Substance

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Illustrative Example from an Article about Generalized Geometric Programs, Third Edition

Jsun Yui Wong

The following computer program tries to solve the small illustrative example on page 283 of Floudas [2].  The source of this example is Avriel, Dembo, and Passy [1, p. 158].

Through line 222 and line 223 a slack variable X(3) is added.  Hence this paper changes the second inequality constraint to an equality constraint while the preceding post changes the first inequality constraint to an equality constraint.

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
76 A(1)=RND
77 A(2)=RND
78 A(3)=RND
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129   FOR KKQQ=1 TO 3
130   X(KKQQ)=A(KKQQ)
131   NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*2))
181 J=1+FIX(RND*2)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
195 REM X(3)=- (1/4)*X(1)-(1/2)*X(2)+(1/16)*X(1)^2+(1/16)*X(2)^2         +1
196 REM IF X(3)<0 THEN 1670
197 REM IF X(2)=0 THEN 1670
198 REM IF ( (1/14)* X(1)^2+(1/14)*X(2)^2+1-(3/7)*X(1)-(3/7)*X(2))>0 THEN 1670
222 X(3)=-(1/14)* X(1)^2-(1/14)*X(2)^2-1+(3/7)*X(1)+(3/7)*X(2)
223 IF X(3)<0 THEN 1670
225 IF ( (1/4)*X(1)+(1/2)*X(2)-(1/16)*X(1)^2-(1/16)*X(2)^2 -1)>0 THEN 1670
454 P1NEWMAY= -X(1)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1461 PP(8)=P(8)
1462 PP(9)=P(9)
1557 GOTO 128
1670 NEXT I
1889 IF M<-1.178 THEN 1999
1900 PRINT A(1),A(2),A(3),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31252 is presented below.  What immediately follows is a hand copy from the computer-monitor screen.

1.177915   2.175375   5.960465E-08   -1.177915
-31949

1.177557   2.176167   5.960465E-08   -1.177557
-31800

1.177415   2.176481   5.960465E-08   -1.177415
-31325

1.177321   2.176689   0   -1.177321   -31252

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31252 was 35 seconds.

References

[1] M. Avriel, R. Dembo, and U. Passy.  Solution of generalized geometric programs.  International Journal for Numerical Methods in Engineering, Vol. 9, pp. 149-168, 1975.      

[2] C. A. Floudas.  Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic Publishers, 2000.

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

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Illustrative Example from an Article about Generalized Geometric Programs, Second Edition

Jsun Yui Wong

The following computer program tries to solve the small illustrative example on page 283 of Floudas [2].  The source of this example is Avriel, Dembo, and Passy [1, p. 158].

Through line 195 and line 196 a slack variable X(3) is added.

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
76 A(1)=RND
77 A(2)=RND
78 A(3)=RND
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129   FOR KKQQ=1 TO 3
130   X(KKQQ)=A(KKQQ)
131   NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*2))
181 J=1+FIX(RND*2)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
195 X(3)=- (1/4)*X(1)-(1/2)*X(2)+(1/16)*X(1)^2+(1/16)*X(2)^2         +1
196 IF X(3)<0 THEN 1670
197 REM IF X(2)=0 THEN 1670
198 IF ( (1/14)* X(1)^2+(1/14)*X(2)^2+1-(3/7)*X(1)-(3/7)*X(2))>0 THEN 1670
454 P1NEWMAY= -X(1)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1461 PP(8)=P(8)
1462 PP(9)=P(9)
1557 GOTO 128
1670 NEXT I
1889 IF M<-1.178 THEN 1999
1900 PRINT A(1),A(2),A(3),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31252 is presented below.  What immediately follows is a hand copy from the computer-monitor screen.

1.177915   2.175375   3.174544E-04   -1.177915
-31949

1.177557   2.176167   1.738072E-04   -1.177557
-31800

1.177415   2.176481   1.167059E-04   -1.177415
-31325

1.177321   2.176689   7.891655E-05   -1.177321
-31252

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31252 was 35 seconds.

References

[1] M. Avriel, R. Dembo, and U. Passy.  Solution of generalized geometric programs.  International Journal for Numerical Methods in Engineering, Vol. 9, pp. 149-168, 1975.      

[2] C. A. Floudas.  Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic Publishers, 2000.

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

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Illustrative Example from an Article about Generalized Geometric Programs

Jsun Yui Wong

The following computer program tries to solve the small illustrative example from Avriel, Dembo, and Passy [1, p. 158].

Through line 195 and line 196 a slack variable is added.

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
76 A(1)=RND
77 A(2)=RND
78 A(3)=RND
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129   FOR KKQQ=1 TO 3
130   X(KKQQ)=A(KKQQ)
131   NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*2))
181 J=1+FIX(RND*2)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
195 X(3)=  1   -X(1)/4  - X(2)/2+X(1)^2/16      + X(2)^2/16
196 IF X(3)<0 THEN 1670
197 IF X(2)=0 THEN 1670
198 IF (  X(1)^2/(6*X(2))  +X(2)/6   +7/(3*X(2))-X(1)/X(2)    )>1 THEN 1670
454 P1NEWMAY= -X(1)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1461 PP(8)=P(8)
1462 PP(9)=P(9)
1557 GOTO 128
1670 NEXT I
1889 IF M<-1.178 THEN 1999
1900 PRINT A(1),A(2),A(3),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31367 is presented below.  What immediately follows is a hand copy from the computer-monitor screen.

1.177604   2.176065   1.922548E-04   -1.177604
-31914

1.177636   2.175991   2.056956E-04   -1.177636
-31873

1.17718   2.177001   2.247095E-05   -1.17718
-31814

1.177275   2.176793   6.014109E-05   -1.177275
-31367

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31367 was  30 seconds.

References

[1] M. Avriel, R. Dembo, and U. Passy.  Solution of generalized geometric programs.  International Journal for Numerical Methods in Engineering, Vol. 9, pp. 149-168, 1975.      

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

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Pooling/Blending Formulation of Adjiman, Androulakis, and Floudas

Jsun Yui Wong

This paper is concerned with solving the nonlinear programming formulation on page 1163 of Adjiman, Androulakis, and Floudas [1].  One notes line 426 through line 454 of each of the three computer programs below.
 
Instance One of the Pooling/Blending Formulation:

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
89 FOR J44=1 TO 10
90 A(J44)=RND*50
91 NEXT J44
126 IMAR=10+FIX(RND*3000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*9))
181 J=1+FIX(RND*10)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
193 IF RND<.5 THEN 201 ELSE 211
201 FOR J44=1 TO 10
202 X(J44)=INT(X(J44))
203 NEXT J44
211     X(6)=X(7)+X(8)
214     X(4)=X(9)-X(8)
215     X(5)=X(3)+X(7)
223    X(1)=X(3)+X(4)-X(2)
224    IF  (X(3)+X(4)   )  =0 THEN 1670
225    X(10)=( 3*X(1)+X(2)  )  /  (X(3)+X(4)   )
231 IF (  X(10)*X(3)+2*X(7)   -2.5*X(5)   )>0 THEN 1670
234 IF (  X(10)*X(4)+2*X(8)   -1.5*X(9)   )>0 THEN 1670
426 FOR J49 =1 TO 10
427 IF X(J49)<0 THEN 1670
428 NEXT J49
431 IF X(1)>500 THEN 1670
432 IF X(2)>500 THEN 1670
433 IF X(3)>500 THEN 1670
434 IF X(4)>500 THEN 1670
435 IF X(5)>100 THEN 1670
436 IF X(6)>500 THEN 1670
437 IF X(7)>500 THEN 1670
438 IF X(8)>500 THEN 1670
439 IF X(9)>200 THEN 1670
440 IF X(10)>500 THEN 1670
454 P1NEWMAY=9*X(5)+15*X(9)-6*X(1)-16*X(2)-10*X(6)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<398 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-30136 is shown below.  What immediately follows is a hand copy from the computer-monitor screen.

1.000015   101   0   102   0
97.99999   0   97.99999   200   1.019608
398.0001   -31905

1   101   0   102   0
98   0   98   200   1.019608
398   -31875

0   100   0   100   0
100   0   100   200   1
400   -31632

1   101   0   102   0
98   0   98   200   1.019608
398   -31502

0   100   0   100   0
100   0   100   200   1
400   -30136

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-30136 was 19 minutes.

Instance Two of the Pooling/Blending Formulation:

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
89 FOR J44=1 TO 10
90 A(J44)=RND*50
91 NEXT J44
126 IMAR=10+FIX(RND*3000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*9))
181 J=1+FIX(RND*10)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
193 IF RND<.5 THEN 201 ELSE 211
201 FOR J44=1 TO 10
202 X(J44)=INT(X(J44))
203 NEXT J44
211     X(6)=X(7)+X(8)
214     X(4)=X(9)-X(8)
215     X(5)=X(3)+X(7)
223    X(1)=X(3)+X(4)-X(2)
224    IF  (X(3)+X(4)   )  =0 THEN 1670
225    X(10)=( 3*X(1)+X(2)  )  /  (X(3)+X(4)   )
231 IF (  X(10)*X(3)+2*X(7)   -2.5*X(5)   )>0 THEN 1670
234 IF (  X(10)*X(4)+2*X(8)   -1.5*X(9)   )>0 THEN 1670
426 FOR J49 =1 TO 10
427 IF X(J49)<0 THEN 1670
428 NEXT J49
431 IF X(1)>500 THEN 1670
432 IF X(2)>500 THEN 1670
433 IF X(3)>500 THEN 1670
434 IF X(4)>500 THEN 1670
435 IF X(5)>600 THEN 1670
436 IF X(6)>500 THEN 1670
437 IF X(7)>500 THEN 1670
438 IF X(8)>500 THEN 1670
439 IF X(9)>200 THEN 1670
440 IF X(10)>500 THEN 1670
454 P1NEWMAY=9*X(5)+15*X(9)-6*X(1)-16*X(2)-10*X(6)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<598 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-30745 is shown below.  What immediately follows is a hand copy from the computer-monitor screen.

299.4216   0   299.4216   0   598.8436
299.422   299.422   0   0   3
598.843   -31904

300.0001   0   300.0001   0   600
299.9999   299.9999   0   0   3
600.0008   -31863

300   0   300   0   600
300   300   0   0   3
600   -30745

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-30745 was 11 minutes.

Instance Three of the Pooling/Blending Formulation:

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
89 FOR J44=1 TO 10
90 A(J44)=RND*50
91 NEXT J44
126 IMAR=10+FIX(RND*3000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*9))
181 J=1+FIX(RND*10)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
193 IF RND<.5 THEN 201 ELSE 211
201 FOR J44=1 TO 10
202 X(J44)=INT(X(J44))
203 NEXT J44
211     X(6)=X(7)+X(8)
214     X(4)=X(9)-X(8)
215     X(5)=X(3)+X(7)
223    X(1)=X(3)+X(4)-X(2)
224    IF  (X(3)+X(4)   )  =0 THEN 1670
225    X(10)=( 3*X(1)+X(2)  )  /  (X(3)+X(4)   )
231 IF (  X(10)*X(3)+2*X(7)   -2.5*X(5)   )>0 THEN 1670
234 IF (  X(10)*X(4)+2*X(8)   -1.5*X(9)   )>0 THEN 1670
426 FOR J49 =1 TO 10
427 IF X(J49)<0 THEN 1670
428 NEXT J49
431 IF X(1)>500 THEN 1670
432 IF X(2)>500 THEN 1670
433 IF X(3)>500 THEN 1670
434 IF X(4)>500 THEN 1670
435 IF X(5)>100 THEN 1670
436 IF X(6)>500 THEN 1670
437 IF X(7)>500 THEN 1670
438 IF X(8)>500 THEN 1670
439 IF X(9)>200 THEN 1670
440 IF X(10)>500 THEN 1670
454 P1NEWMAY=9*X(5)+15*X(9)-6*X(1)-13*X(2)-10*X(6)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<748 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31963 is shown below.  What immediately follows is a hand copy from the computer-monitor screen.

50   150   0   200   0
0   0   0   200   1.5
750   -31999

50   150   0   200   0
0   0   0   200   1.5
750   -31998

49.89852   149.6977   0   199.5962   0
0   0   0   199.5962   1.499995
748.482   -31980

49.99509   149.9873   0   199.9824   0
0   0   0   199.9824   1.499995
749.9305   -31972

50   150   0   200   0
0   0   0   200   1.5
750   -31963

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31963 was 17 seconds..

References

[1] C. S. Adjiman, I. P. Androulakis, and  C. A. Floudas.  A global optimization method, alphaBB, for general twice-differentiable constrained NLPs--II.  Implementation and computational results. Computers and Chemical Engineering Vol. 22, No. 9. pp. 1159-1179, 1998.      

[2] C. A. Floudas and P. M. Pardalos.  A collection of test problems for constrained optimization algorithms.  Springer-Verlag, 1990.

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

[4] H. S. Ryoo and N. V. Sahinidis.  Global optimization of nonconvex NLPs and MINLPs with applications in process design.  Computers and Chemical Engineering Vol. 19, No. 5. pp. 551-566, 1995.      

[5] V. Visweswaran and C.A. Floudas .  A global optimization algorithm (GOP) for certain classes of nonconvex NLPs--II.  Application of theory and test problems.  Computers and Chemical Engineering Vol. 14, No. 12. pp. 1419-1434, 1990.      

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem

Jsun Yui Wong

Based on the publication of Floudas and Pardalos [2], of Visweswaran and Floudas [5], of Ryoo and Sahinidis [4], and of Adjiman, Androulakis, and Floudas [1], the present paper tries to solve three instances of the Haverly pooling problem.  One notes line 435, line 439, and line 454 of each of the three computer programs.

Instance One of the Haverly Pooling Problem:

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
88 A(10)=1+RND*2
89 FOR J44=1 TO 9
90 A(J44)=RND*50
91 NEXT J44
126 IMAR=10+FIX(RND*3000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*9))
181 J=1+FIX(RND*10)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
193 IF RND<.5 THEN 201 ELSE 211
201 FOR J44=1 TO 9
202 X(J44)=INT(X(J44))
203 NEXT J44
211     X(6)=X(7)+X(8)
214     X(4)=X(9)-X(8)
215     X(5)=X(3)+X(7)
223    X(1)=X(3)+X(4)-X(2)
224    IF  (X(3)+X(4)   )  =0 THEN 1670
225    X(10)=( 3*X(1)+X(2)  )  /  (X(3)+X(4)   )
231 IF (  X(10)*X(3)+2*X(7)   -2.5*X(5)   )>0 THEN 1670
234 IF (  X(10)*X(4)+2*X(8)   -1.5*X(9)   )>0 THEN 1670
426 FOR J49 =1 TO 9
427 IF X(J49)<0 THEN 1670
428 NEXT J49
429 IF X(10)<1 THEN 1670
430 IF X(10)>3 THEN 1670
431 IF X(1)>500 THEN 1670
432 IF X(2)>500 THEN 1670
433 IF X(3)>500 THEN 1670
434 IF X(4)>500 THEN 1670
435 IF X(5)>100 THEN 1670
436 IF X(6)>500 THEN 1670
437 IF X(7)>500 THEN 1670
438 IF X(8)>500 THEN 1670
439 IF X(9)>200 THEN 1670
454 P1NEWMAY=9*X(5)+15*X(9)-6*X(1)-16*X(2)-10*X(6)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<398 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through -30476 is shown below .  What immediately follows is a hand copy from the computer-monitor screen.

0   100   0   100   0
100   0   100   200   1
400   -31541

0   100   0   100   0
100   0   100   200   1
400   -31374

0   100   0   100   0
100   0   100   200   1
400   -30890

7.629395E-06   99.99999   0   100  
0   100   0   100   200
1   400.0001   -30476

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-30476 was 16 minutes.

Instance Two of the Haverly Pooling Problem:

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
88 A(10)=1+RND*2
89 FOR J44=1 TO 9
90 A(J44)=RND*50
91 NEXT J44
126 IMAR=10+FIX(RND*3000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*9))
181 J=1+FIX(RND*10)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
193 IF RND<.5 THEN 201 ELSE 211
201 FOR J44=1 TO 9
202 X(J44)=INT(X(J44))
203 NEXT J44
211     X(6)=X(7)+X(8)
214     X(4)=X(9)-X(8)
215     X(5)=X(3)+X(7)
223    X(1)=X(3)+X(4)-X(2)
224    IF  (X(3)+X(4)   )  =0 THEN 1670
225    X(10)=( 3*X(1)+X(2)  )  /  (X(3)+X(4)   )
231 IF (  X(10)*X(3)+2*X(7)   -2.5*X(5)   )>0 THEN 1670
234 IF (  X(10)*X(4)+2*X(8)   -1.5*X(9)   )>0 THEN 1670
426 FOR J49 =1 TO 9
427 IF X(J49)<0 THEN 1670
428 NEXT J49
429 IF X(10)<1 THEN 1670
430 IF X(10)>3 THEN 1670
431 IF X(1)>500 THEN 1670
432 IF X(2)>500 THEN 1670
433 IF X(3)>500 THEN 1670
434 IF X(4)>500 THEN 1670
435 IF X(5)>600 THEN 1670
436 IF X(6)>500 THEN 1670
437 IF X(7)>500 THEN 1670
438 IF X(8)>500 THEN 1670
439 IF X(9)>200 THEN 1670
454 P1NEWMAY=9*X(5)+15*X(9)-6*X(1)-16*X(2)-10*X(6)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<598 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through -28477 is shown below .  What immediately follows is a hand copy from the computer-monitor screen.

299   0   299   0   598
299   299   0   0   3
598   -31990

299.0001   0   299.0001   0   598
298.9999   298.9999   0   0   3
598.0008   -31880

300   0   300   0   600
300   300   0   0   3
600   -31642

299.0001  0   299.0001  0   598
298.9999   298.9999   0   0   3
598.0008   -31625

299   0   299   0   598
299   299   0   0   3
598   -31156

299.0002   0   299.0002   0   598.0002
299   299   0   0   3
598.0003   -31031

299.0002   0   299.0002   0   598.0002
299   299   0   0   3
598.0005   -30836

299.0332   0   299.0332   0   598.0662
299.033   299.033   0   0   3
598.0671   -30599

299   0   299   0   598
299   299   0   0   3
598   -30413

299   0   299   0   598
299   299   0   0   3
598   -29513

300   0   300   0   599.9998
299.9998   299.9998   0   0   3
600.0008   -29374

299.0332   0   299.0332   598.0662
299.033   299.033   0   0   3
598.0671   -28862

300   0   300   0   599.9999
299.9999   299.9999   300   0   0   3
600.0003   -28477

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-28477 was 18 minutes.

Instance Three of the Haverly Pooling Problem:

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
88 A(10)=1+RND*2
89 FOR J44=1 TO 9
90 A(J44)=RND*50
91 NEXT J44
126 IMAR=10+FIX(RND*3000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*9))
181 J=1+FIX(RND*10)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
193 IF RND<.5 THEN 201 ELSE 211
201 FOR J44=1 TO 9
202 X(J44)=INT(X(J44))
203 NEXT J44
211     X(6)=X(7)+X(8)
214     X(4)=X(9)-X(8)
215     X(5)=X(3)+X(7)
223    X(1)=X(3)+X(4)-X(2)
224    IF  (X(3)+X(4)   )  =0 THEN 1670
225    X(10)=( 3*X(1)+X(2)  )  /  (X(3)+X(4)   )
231 IF (  X(10)*X(3)+2*X(7)   -2.5*X(5)   )>0 THEN 1670
234 IF (  X(10)*X(4)+2*X(8)   -1.5*X(9)   )>0 THEN 1670
426 FOR J49 =1 TO 9
427 IF X(J49)<0 THEN 1670
428 NEXT J49
429 IF X(10)<1 THEN 1670
430 IF X(10)>3 THEN 1670
431 IF X(1)>500 THEN 1670
432 IF X(2)>500 THEN 1670
433 IF X(3)>500 THEN 1670
434 IF X(4)>500 THEN 1670
435 IF X(5)>100 THEN 1670
436 IF X(6)>500 THEN 1670
437 IF X(7)>500 THEN 1670
438 IF X(8)>500 THEN 1670
439 IF X(9)>200 THEN 1670
454 P1NEWMAY=9*X(5)+15*X(9)-6*X(1)-13*X(2)-10*X(6)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<748 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through -31968 is shown below .  What immediately follows is a hand copy from the computer-monitor screen.

49.9287   149.7706   0   199.6935   0
0   0   0   199.6935   1.499995
748.8473   -31997

50   150   0   200   0
0   0   0   200   1.5
750   -31987

49.96364   149.891   0   199.8574   0
0   0   0   199.8574   1.5
749.4547   -31984

50   150   0   200   0
0   0   0   200   1.5
750   -31981

50   150   0   200   0
0   0   0   200   1.5
750   -31975

50   150   0   200   0
0   0   0   200   1.5
750   -31968

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31968 was 25 seconds.

References

[1] C. S. Adjiman, I. P. Androulatis, and  C. A. Floudas.  A global optimization method, alphaBB, for general twice-differentiable constrained NLPs--II.  Implementation and computational results. Computers and Chemical Engineering Vol. 22, No. 9. pp. 1159-1179, 1998.      

[2] C. A. Floudas and P. M. Pardalos.  A collection of test problems for constrained optimization algorithms.  Springer-Verlag, 1990.

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

[4] H. S. Ryoo and N. V. Sahinidis.  Global optimization of nonconvex NLPs and MINLPs with applications in process design.  Computers and Chemical Engineering Vol. 19, No. 5. pp. 551-566, 1995.      

[5] V. Visweswaran and C.A. Floudas .  A global optimization algorithm (GOP) for certain classes of nonconvex NLPs--II.  Application of theory and test problems.  Computers and Chemical Engineering Vol. 14, No. 12. pp. 1419-1434, 1990.      

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Case 2 of Haverly's Pooling Problem

Jsun Yui Wong

The following computer program tries to solve a problem based on page 564 of Ryoo and Sahinidis [1] and on case 2 of Visweswaran and Floudas [2, pages 1427-1428].  It is the case of the upper bound on product x is 600 and the upper bound on product y is 200 as shown in line  435 and line 439  below.

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
88 A(10)=1+RND*2
89 FOR J44=1 TO 9
90 A(J44)=RND*50
91 NEXT J44
126 IMAR=10+FIX(RND*3000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*9))
181 J=1+FIX(RND*10)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
193 IF RND<.5 THEN 201 ELSE 211
201 FOR J44=1 TO 9
202 X(J44)=INT(X(J44))
203 NEXT J44
211     X(6)=X(7)+X(8)
214     X(4)=X(9)-X(8)
215     X(5)=X(3)+X(7)
223    X(1)=X(3)+X(4)-X(2)
224    IF  (X(3)+X(4)   )  =0 THEN 1670
225    X(10)=( 3*X(1)+X(2)  )  /  (X(3)+X(4)   )
231 IF (  X(10)*X(3)+2*X(7)   -2.5*X(5)   )>0 THEN 1670
234 IF (  X(10)*X(4)+2*X(8)   -1.5*X(9)   )>0 THEN 1670
426 FOR J49 =1 TO 9
427 IF X(J49)<0 THEN 1670
428 NEXT J49
429 IF X(10)<1 THEN 1670
430 IF X(10)>3 THEN 1670
431 IF X(1)>500 THEN 1670
432 IF X(2)>500 THEN 1670
433 IF X(3)>500 THEN 1670
434 IF X(4)>500 THEN 1670
435 IF X(5)>600 THEN 1670
436 IF X(6)>500 THEN 1670
437 IF X(7)>500 THEN 1670
438 IF X(8)>500 THEN 1670
439 IF X(9)>200 THEN 1670
454 P1NEWMAY=9*X(5)+15*X(9)-6*X(1)-16*X(2)-10*X(6)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<598 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),A(10),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through -28477 is shown below .  What immediately follows is a hand copy from the computer-monitor screen.

299   0   299   0   598
299   299   0   0   3
598   -31990

299.0001   0   299.0001   0   598
298.9999   298.9999   0   0   3
598.0008   -31880

300   0   300   0   600
300   300   0   0   3
600   -31642

299.0001  0   299.0001  0   598
298.9999   298.9999   0   0   3
598.0008   -31625

299   0   299   0   598
299   299   0   0   3
598   -31156

299.0002   0   299.0002   0   598.0002
299   299   0   0   3
598.0003   -31031

299.0002   0   299.0002   0   598.0002
299   299   0   0   3
598.0005   -30836

299.0332   0   299.0332   0   598.0662
299.033   299.033   0   0   3
598.0671   -30599

299   0   299   0   598
299   299   0   0   3
598   -30413

299   0   299   0   598
299   299   0   0   3
598   -29513

300   0   300   0   599.9998
299.9998   299.9998   0   0   3
600.0008   -29374

299.0332   0   299.0332   598.0662
299.033   299.033   0   0   3
598.0671   -28862

300   0   300   0   599.9999
299.9999   299.9999   300   0   0   3
600.0003   -28477

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-28477 was 18 minutes.

References

[1] H. S. Ryoo and N. V. Sahinidis.  Global optimization of nonconvex NLPs and MINLPs with applications in process design.  Computers and Chemical Engineering Vol. 19, No. 5. pp. 551-566, 1995.      

[2] V. Visweswaran and C.A. Floudas .  A global optimization algorithm (GOP) for certain classes of nonconvex NLPs--II.  Application of theory and test problems.  Computers and Chemical Engineering Vol. 14, No. 12. pp. 1419-1434, 1990.      

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

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Design of Two-Stage Process System

Jsun Yui Wong

The following computer program tries to solve the design problem on pages 564-565 of Ryoo and Sahinidis [1].

0 REM DEFSNG
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
79 A(1)=RND*3
80 A(2)=RND*4
81 A(3)=RND*2
82 A(4)=RND
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*3))
181 J=1+FIX(RND*4)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
312 X(2)=3*X(1)+3*X(3)
314 IF X(1)+2*X(3)-4>0 THEN 1670
316 IF X(2)+2*X(4)-4>0 THEN 1670
318 FOR J43=1 TO 4
319 IF X(J43)<0 THEN 1670
320 NEXT J43
321 IF X(1)>3 THEN 1670
322 IF X(2)>4 THEN 1670
323 IF X(3)>2 THEN 1670
324 IF X(4)>1 THEN 1670
454 P1NEWMAY= -X(1)^.6-X(2)^.6+6*X(1)+4*X(3)-3*X(4)
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<4.5 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4)
1933 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through -31993 is presented below.  What immediately follows is a hand copy from the computer-monitor screen.

1.333333   4   0   0
4.514202   -32000

1.33333   3.999991   0   0
4.514189   -31998

1.333332   3.999995   0   0
4.514195   -31996

1.333333   4   0   0
4.514202   -31994

1.333333   4   0   0
4.514202   -31993

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31993 was four seconds.

References

[1] H. S. Ryoo and N. V. Sahinidis.  Global optimization of nonconvex NLPs and MINLPs with applications in process design.  Computers and Chemical Engineering Vol. 19, No. 5. pp. 551-566, 1995.      

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

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Test Problem from the Literature

Jsun Yui Wong

The following computer program tries to solve the nonlinear programming formulation on page 565 of Ryoo and Sahinidis [1].

0 REM DEFSNG
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
79 A(1)=RND*1.12
80 A(2)=RND*2.1
81 A(3)=RND
82 A(4)=RND
83 A(5)=RND
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*3))
181 J=2+FIX(RND*4)
183 R=(1-RND*2)*A(J)
185 IF J>2 THEN 197
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
194 GOTO 413
197 X(J)=FIX(RND*2)
413 X(1)= ( 1.25  -X(3)  ) ^.5
418 P(2)= X(2)^1.5 +1.5*X(4)-3
419 IF (X(1)+X(3) -1.6)  >0 THEN 1670
422 IF (  1.333*X(2)+X(4)     -3  )> 0 THEN 1670
425 IF (-X(3)-X(4)+X(5)   )>0 THEN 1670
426 FOR J49 =1 TO 5
427 IF X(J49)<0 THEN 1670
428 NEXT J49
431 IF X(1)>1.12 THEN 1670
432 IF X(2)>2.1 THEN 1670
433 IF X(3)>1 THEN 1670
434 IF X(4)>1 THEN 1670
435 IF X(5)>1 THEN 1670
454 P1NEWMAY= -2*X(1)-3*X(2)-1.5*X(3)-2*X(4)+.5*X(5) -5000000!*(P(2))^2
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1461 PP(2)=P(2)
1557 GOTO 128
1670 NEXT I
1889 IF M<-7.6672 THEN 1999
1929 PRINT A(1),A(2),A(3),A(4),A(5)
1933 PRINT M,JJJJ,PP(2)
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through -31979 is presented below.  What immediately follows is a hand copy from the computer-monitor screen.

1.118034   1.310371   0   1   1
-7.66718   -31996   -4.768372E-07

1.118034   1.310371   0   1   1
-7.66718   -31992   -4.768372E-07

1.118034   1.310371   0   1   1
-7.66718   -31989   -4.768372E-07

1.118034   1.310371   0   1   1
-7.66718   -31983   -4.768372E-07

1.118034   1.31037   0   1   1
-7.667182   -31979   -7.152558E-07

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31979 was five seconds.

References

[1] H. S. Ryoo and N. V. Sahinidis.  Global optimization of nonconvex NLPs and MINLPs with applications in process design.  Computers and Chemical Engineering Vol. 19, No. 5. pp. 551-566, 1995.      

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

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Model of a Water Pumping System

Jsun Yui Wong

The following computer program tries to solve the nonlinear programming formulation on page 563 of Ryoo and Sahinidis [1].

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
65  A(1)=RND*9.422
66  A(2)=RND*5.903
68  A(3)=RND*267.42
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129   FOR KKQQ=1 TO 3
130   X(KKQQ)=A(KKQQ)
131   NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*0))
181 J=1+FIX(RND*2)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
223 X(3)=250+30*X(1)-6*X(1)^2
226 P(2)=X(3)-300-20*X(2)+12*X(2)^2
229 P(3)=X(3)-150-.5*(X(1)+X(2))^2
454 P1NEWMAY=  -X(3)   -100000!*P(2)^2   -100000!*P(3)^2
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-555 THEN 1999
1929 PRINT A(1),A(2),A(3),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31988 is shown below.  What immediately follows is a hand copy from the computer-monitor screen.
..
6.293453   3.821861   201.1583   -201.405   -32000

6.293435   3.821858   201.1591   -201.3072   -31998

6.293429   3.821849   201.1594   -201.2131   -31997

6.293432   3.82184   201.1593   -201.1603   -31996

6.29343   3.821838   201.1594   -201.1596   -31995

6.29343   3.821839   201.1594   -201.1596   -31994

6.293431   3.821839   201.1593   -201.1594   -31993

6.293431   3.821839   201.1593   -201.1593   -31992

6.29343   3.821839   201.1593   -201.1593   -31988

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31988 was three seconds.

References

[1] H. S. Ryoo and N. V. Sahinidis.  Global optimization of nonconvex NLPs and MINLPs with applications in process design.  Computers and Chemical Engineering Vol. 19, No. 5. pp. 551-566, 1995.      

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

 

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Test Problem from the Literature

Jsun Yui Wong

The following computer program tries to solve the test problem on page 564 of Ryoo and Sahinidis [1].

Line 195 and line 199 generate domino effect/chain reaction.

0 REM DEFSNG A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PP(22)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
65  A(1)=RND*34
66  A(2)=RND*17
68  A(3)=100+RND*200
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129   FOR KKQQ=1 TO 3
130   X(KKQQ)=A(KKQQ)
131   NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*0))
181 J=3+FIX(RND*0)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
192 NEXT IPP
195 X(2)=(-50*X(3)+15000)/600
199 X(1)=( -5000+50*X(3))/    (600-X(3)   )
452 IF X(1) <0 THEN 1670
453 IF X(2) <0 THEN 1670
454 P1NEWMAY=  -35*X(1)^.6        -35*X(2)^.6
455 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-555 THEN 1999
1929 PRINT A(1),A(2),A(3),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31996 is shown below.  What immediately follows is a hand copy from the computer-monitor screen.

0   16.66667   100   -189.3116   -32000

0   16.66667   100   -189.3116   -31999

0   16.66667   100   -189.3116   -31998

0   16.66667   100   -189.3116   -31997

0   16.66667   100   -189.3116   -31996

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time (throughput time) from JJJJ=-32000 through JJJJ=-31996 was two seconds.

References

[1] H. S. Ryoo and N. V. Sahinidis.  Global optimization of nonconvex NLPs and MINLPs with applications in process design.  Computers and Chemical Engineering Vol. 19, No. 5. pp. 551-566, 1995.      

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