=( 1+SONE )^(1/2)
410 STWO=0
411 FOR J44=1 TO 3000
413 STWO=STWO+J44*(X(J44)^2+X(J44)^4 )
415 NEXT J44
457 PD1= - STWO
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 3000
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-999999999# THEN 1999
1911 GOTO 1935
1922 LPRINT A(1),A(2),A(3),A(4),A(5)
1923 LPRINT A(6),A(7),A(8),A(9),A(10)
1924 LPRINT A(11),A(12),A(13),A(14),A(15)
1925 LPRINT A(16),A(17),A(18),A(19),A(20)
1926 LPRINT A(21),A(22),A(23),A(24),A(25)
1927 LPRINT A(26),A(27),A(28),A(29),A(30)
1928 LPRINT A(31),A(32),A(33),A(34),A(35)
1929 LPRINT A(36),A(37),A(38),A(39),A(40)
1930 LPRINT A(41),A(42),A(43),A(44),A(45)
1931 LPRINT A(46),A(47),A(48),A(49),A(50)
1935 PRINT A(1),A(2),A(3),A(4),A(5)
1937 PRINT A(2996),A(2997),A(2998),A(2999),A(3000)
1939 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=-32000 is summarized below:

.9129742 -.4080171 -6.71842E-05 -8.418951E-05
2.882922E-05
-4.306535E-06 3.69088E-06 2.975576E-06
-2.237746E-06 -3.466139E-06
-1.916674 -32000

Of the 3000 A's, only the ten A's of line 1935 and line 1937 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=-32000 was five hours 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] 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 24, 2014

A Unified Computer Program for Schittkowski's Test Problem 395 but with 1500 Unknowns instead of 50 Unknowns

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's last test problem [14, p. 213, Test Problem 395] but with 1500 unknowns instead of 50 unknowns. The source of this Test Problem 395 is given in Schittkowski [14]. Thus, the problem is to minimize the following:

1500
SIGMA i*(X(i)^2+X(i)^4 )
i=1

subjec to

1500
SIGMA X(i)^2 =1.
i=1

0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(2001),X(2001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 1500
112 A(J44)=-.01+RND*.02
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 1500
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*1500)
144 REM GOTO 167
145 IF RND<.33 THEN 150 ELSE IF RND<.5 THEN 163 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
170 GOTO 400
396 FOR J44=1 TO 50
397 REM IF X(J44)<-1 THEN X(J44)=A(J44)
398 REM IF X(J44)>1 THEN X(J44)=A(J44)
399 NEXT J44
400 SONE=0
401 FOR J44=2 TO 1500
403 SONE=SONE-X(J44)^2
404 NEXT J44
405 IF ( 1+SONE )<.0000001 THEN 1670
406 X(1)=( 1+SONE )^(1/2)
410 STWO=0
411 FOR J44=1 TO 1500
413 STWO=STWO+J44*(X(J44)^2+X(J44)^4 )
415 NEXT J44
457 PD1= - STWO
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 1500
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-999999999# THEN 1999
1911 GOTO 1935
1922 LPRINT A(1),A(2),A(3),A(4),A(5)
1923 LPRINT A(6),A(7),A(8),A(9),A(10)
1924 LPRINT A(11),A(12),A(13),A(14),A(15)
1925 LPRINT A(16),A(17),A(18),A(19),A(20)
1926 LPRINT A(21),A(22),A(23),A(24),A(25)
1927 LPRINT A(26),A(27),A(28),A(29),A(30)
1928 LPRINT A(31),A(32),A(33),A(34),A(35)
1929 LPRINT A(36),A(37),A(38),A(39),A(40)
1930 LPRINT A(41),A(42),A(43),A(44),A(45)
1931 LPRINT A(46),A(47),A(48),A(49),A(50)
1935 PRINT A(1),A(2),A(3),A(4),A(5)
1937 PRINT A(496),A(497),A(498),A(499),A(500)
1939 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the output through
JJJJ=-31999 is summarized below:

.9127843 -.4084416 -8.869726E-05 -1.162507E-04
-3.313397E-05
-4.366472E-06 -1.032774E-05 8.681908E-06
1.707782E-06 -7.17702E-06
-1.916668 -32000

.912933 -.4081091 -2.284031E-04 1.038767E-04
9.981354E-05
-5.406792E-06 -8.637368E-06 -7.518956E-06
-3.095325E-06 -9.482914E-06
-1.916667 -31999

Of the 1500 A's, only the ten A's of line 1935 and line 1937 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/

Sunday, June 22, 2014

A Unified Computer Program for Schittkowski's Last Test Problem

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Schittkowski's last test problem [14, p. 213, Test Problem 395]. The problem is to minimize the following:

50
SIGMA i*(X(i)^2+X(i)^4 )
i=1

subjec to

50
SIGMA X(i)^2 =1.
i=1

0 REM DEFDBL A-Z
1 DEFINT J,K,B
2 DIM A(1001),X(1001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 50
112 A(J44)=-.1+RND*.2
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 50
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*50)
144 REM GOTO 167
145 IF RND<.33 THEN 150 ELSE IF RND<.5 THEN 163 ELSE 167
150 R=(1-RND*2)*A(B)
160 X(B)=(A(B) +RND^3*R)
162 GOTO 168
163 IF RND<.5 THEN X(B)=(A(B)-.001) ELSE X(B)=(A(B) +.001 )
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1 )
168 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
170 GOTO 400
396 FOR J44=1 TO 50
397 IF X(J44)<-1 THEN X(J44)=A(J44)
398 IF X(J44)>1 THEN X(J44)=A(J44)
399 NEXT J44
400 SONE=0
401 FOR J44=2 TO 50
403 SONE=SONE-X(J44)^2
404 NEXT J44
405 IF ( 1+SONE )<.0000001 THEN 1670
406 X(1)=( 1+SONE )^(1/2)
410 STWO=0
411 FOR J44=1 TO 50
413 STWO=STWO+J44*(X(J44)^2+X(J44)^4 )
415 NEXT J44
457 PD1= - STWO
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 50
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-999999999# THEN 1999
1911 GOTO 1945
1922 LPRINT A(1),A(2),A(3),A(4),A(5)
1923 LPRINT A(6),A(7),A(8),A(9),A(10)
1924 LPRINT A(11),A(12),A(13),A(14),A(15)
1925 LPRINT A(16),A(17),A(18),A(19),A(20)
1926 LPRINT A(21),A(22),A(23),A(24),A(25)
1927 LPRINT A(26),A(27),A(28),A(29),A(30)
1928 LPRINT A(31),A(32),A(33),A(34),A(35)
1929 LPRINT A(36),A(37),A(38),A(39),A(40)
1930 LPRINT A(41),A(42),A(43),A(44),A(45)
1931 LPRINT A(46),A(47),A(48),A(49),A(50)
1939 LPRINT M,JJJJ
1945 PRINT A(1),A(2),A(3),A(4),A(5)
1947 PRINT A(46),A(47),A(48),A(49),A(50)
1949 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See the BASIC manual [11]. Copied by hand from the screen, the output through
JJJJ=-31999 is summarized below:

.9128926 -.4081998 8.617702E-05 -7.757013E-05
-7.474125E-05
2.186212E-05 -3.765578E-06 2.219921E-05
2.747354E-05 4.096901E-06
-1.916666 -32000

.9128328 -.4083324 -8.775965E-04 2.634784E-04
-3.219565E-05
-1.469409E-05 2.279203E-05 1.603028E-05
-1.14861E-05 -3.286808E-05
-1.916667 -31999

Of the fifty A's, only the ten A's of line 1945 and line 1947 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output shown above was 30 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

Saturday, June 21, 2014

Unified Computer Programs for Nonlinear Integer/Continuous/Discrete Programming Problems Including One Here with Mixed Integer Variables and Continuous Variables

Jsun Yui Wong

Case One: Nonlinear Programming Problem with Six Continuous Variables and Six Equality Constraints

Similar to the computer programs of the preceding papers, the computer program below seeks to solve Hock and Schittkowski's Problem 55 [6, p. 78]. The problem is to minimize the following:

X(1) + 2*X(2) + 4*X(5) + EXP(X(1)*X(4) )

subjec to

X(1)+ 2*X(2) +5* X(5) -6 =0
X(1)+ X(2) + X(3) -3 =0
X(4)+ X(5) + X(6) -2 =0
X(1)+X(4)-1 =0
X(2)+X(5)-2 =0
X(3)+X(6)-2 =0
0<=X(i), i=1,2,3,4,5,6, X(1)<=1, X(4)<=1.

One notes that X(7), X(8), and X(9) of line 391, line 393 , and line 395, respectively, are artificial variables.

One also notes line 422, 424, and line 426, which are 422 IF ABS(X(7))<.000001 THEN X(7)=0,
424 IF ABS(X(8))<.000001 THEN X(8)=0, and 426 IF ABS(X(9))<.000001 THEN X(9)=0, respectively.

The arrangement of line 385 through line 395 is to induce domino effect.

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 6
112 A(J44)= RND*(10)
114 NEXT J44
115 A(1)=RND
116 A(4)=RND
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*6)
144 REM GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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
370 FOR J44=1 TO 6
371 IF X(J44)<0 THEN X(J44)=A(J44)
375 NEXT J44
376 IF X(1)>1 THEN X(1)=A(1)
377 IF X(4)>1 THEN X(4)=A(4)
385 X(6)=-X(3)+2
387 X(5)=-X(2)+2
389 X(4)=-X(1)+1
391 X(7 ) =6- X(1)- 2*X(2) -5* X(5)
393 X(8 )=3- X(1)- X(2) - X(3)
395 X(9 )=2- X(4)- X(5) - X(6)
422 IF ABS(X(7))<.000001 THEN X(7)=0
424 IF ABS(X(8))<.000001 THEN X(8)=0
426 IF ABS(X(9))<.000001 THEN X(9)=0
557 PD1= - X(1) -2*X(2) - 4*X(5) - EXP(X(1)*X(4) ) -10000*(ABS( X(7) )+ABS( X(8) )+ABS( X(9) ) )
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-6.5 THEN 1999
1922 PRINT A(1),A(2),A(3),A(4)
1923 PRINT A(5),A(6),A(7),A(8)
1929 PRINT A(9),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=-31991 is as follows:

4.342672E-03 1.334781 1.660876 .9956573
.665219 .3391246 0 0
0 -6.339114 -32000

2.411294E-02 1.341371 1.634515 .9758871
.658629 .3654848 0 0
0 -6.365181 -31994

.1135881 1.371196 1.515215 .8864119
.6288036 .4847851 0 0
0 -6.477124 -31991

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=-31991 was 25 seconds.

Case Two: Nonlinear Programming Problem with Two Integer Variables, Four Continuous Variables, and Six Equality Constraints

The problem here is the problem above plus the two restrictions that X(1) and X(4) are integer variables; these two restrictions are enforced by line 378 and line 379, which are
378 X(1)=CINT( X(1)) and 379 X(4)=CINT( X(4)).

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 6
112 A(J44)= RND*(10)
114 NEXT J44
115 A(1)=RND
116 A(4)=RND
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*6)
144 REM GOTO 167
145 IF RND<.5 THEN 150 ELSE 163
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
370 FOR J44=1 TO 6
371 IF X(J44)<0 THEN X(J44)=A(J44)
375 NEXT J44
376 IF X(1)>1 THEN X(1)=A(1)
377 IF X(4)>1 THEN X(4)=A(4)
378 X(1)=CINT( X(1))
379 X(4)=CINT( X(4))
385 X(6)=-X(3)+2
387 X(5)=-X(2)+2
389 X(4)=-X(1)+1
391 X(7 ) =6- X(1)- 2*X(2) -5* X(5)
393 X(8 )=3- X(1)- X(2) - X(3)
395 X(9 )=2- X(4)- X(5) - X(6)
422 IF ABS(X(7))<.000001 THEN X(7)=0
424 IF ABS(X(8))<.000001 THEN X(8)=0
426 IF ABS(X(9))<.000001 THEN X(9)=0
557 PD1= - X(1) -2*X(2) - 4*X(5) - EXP(X(1)*X(4) ) -10000*(ABS( X(7) )+ABS( X(8) )+ABS( X(9) ) )
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 9
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-6.5 THEN 1999
1922 PRINT A(1),A(2),A(3),A(4)
1923 PRINT A(5),A(6),A(7),A(8)
1929 PRINT A(9),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=-31998 is as follows:

0 1.333333 1.666667 1
.6666666 .3333329 0 0
0 -6.333333 -32000

0 1.333333 1.666668 1
.6666668 .3333323 0 0
0 -6.333334 -31999

0 1.333333 1.666666 1
.6666666 .3333342 0 0
0 -6.333333 -31998

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 8 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[5] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] Jack Lashover (November 12, 2012). Monte Carlo Marching. www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[9] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[10] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Publisher: Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[12] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[13] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[15] S. Surjanovic, Zakharov Function. www.sfu.ca/~ssurjano/zakharov.html

[16] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[17] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

Tuesday, June 17, 2014

Unified Computer Programs for Nonlinear Integer/Continuous/Discrete Programming Problems Including One Here with Nine Thousand Binary 0-1 Integer Variables

Jsun Yui Wong

Case One: Four General Integer Variables with Lower Bounds of -1000's and Upper Bounds of
+1000's

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 -1000's and +1000's instead of -10's and +10's and of the variables here being 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

-1000<=X(i)<=1000, X(i) integer, i=1, 2, 3, 4. See line 112, line 213, and line 214 below.

One notes that 2001*2001*2001*2001 equals 16032024008001.

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(J44)=-1000+ RND*2000
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)<-1000 THEN X(J44)=A(J44)
214 IF X(J44)>1000 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=-31995 is as follows:

1 1 1 1
0 -32000

1 1 1 1
0 -31999

-1 1 -1 1
-8 -31998

1 1 1 1
0 -31997

1 1 -1 1
-4 -31996

1 1 1 1
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 twelve seconds.

Case Two: Nine 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 9000 unknowns instead of 100 unknowns and with the modification that the 9000 unknowns are 0-1 variables. Thus, the problem is to minimize the following:

9000 9000 9000
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,..., 9000. 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(9001),X(9001)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 9000
112 A(J44)=FIX( RND*2)
114 NEXT J44
128 FOR I=1 TO 32000
129 FOR KKQQ=1 TO 9000
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*9000)
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 9000
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 9000
305 SONE=SONE+X(J44)^2
309 NEXT J44
311 SONEONE=SONE
321 STWO=0
323 FOR J44=1 TO 9000
325 STWO=STWO+(1/2)*J44*X(J44)
329 NEXT J44
331 STWOTWO= ( STWO)^2
341 STHR=0
343 FOR J44=1 TO 9000
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 9000
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(8976),A(8977),A(8978),A(8979),A(8980)
1930 PRINT A(8981),A(8982),A(8983),A(8984),A(8985)
1931 PRINT A(8986),A(8987),A(8988),A(8989),A(8990)
1932 PRINT A(8991),A(8992),A(8993),A(8994),A(8995)
1933 PRINT A(8996),A(8997),A(8998),A(8999),A(9000)
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
-4.52189E+07 -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
0 -31999

Immediately above there is no rounding by hand.

One notes that M=-4.52189E+07 at JJJJ=-32000 and M=0 at JJJJ=-31999 and that only 50 A's of 9000 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 five hour 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] 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

Unified Computer Programs for Nonlinear Integer/Continuous/Discrete Programming Problems Including One Here with Seven Thousand Binary 0-1 Integer Variables

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/

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/

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/

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/