Jsun Yui Wong
The following computer programs try to solve a transformer design model based on the transformer design model on page 265 of Ballard, Jelinek, and Schinzinger [1, p. 265]. For the present paper the first four variables are required to be integer; X(5) and X(6) are continuous, representing magnetic-flux density and electric-current density, respectively; see Papalambros and Wilde [2, p. 258] and Ballard, Jelinet and Schinzinger [1, p. 265]. Thus the present model has four integer variables and two continuous variables.
The transformer design model on page 265 of Ballard, Jelinek, and Schinzinger [1] has two constraints, one much longer than the other. It seems that at least one of these two inequality constraints is active/holding, but which one is active/holding is a question. To find out one can treat one inequality constraint at a time as an equality constraint. Hence the first computer program below treats the shorter constraint of the transformer design model as an equality constraint; line 354 below is for that purpose. Similarly the second computer program below treats the longer constraint as an equality constaint. As usual, the user normally goes with the best solution produced.
One notes that the right-hand side of the longer constraint is 1, Ballard et al. [1, p. 265], not 1.2.
Case One: The shorter constraint is active/holding:
0 DEFSNG A-Z
2 DEFINT 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
61 FOR KLQ=1 TO 6
62 A(KLQ)=RND*30
63 NEXT KLQ
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR STEP .5
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=1+FIX(RND*6)
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))
327 X(1)=INT(X(1))
328 X(2)=INT(X(2))
329 X(3)=INT(X(3))
330 X(4)=INT(X(4))
331 FOR J42=1 TO 6
333 IF X(J42)<.5 THEN 1670
336 NEXT J42
354 X(6)=2.07/(.001*X(1)*X(2)*X(3)*X(4)*X(5) )
425 L2=.00062*X(1)*X(4)*X(5)^2*(X(1)+X(2)+X(3))+.00058*X(2)*X(3)*X(6)^2*(X(1)+1.57*X(2)+X(4))-1
426 IF L2>0 THEN 1670
444 P1NEWMAY=-.0204*X(1)*X(4)*(X(1)+X(2)+X(3))-.0187*X(2)*X(3)*(X(1)+1.57*X(2)+X(4))-.0607*X(1)*X(4)*X(5)^2*(X(1)+X(2)+X(3))-.0437*X(2)*X(3)*X(6)^2*(X(1)+1.57*X(2)+X(4))
448 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 6
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-138 THEN 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=32000 is shown below. What immediately follows is a hand copy from the computer monitor.
5 5 11 13 .7081461
.8176574 -137.5205 -30999
5 5 11 13 .7080245
.8177978 -137.5205 -29097
5 5 11 12 .7444006
.8426546 -136.0535 -17682
5 5 12 12 .7200568
.7985481 -137.9016 -17650
5 5 11 12 .7445957
.8424338 -137.0535 -15859
5 5 10 14 .7010605
.8436198 -137.5205 -7566
6 5 9 13 .6635701
.8887434 -137.8188 -869
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=32000 was one hour.
Case Two: The longer constraint is active/holding:
0 DEFSNG A-Z
2 DEFINT 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
61 FOR KLQ=1 TO 6
62 A(KLQ)=RND*30
63 NEXT KLQ
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR STEP .5
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=1+FIX(RND*6)
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))
327 X(1)=INT(X(1))
328 X(2)=INT(X(2))
329 X(3)=INT(X(3))
330 X(4)=INT(X(4))
334 FOR J42=1 TO 6
335 IF X(J42)<.1 THEN 1670
336 NEXT J42
345 DOWN=.00062*X(1)*X(4)* (X(1)+X(2)+X(3))
348 IF ( ( -.00058*X(2)*X(3)*X(6)^2*(X(1)+1.57*X(2)+X(4))+1 ) /DOWN )<0 THEN 1670
352 X(5)= ( ( -.00058*X(2)*X(3)*X(6)^2*(X(1)+1.57*X(2)+X(4))+1 ) /DOWN )^.5
433 L1=-.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6)+2.07
434 IF L1>0 THEN 1670
444 P1NEWMAY=-.0204*X(1)*X(4)*(X(1)+X(2)+X(3))-.0187*X(2)*X(3)*(X(1)+1.57*X(2)+X(4))-.0607*X(1)*X(4)*X(5)^2*(X(1)+X(2)+X(3))-.0437*X(2)*X(3)*X(6)^2*(X(1)+1.57*X(2)+X(4))
448 P=P1NEWMAY
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 6
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-135.8 THEN 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=32000 is shown below. What immediately follows is a hand copy from the computer monitor.
5 5 10 13 .7305032
.8718993 -135.7372 -30900
5 5 10 13 .7304968
.8719051 -135.737 -29482
5 5 10 13 .7304941
.8719076 -135.7369 -28262
5 5 10 13 .7304928
.8719087 -135.7369 -24142
5 5 10 13 .7304967
.8719052 -135.737 -19194
5 5 10 13 .7304926
.8719089 -135.7369 -19066
5 5 10 13 .730497
.8719049 -135.737 -12277
5 5 10 13 .7305082
.8718948 -135.7373 -8203
5 5 10 13 .7304963
.8719055 -135.737 -1702
5 5 10 13 .7305092
.8718939 -135.7373 8294
5 5 10 13 .7304939
.8719076 -135.7369 10207
5 5 10 13 .7304928
.8719087 -135.7369 18893
5 5 10 13 .7304925
.871909 -135.7369 20486
5 5 10 13 .7308841
.8715561 -135.7473 28720
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=32000 was two hours.
References
[1] D. H. Ballard, C. O. Jelinek, and R. Schinzinger (1974). An algorithm for the solution of constrained generalised polynomial programming problems. The Computer Journal, Vol. 17, Number 3, pp. 261-266.
[2] Panos Y. Papalambros, Douglass J. Wilde. Principles of Optimal Design, Second Edition. Cambridge University Press, 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.
« May 2011 | Main | July 2011 »
June 2011
06/29/2011
A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Transformer Design Model with Integer Variables and Continuous Variables, Third Edition
Posted at 12:17 PM | Permalink | Comments (0)
06/25/2011
A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Nonlinear Programming Test Problem
Jsun Yui Wong
The following computer program tries to solve the nonlinear programming test problem on page 28 of Floudas and Pardalos [1].
0 REM DEFDBL 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
61 FOR KLQ=1 TO 4
62 A(KLQ)=RND*5
63 NEXT KLQ
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
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))
197 X(2)=3*X(1)+3*X(3)
199 IF X(1)+2*X(3)>4 THEN 1670
201 IF X(2)+2*X(4)>4 THEN 1670
216 IF X(1)>3 THEN 1670
229 IF X(4)>1 THEN 1670
315 IF X(1)<0 THEN 1670
317 IF X(2)<0 THEN 1670
322 IF X(3)<0 THEN 1670
327 IF X(4)<0 THEN 1670
1446 P=-X(1)^.6-X(2)^.6+6*X(1) +4*X(3)-3*X(4)
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<4.45 THEN 1999
1928 PRINT A(1),A(2),A(3),A(4),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-30854 is shown below. What immediately follows is a hand copy from the computer monitor.
1.333333 3.999998 0 0 4.5142
-31877
1.333331 3.999994 0 0 4.514193
-31855
1.332904 3.998713 0 0 4.5123
-31202
1.312891 4 2.044251E-02 0
4.484283 -30854
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=-30854 was thirty seconds.
References
[1] Floudas, C.A., Pardalos, P.M. (1990). A collection of test problems for constrained global optimization algorithms, Springer-Verlag, Berlin.
[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.
Posted at 03:19 AM | Permalink | Comments (0)
06/18/2011
A Nonlinear Integer/Discrete/Continuous Programming Solver Finding All Solutions to a Standard Test Problem in the Literature
Jsun Yui Wong
The following computer program tries to find all nine solutions to the system of two nonlinear equations on page 326 of Floudas et al. [1] and on page 657 of Floudas [2]. These two equations are used in line 1221 and line 1226 below. This is a standard test problem [2, p. 657].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
5 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
61 FOR KLQ=1 TO 2
62 A(KLQ)=-5+RND*10
63 NEXT KLQ
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 2
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
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))
1221 P(1)= 4*X(1)^3+4*X(1)*X(2)+2*X(2)^2-42*X(1)-14
1226 P(2)= 4*X(2)^3+2*X(1)^2+4*X(1)*X(2)-26*X(2)-22
1444 P= -(P(1))^2 -(P(2))^2
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 2
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-1E-09 THEN 1999
1928 PRINT A(1),A(2),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. Selected solutions through JJJJ=-31937 are shown below. What immediately follows is a hand copy from the computer monitor.
-3.073026 -8.135308E-02 -7.275958E-11
-31998
.0866775 2.884255 -5.820766E-11 -31997
3.584428 -1.848126 -2.910383E-10 -31982
-.1279614 -1.953715 0 -31981
-2.805118 3.131312 -2.910383E-10 -31976
-3.77931 -3.283186 -2.473828E-10 -31974
-.2708446 -.9230386 -3.637979E-12 -31973
3.385154 .0738525 -9.094947E-11 -31955
3 2 0 -31937
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31937 was eight seconds.
References
[1] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, and C. A. Schweiger. Handbook of Test Problems in Local and Global Optimization. Dordrecht/Boston/London: Kluwer Academic Publishers, 1999.
[2] C. A. Floudas. Deterministic Global Optimization. Dordrecht/Norwell, MA: 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.
Posted at 10:54 AM | Permalink | Comments (0)
06/16/2011
A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Circuit Design Problem
Jsun Yui Wong
The following computer program tries to solve the circuit design problem in Ratschek and Rokne [3], on page 331 of Floudas et al. [1], and on pages 663-664 of Floudas [2]. According to Floudas et al. [1, p. 326], this is a challenging problem.
0 REM DEFDBL 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)
5 G11=.485:G12=.752:G13=.869:G14=.982
6 G21=.369:G22=1.254:G23=.703:G24=1.455
7 G31=5.2095:G32=10.0677:G33=22.9274:G34=20.2153
8 G41=23.3037:G42=101.779:G43=111.461:G44=191.267
9 G51=28.5132:G52=111.8467:G53=134.3884:G54=211.4823
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
61 FOR KLQ=1 TO 9
62 A(KLQ)=RND*10
63 NEXT KLQ
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 9
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=1+FIX(RND*9)
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))
233 FOR J43=1 TO 9
237 IF X(J43)>10 THEN 1670
244 NEXT J43
341 IF X(4)<.00001 THEN 1670
344 X(2)=X(1)*X(3)/X(4)
551 P(1)=(1-X(1)*X(2))*X(3)*(EXP(X(5)*(G11-G31*X(7)*.001-G51*X(8)*.001))-1)-G51+G41*X(2)
552 P(2)=(1-X(1)*X(2))*X(3)* ( EXP( X(5)*(G12-G32*X(7)*.001-G52*X(8)*.001) ) -1 ) -G52+G42*X(2)
553 P(3)=(1-X(1)*X(2))*X(3)* ( EXP( X(5)*(G13-G33*X(7)*.001-G53*X(8)*.001) ) -1 ) -G53+G43*X(2)
554 P(4)=(1-X(1)*X(2))*X(3)* ( EXP( X(5)*(G14-G34*X(7)*.001-G54*X(8)*.001) ) -1 ) -G54+G44*X(2)
571 P(5)=(1-X(1)*X(2))*X(4)*(EXP(X(6)*(G11-G21-G31*X(7)*.001+G41*X(9)*.001))-1)-G51*X(1)+G41
572 P(6)=(1-X(1)*X(2))*X(4)* ( EXP( X(6)*(G12-G22 -G32*X(7)*.001+G42*X(9)*.001) ) -1 ) -G52*X(1)+G42
573 P(7)=(1-X(1)*X(2))*X(4)* ( EXP( X(6)*(G13-G23 -G33*X(7)*.001+G43*X(9)*.001) ) -1 ) -G53*X(1)+G43
574 P(8)=(1-X(1)*X(2))*X(4)* ( EXP( X(6)*(G14-G24 -G34*X(7)*.001+G44*X(9)*.001) ) -1 ) -G54*X(1)+G44
1444 P=-(P(1))^2-(P(2))^2-(P(3))^2-(P(4))^2-(P(5))^2-(P(6))^2-(P(7))^2-(P(8))^2
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 9
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.1 THEN 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. Selected candidate solutions through JJJJ=-31753 are shown below. What immediately follows is a hand copy from the computer monitor.
.9009521 .5758661 1.181847 1.849019 7.667813
9.874598 4.547611 1.009119 1.689384 -2.653317E-02
-31893
.8999669 .4425691 1.001143 2.035831 7.997395
7.766736 5.017677 .9978938 2.047545 -1.492426E-03
-31753
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31753 was two hours and half an hour.
References
[1] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, and C. A. Schweiger. Handbook of Test Problems in Local and Global Optimization. Dordrecht/Boston/London: Kluwer Academic Publishers, 1999.
[2] C. A. Floudas. Deterministic Global Optimization. Dordrecht/Norwell, MA: Kluwer Academic Publishers, 2000.
[3] H. Ratschek and J. Rokne. Experiments using interval analysis for solving a circuit design problem. Journal of Global Optimization 3: 501-518, 1993.
[4] 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.
Posted at 12:02 PM | Permalink | Comments (0)
06/12/2011
A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Test Problem from the Literature
Jsun Yui Wong
The following computer program tries to solve Test Problem 1 on pages 264-265 of Floudas et al. [1].
0 REM DEFSNG A-Z
2 DEFINT I,J,K
5 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
51 A(1)=RND*3
53 A(2)=RND*3
61 FOR KLQ=3 TO 5
62 A(KLQ)=FIX(RND*2)
63 NEXT KLQ
126 IMAR=10+FIX(RND*500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=3+FIX(RND*3)
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))
331 FOR J44=3 TO 5
335 X(J44)=INT(X(J44))
338 NEXT J44
441 X(1)=( 1.25-X(3) )^.5
445 X(2)= ( 3-1.5*X(4) )^.6666666
988 IF (X(1)+ X(3) ) >1.6 THEN 1670
989 IF 1.333*X(2)+X(4)>3 THEN 1670
990 IF -X(3)-X(4)+X(5)>0 THEN 1670
1450 P=-2*X(1)-3*X(2)-1.5*X(3)-2*X(4)+.5*X(5)
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1888 IF M<-8.22 THEN 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31979 is shown below. What immediately follows is a hand copy from the computer monitor.
1.118034 1.310371 0 1 1
-7.66718 -31998
1.118034 1.310371 0 1 0
-8.16718 -31995
1.118034 1.310371 0 1 0
-8.16718 -31995
1.118034 1.310371 0 1 0
-8.16718 -31993
1.118034 1.310371 0 1 0
-8.16718 -31992
1.118034 1.310371 0 1 0
-8.16718 -31991
1.118034 1.310371 0 1 0
-8.16718 -31989
1.118034 1.310371 0 1 0
-8.16718 -31987
1.118034 1.310371 0 1 0
-8.16718 -31982
1.118034 1.310371 0 1 0
-8.16718 -31980
1.118034 1.310371 0 1 1
-7.66718 -31979
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31979 was two seconds.
References
[1] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, and C. A. Schweiger. Handbook of Test Problems in Local and Global Optimization. Dordrecht/Boston/London: Kluwer Academic Publisher, 1999.
[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.
Posted at 03:28 AM | Permalink | Comments (0)
06/11/2011
sA Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Test Problem from Floudas et Al.
Jsun Yui Wong
The following computer program tries to solve Test Problem 5 on pages 267-268 of Floudas et al. [1]; this problem is Example 1 of Porn et al. [2, p. 441-442].
0 REM DEFDBL A-Z
2 DEFINT I,J,K,A,X
5 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
61 REM FOR KLQ=1 TO 2
62 REM A(KLQ)=1+RND*4
63 REM NEXT KLQ
67 A(1)=1+FIX(RND*5)
68 A(2)=1+FIX(RND*5)
126 IMAR=10+FIX(RND*500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 2
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
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))
944 IF ( X(1) ) <1 THEN 1670
955 IF ( X(1) ) >5 THEN 1670
966 IF ( X(2) ) <1 THEN 1670
977 IF ( X(2) ) >5 THEN 1670
1111 IF (X(1)^1.2) * ( X(2)^1.7 ) -7*X(1) -9*X(2) >-24 THEN 1670
1225 IF ( - X(1) -2*X(2) ) >-5 THEN 1670
1228 IF ( -3*X(1) +X(2) ) > 1 THEN 1670
1233 IF ( 4*X(1) -3*X(2) ) >11 THEN 1670
1450 P= - 7*X(1)-10*X(2)
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 2
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1890 IF M<-34 THEN 1999
1928 PRINT A(1),A(2),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31990 is shown below. What immediately follows is a hand copy from the computer monitor.
2 2 -34 -32000
2 2 -34 -31997
3 1 -31 -31996
2 2 -34 -31995
2 2 -34 -31992
3 1 -31 -31990
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31990 was two seconds.
References
[1] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, and C. A. Schweiger. Handbook of Test Problems in Local and Global Optimization. Dordrecht/Boston/London: Kluwer Academic Publisher, 1999.
[2] R. Porn, I. Harjunkoski, and T. Westerlund. Convexification of different classes of non-convex MINLP problems. Computers and Chemical Engineering 23 (1999) 439-448.
[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.
Posted at 04:54 PM | Permalink | Comments (0)
06/08/2011
The Domino Method Solving a Nonlinear System of Equations from the Literature
Jsun Yui Wong
The following computer program tries to solve Test Problem 3 of two nonlinear equations on page 328 of Floudas et al. [1]. These two equations are used in line 1003 and line 1005. X(2) of line 1003 of the following computer program is like a domino of the falling domino principle.
0 DEFDBL A-Z
2 DEFINT I,J,K
5 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
61 REM FOR KLQ=1 TO 2
62 REM A(KLQ)=.000005+18.21
63 REM NEXT KLQ
77 A(1)=5.49E-06+RND*4.55299451#
78 A(2)=.002196+RND*18.207804#
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 2
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=1+FIX(RND*1)
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))
1001 IF X(1)<5.49E-06 THEN 1670
1002 IF X(1)>4.553 THEN 1670
1003 X(2)=1/(10000*X(1))
1005 P(2)= (1/(2.718281828#^X(1))) +( 1/(2.718281828#^X(2)) ) -1.001
1450 P= -(P(2))^2
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 2
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.000000000000001# THEN 1999
1928 PRINT A(1),A(2),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31965 is presented below. What immediately follows is a hand copy from the computer monitor.
1.450682584474078D-05 6.893306714387381 -2.631900750731765D-26
-31971
1.450682582172361D-05 6.893306725324607 -1.190528257035044D-22
-31965
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31965 was ten seconds.
References
[1] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, and C. A. Schweiger. Handbook of Test Problems in Local and Global Optimization. Dordrecht/Boston/London: Kluwer Academic Publisher, 1999.
[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.
Posted at 06:16 PM | Permalink | Comments (0)
06/04/2011
A Nonlinear Integer/Discrete/Continuous Programming Solver To Solve Nonlinear Systems of Equations, Fifth Edition
Jsun Yui Wong
"Isn't minimization equivalent to finding a zero of an N-directional gradient vector, which is not so different from zeroing an N-dimensional function? No!" (Press, Teukolsky, Vetterling, and Flannery [1, p. 476]).
The following computer program tries to solve the following nonlinear system representing a problem of robotics [2, p. 347] and coming from Bellido [2, p. 348; 3, p. I-14].
X(1)^2+X(2)^2+X(3)^2-12*X(1)-68=0
X(4)^2+X(5)^2+X(6)^2-12*X(5)-68=0
X(7)^2+X(8)^2+X(9)^2-24*X(8)-12*X(9)+100=0
X(1)*X(4)+X(2)*X(5)+X(3)*X(6)-6*X(1)-6*X(5)-52=0
X(1)*X(7)+X(2)*X(8)+X(3)*X(9)-6*X(1)-12*X(8)-6*X(9)+64=0
X(4)*X(7)+X(5)*X(8)+X(6)*X(9)-6*X(5)-12*X(8)-6*X(9)+32=0
2*X(2)+2*X(3)-X(4)-X(5)-2*X(6)-X(7)-X(9)+18=0
X(1)+X(2)+2*X(3)+2*X(4)+2*X(6)-2*X(7)+X(8)-X(9)-38=0
X(1)+X(3)-2*X(4)+X(5)-X(6)+2*X(7)-2*X(8)+8=0
Line 1450 below is noteworthy. Like line 1450 of the original edition posted on May 20, 2011, line 1450 below has the purpose of making all the left-hand sides of the nonlinear system above to reach approximate zeros at about the same time. Some left-hand sides go toward zero faster than other left-hand sides. Exponent 2 of line 1450 slows down the march to zero of a fast-to-zero left-hand side more than a slow-to-zero left-hand side. Exponent 4 of line 1450 of the May 20 edition works similarly.
0 REM DEFDBL A-Z
2 DEFINT I,J,K
5 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+17
61 FOR KLQ=1 TO 9
62 A(KLQ)=-20+RND*40
63 NEXT KLQ
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 9
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=1+FIX(RND*9)
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))
504 P(1)=X(1)^2 +X(2)^2+X(3)^2 -12*X(1) -68
516 P(2)= X(4)^2 +X(5)^2+X(6)^2 -12*X(5) -68
555 P(3)=X(7)^2+X(8)^2+X(9)^2 -24*X(8)-12*X(9) +100
622 P(4)= X(1)*X(4)+X(2)*X(5)+X(3)*X(6)-6*X(1)-6*X(5)-52
624 P(5)= X(1)*X(7)+X(2)*X(8)+X(3)*X(9)-6*X(1)-12*X(8)-6*X(9)+64
626 P(6)= X(4)*X(7)+X(5)*X(8)+X(6)*X(9)-6*X(5)-12*X(8)-6*X(9)+32
628 P(7)= 2* X(2)+2*X(3)-X(4)-X(5)-2*X(6)-X(7) -X(9)+18
665 P(8)=X(1)+X(2)+2*X(3)+ 2*X(4) +2*X(6)-2*X(7) +X(8) -X(9)-38
672 P(9)= X(1)+X(3)-2*X(4) +X(5) -X(6)+2*X(7)-2*X(8) +8
1450 P=-(P(1))^2 -(P(2))^2-(P(3))^2 -( P(4))^2-(P(5))^2 -(P(6))^2 -P(7)^2-P(8)^2-P(9)^2
1451 IF P<=M THEN 1670
1452 M=P
1453 PP(1)=P(1):PP(2)=P(2):PP(3)=P(3):PP(4)=P(4):PP(5)=P(5):PP(6)=P(6):PP(7)=P(7):PP(8)=P(8):PP(9)=P(9)
1454 FOR KLX=1 TO 9
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1888 IF M<-1111 THEN 1999
1895 FOR J45=1 TO 9
1900 IF ABS(PP(J45))>.1 THEN 1999
1911 NEXT J45
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),A(8),A(9)
1933 PRINT M,JJJJ,PP(1),PP(2),PP(3),PP(4),PP(5),PP(6),PP(7),PP(8),PP(9)
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. Selected solutions produced through JJJJ=-23625 are presented below. What immediately follows is a hand copy from the computer monitor.
3.999556 -5.196781E-08 9.999921 -6.742994E-09
4.000358 10.00006 0 8.000212 14.00011
-1.672172E-06 -31374 1.907349E-04
-2.975464E-04 -1.525879E-05 2.937317E-04
-5.569458E-04 2.212525E-04 -7.343292E-04
-3.814697E-04 -6.456375E-04
9.394526 9.247757 2.637445 7.967089 5.193204
6.314771 5.014449 6.499678 10.95995
-1.456051E-4 -28184 -6.103516E-05
1.739502E-03 -6.408692E-04 9.918213E-04
-9.155273E-04 -7.781983E-04 6.170273E-03
-8.270264E-03 5.767823E-03
2.74995 2.546467 9.325024 7.41096 4.616787
6.8687 8.640625 10.12129 7.34499
-2.209818E-03 -23625 3.379822E-03
1.464844E-02 -1.029968E-03 -1.335907E-02
1.934052E-03 -1.240921E-02 -7.781983E-03
8.354187E-04 .0398159
Bellido's computational results are on pages 348-349 of Bellido [2].
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-23625 was four hours.
In general, a usable solution for another application of this algorithm may come early, late, or worse. To alleviate this shortcoming, one can simultaneously run independent computers with different line 126s. Here is one example: 126 IMAR=10+FIX(RND*1000), 126 IMAR=10+FIX(RND*1200), and 126 IMAR=10+FIX(RND*1500). This multicomputer approach can considerably reduce the length of time to obtain one usable solution.
References
[1] William H. Press, Saul A. Teukosky, William T. Vetterling, and Brian P. Flannery. Numerical recipes: the art of scientific computing, Third Edition. New York: Cambridge University Press, 2007.
[2] A.-M. Bellido. Construction of iteration functions for the simultaneous computation of the solutions of equations and algebaric systems. Numerical Algorithms 6 (1994) 317-351.
[3] A.-M. Bellido. "Construction of iteration functions for the simultaneous computation of the solutions of equations and algebaric systems," Seminaire d'ANALYSE NUMEROQUE, Annee 1991-1992, PP. i-1 to i-16. Universite Paul Sabatier de Toulouse, U.F.R. Mathematiques, Informatique, Gestion.
[4] 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.
Posted at 06:07 AM | Permalink | Comments (0)
