Substance

The Domino Method of General Integer Nonlinear Programming Applied to Waste Treatment Plant Design

Jsun Yui Wong

The computer program below tries to solve the waste treatment plant design problem in Ballard, Jelinek, and Schinzinger [1].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [2] and after the domino method of solving nonlinear systems of equations [12].  Line 251 through line 459 are illustrative.  Domino one is line 251's X(4), the slack variable of the given constraint.  

0 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),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 3
42 A(J44)=RND
43 NEXT J44
126 IMAR=10+FIX(RND*5000)
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*3)
182 REM      IF J>4 THEN 183 ELSE 191
183  R=(1-RND*2)*A(J)
185 REM IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
187    X(J)=A(J)+  (RND^(RND*10))*R
188 GOTO 192
189 REM   X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
192 NEXT IPP
228    FOR J44=1 TO 3
230    IF X(J44)<.000001 THEN 1670
231    NEXT J44
251 X(4)=1 -.0000033333333#*X(3)
331 IF  X(4) <0 THEN PS1=ABS(X(4)) ELSE PS1=0
332 REM   IF  X(8) <0 THEN PS2=ABS(X(8)) ELSE PS2=0
403 POBA2= -.000000000021#*X(2)^2.55-62900000#*X(2)^5/X(3)^6  -85000000000#/(X(2)*X(3)^.2*X(1)^2    )-160000#*X(1)^2.5    *X(3)/X(2)
411 POBA= POBA2
459 POB1=POBA -999999999#*( + PS1 )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA=POBA
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-999999! THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),M,JJJJ,PPOBA
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31998 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

.6168611557253477    581517.9425713552   300000.00299089
3.036670914724482D-11   -71758.44587308557   -32000
-71758.44587308557 

.6168611552372779    581517.9448577983   300000.003
2.081668171172169D-16   -71758.44587260198   -31999
-71758.44587260198  

.6168611548868333    581517.9461914927   300000.0029995419
1.527084014796287D-12   -71758.4458726263   -31998
-71758.4458726263   

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-31998 was thirteen seconds.

References

[1] Dana H. Ballard, C. O. Jelinek, R. Schinzinger, An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Vol. 17, Number 3 (1974), pp. 261-266.

[2] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[3] William Conley, Computer Optimization Techniques.  New York/Princeton: Petrocelli Books, Inc., Copyright 1980.

[4] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
 
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[9] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

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

[11] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge, UK: Cambridge University Press, 2000.

[12] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/

The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Continuous Variables

Jsun Yui Wong

The computer program below tries to solve the Duffin, Peterson, and Zener problem in Ballard, Jelinek, and Schinzinger [1].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [2] and after the domino method of solving nonlinear systems of equations [11].  Line 301 through line 459 are illustrative.  Line 301's X(4), a slack variable, is proxy domino one.

0 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),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 3
42 A(J44)=RND
43 NEXT J44
126 IMAR=10+FIX(RND*5000)
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*3)
182 REM     GOTO 191
183  R=(1-RND*2)*A(J)
185 REM IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
187    X(J)=A(J)+  (RND^(RND*10))*R
189 REM   X(J)=A(J)+PA
190 REM  X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
192 NEXT IPP
228    FOR J44=1 TO 3
230    IF X(J44)<1E-09 THEN 1670
231    NEXT J44
301 X(4)=1-440.98*X(1)-28460000#*X(1)^2-615840000000000#*X(1)^2*X(2)-370.18*X(3)-54474000000#*X(3)^2-3223600#*X(1)*X(3)-29200000000#*X(2)*X(3)-44712#*X(2)-379640000000#*X(2)^2-4287600000#*X(1)*X(2)
331 IF  X(4) <0 THEN PS1=ABS(X(4)) ELSE PS1=0
402 POBA2=-1/(X(1)^2*X(2)*X(3)    )
411 POBA= POBA2
459 POB1=POBA -1D+21*( + PS1 )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA=POBA
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889    IF M<-5.52532D+20 THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),M,JJJJ,PPOBA
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31973 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

5.630587680387436D-05   2.449724195244166D-07   2.330336022283708D-06
-3.226585665316861D-16   -5.525312022758336D+20   -31999
-5.525312022758333D+20

5.625444709023255D-05   2.452647001363708D-07   2.331817658777324D-06
-6.749982517373354D-14   -5.52530995189131D+20   -31973
-5.525309951890635D+20

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-31973 was two minutes.

References

[1] Dana H. Ballard, C. O. Jelinek, R. Schinzinger, An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Vol. 17, Number 3 (1974), pp. 261-266.

[2] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[3] William Conley, Computer Optimization Techniques.  New York/Princeton: Petrocelli Books, Inc., Copyright 1980.

[4] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
 
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[9] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
 
[11] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/

The Domino Method of General Integer Nonlinear Programming Applied to an Oil Refinery Problem with Nonlinear Objective Function, Equality Constraints, and Continuous Variables, Second Edition

Jsun Yui Wong

The computer program below tries to solve the oil refinery problem on pages 198-199 of Conley [2].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [10].  Line 206, which is 206 X(1)=(  -.25*   X(2)-.3*X(3)- .25*X(4)- .35*X(5)    +400    )/.3 from one of the given constraints, through line 459 are illustrative.  Here X(1), X(6), and X(7) of line 206, line 321, and line 322, respectively, are proxy domino one, proxy domino 2, and proxy domino three, respectively.  When X(1) is optimized/pushed-down, X(6) is, in this case, optimized/pushed-down.  Similarly, X(7) is, in thise case, optimized/pushed-down.

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),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 5
42 A(J44)=FIX(RND*1000)
43 NEXT J44
126 IMAR=10+FIX(RND*5000)
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*4))
181 J=1+FIX(RND*5)
182 REM     GOTO 191
183  R=(1-RND*2)*A(J)
185 REM IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
187    X(J)=A(J)+  (RND^(RND*10))*R
189 REM   X(J)=A(J)+PA
190 REM  X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
192 NEXT IPP
206 X(1)=(  -.25*   X(2)-.3*X(3)- .25*X(4)- .35*X(5)    +400    )/.3
221    IF X(1)>600 THEN 1670
222    IF X(2)>1500 THEN 1670
223    IF X(3)>600 THEN 1670
224    IF X(4)>200 THEN 1670
225    IF X(5)>400 THEN 1670
228    FOR J44=1 TO 5
230    IF X(J44)<0 THEN 1670
231    NEXT J44
321 X(6)=+.35*X(1)+.3*   X(2)+.25*X(3)+.25*X(4)+ .25*X(5)    -450
322 X(7)= +.3*X(1)+.4*   X(2)+.4*X(3)+.4*X(4)+ .3*X(5)      -550
331 IF  ABS(X(6) )>.00001 THEN PS1=ABS(X(6)  )   ELSE PS1=0
332 IF  ABS(X(7) )>.00001 THEN PS2=ABS(X(7)  )   ELSE PS2=0
402 POBA2=-3.58*X(1)^.88-5.92*   X(2)^1.04-2.95*X(3)^.98-4.01*X(4)^1.02-5.02*X(5)^1.08
411 POBA= POBA2
459 POB1=POBA -999999999999#*( + PS1+PS2 )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7452.9 THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7)
1903 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-27498 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

436.8398   705.2708   142.1107   199.9887   1.039841E-05
0   0
-7452.9   -28617

436.842   705.2644   142.1061   199.9979   7.631095E-06
0   0
-7452.879   -27498

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-27498 was 38 minutes.

References

[1] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[2] William Conley, Computer Optimization Techniques.  New York/Princeton: Petrocelli Books, Inc., Copyright 1980.

[3] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[4] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[5] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[6] 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.
 
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[8] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[9] 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.
 
[10] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/

Testing the Domino Method of General Nonlinear Integer Programming with a Problem Involving a Nonlinear Objective Function, One Nonlinear Inequality Constraint, Five Linear Inequality Constraints, and Three Big Integer Variables, Second Edition

Jsun Yui Wong

The computer program below tries to solve Example 5.1 on page 119-120 of Conley [2].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [10].  Line 221, which is 221 X(6)=-X(1)-X(2)-2*X(3)+2000 from one of the given constraints, through line 459 are illustrative.  Here X(4) through X(9) of line 219 through line 244, respectively, are proxy domino one through proxy domino six, respectively.  When X(4) is optimized/pushed-down, X(5) is relatively easily optimized/pushed-down.  Similarly, X(6), X(7), x(8), and X(9) are relatively easily optimized/pushed-down.

0 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),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 3
42 A(J44)=FIX(RND*1000)
43 NEXT J44
126 IMAR=10+FIX(RND*2000)
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*3)
182 GOTO 191
183  R=(1-RND*2)*A(J)
185  IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
189  X(J)=A(J)+PA
190 REM  X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191     X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
192 NEXT IPP
197    FOR J44=1 TO 3
201    IF X(J44)>999 THEN 1670
202    IF X(J44)<0 THEN 1670
203    NEXT J44
219 X(4)=-X(1)-17*X(2)+8000
220 X(5)=-X(2)-5*X(3)+4000
221 X(6)=-X(1)-X(2)-2*X(3)+2000
224 X(7)=X(1) +X(2)   +X(3)  -200
243 X(8)= X(1)+7*X(2)+2*X(3)     -200
244 X(9)= X(1)^2+  X(2)*  X(3)    -900
250 FOR J44=4 TO 9
251 IF X(J44)<0 THEN PR(J44)=ABS(X(J44)) ELSE PR(J44)=0
255 NEXT J44
402 POBA2=-6*X(1)^2-18*X(2)^2-7*X(3)^2+2*X(1)+16*X(2)+31*X(3)+12*X(1)*X(2)*X(3)
411 POBA= POBA2
459 POB1=POBA -999999999#*(PR(4)+PR(5)+PR(6)+PR(7)+PR(8)+PR(9)  )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 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<1573000000# THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),A(5)
1902 PRINT A(6),A(7),A(8),A(9)
1903 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31730 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

758   426   408   0   1534
0   1392   4356   747472
1573099348   -31970

775   425   400   0   1575
0   1400   4350   769725
1573045750   -31827

775   425   400   0   1575
0   1400   4350   769725
1573045750   -31813

758   426   408   0   1534
0   1392   4356   747472
1573099348   -31805

775   425   400   0   1575
0   1400   4350   769725
1573045750   -31751

758   426   408   0   1534
0   1392   4356   747472
1573099348   -31740

758   426   408   0   1534
0   1392   4356   747472
1573099348   -31730

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-31730 was five minutes.

References

[1] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[2] William Conley, Computer Optimization Techniques, Revised Edition.  New York/Princeton: Petrocelli Books, Inc., Copyright 1984.

[3] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[4] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[5] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[6] 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.
 
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[8] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[9] 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.
 
[10] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/

The Domino Method of Nonlinear Discrete Programming Applied to a Transportation Problem with Nonlinear Objective Function, Eight Equality Constraints, and General Integer Variables, Second Edition

Jsun Yui Wong

The computer program below tries to solve the formulation on page 183 of Conley [2].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [10].  Line 221 through line 459 are illustrative.  Proxy domino one through proxy domino seven are shown in line 221 through line 227, respectively; these expressions are from the given constraints [2, page 183].

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),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 9
42 A(J44)=FIX(RND*200)
43 NEXT J44
126 IMAR=10+FIX(RND*500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 9
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*8))
181 J=1+FIX(RND*9)
182 GOTO 191
183  R=(1-RND*2)*A(J)
185  IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
189  X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
192 NEXT IPP
197    FOR J44=1 TO 9
201    IF X(J44)>200 THEN 1670
202    IF X(J44)<0 THEN 1670
203    NEXT J44
221 X(5)=    60  -X(6)
222 X(4) =70-X(5) - X(6)
223  X(8)= -X(9)     +30
224 X(7)    =-X(8) - X(9)    +40
225   X(1)   = - X(4)  - X(7)    +41
226    X(2)  =-X(5)  - X(8)     +52
227 X(3) = -X(6) - X(9)  +47
228    FOR J44=1 TO 9
229    IF X(J44)>200 THEN 1670
230    IF X(J44)<0 THEN 1670
231    NEXT J44
233 L= X(1)+X(2)+X(3)       -30
311 IF ABS(L  )>.00001 THEN PS=ABS(L)   ELSE PS=0
402 POBA2=-110*X(1)-158*   X(2)-92*X(3)-118*X(4)- 103*X(5)  -88* X(6)  -135* X(7)     -116* X(8)     - 130*X(9)^1.5
411 POBA= POBA2
459 POB1=POBA -999999999#*(PS )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 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<-991735! THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),A(5)
1901 PRINT A(6),A(7),A(8),A(9)
1903 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31733 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

21   0   9   10   22
38   10   30   0
-14758   -31957

21   0   9   10   40
20   10   12   18
-22867.78   -31909

21   0   9   10   22
38   10   30   0
-14758   -31801

21   0   9   10   22
38   10   30   0
-14758   -31759

21   0   9   10   22
38   10   30   0
-14758   -31733

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-31733 was ten seconds.

References

[1] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[2] William Conley, Computer Optimization Techniques.  New York/Princeton: Petrocelli Books, Inc., Copyright 1980.

[3] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[4] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[5] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[6] 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.
 
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[8] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[9] 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.
 
[10] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/

The Domino Method of General Integer Nonlinear Programming Applied to a Problem with Nonlinear Objective Function, Linear Inequality Constraints, and General Integer Variables

Jsun Yui Wong

The computer program below tries to solve Example 10.2 on pages 192-194 of Conley [2].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [10].  Line 301, which is 301 X(6)=-X(1)-7*X(2)-3*X(3)-2*X(4)-X(5)+360, through line 459 are illustrative.  Line 301's X(6), a slack variable, is proxy domino one; this line 301 is an expression of the first given nonequality constraint [2, page 192].

0 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),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 8
42 A(J44)=FIX(RND*35)
43 NEXT J44
126 IMAR=10+FIX(RND*1000)
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*4))
181 J=1+FIX(RND*5)
182 GOTO 191
183  R=(1-RND*2)*A(J)
185  IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
189  X(J)=A(J)+PA
190 REM  X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191     X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
192 NEXT IPP
197    FOR J44=1 TO 5
201    IF X(J44)>35 THEN 1670
202    IF X(J44)<0 THEN 1670
203    NEXT J44
301 X(6)=-X(1)-7*X(2)-3*X(3)-2*X(4)-X(5)+360
302 L2=X(1)+2*X(2)+3*X(3)+4*X(4)+5*X(5)   -400
303 L3=5*X(1)+4*X(2)+3*X(3)+2*X(4)+X(5)   -400
307 IF X(6)<0 THEN PS(1)=ABS(X(6)) ELSE PS(1)=0
308 IF L2>0 THEN PS(2)=L2 ELSE PS(2)=0
309 IF L3>0 THEN PS(3)=L3 ELSE PS(3)=0
402 POBA2=2*X(1)+X(2)^2+6*X(3)+5*X(4)+9*X(5)
411 POBA= POBA2
459 POB1=POBA -999999999#*(PS(1)+PS(2)+PS(3)  )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 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<1735 THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),A(5)
1903 PRINT 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=-31422 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

1   35   3   35   35
0   1735   -31993

4   35   2   35   35
0   1735   -31989

7   35   1   35   35
0   1735   -31890

1   35   3   35   35
0   1735   -31821

10   35   0   35   35
0   1735   -31731

10   35   0   35   35
0   1735   -31523

4   35   2   35   35
0   1735   -31422

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-31422 was two minutes.

References

[1] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[2] William Conley, Optimization: A Simplified Approach.  New York/Princeton: Petrocelli Books, Inc., Copyright 1981.

[3] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[4] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[5] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[6] 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.
 
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[8] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[9] 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.
 
[10] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/

The Domino Method of General Integer Nonlinear Programming Applied to an Electrical Resistance Network

Jsun Yui Wong

The computer program below tries to solve the electrical resistance network problem in Conley [4].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [2] and after the domino method of solving nonlinear systems of equations [13].  Line 310, which is 310 X(11)=1/ (1/X(1)+1/X(2) )   + 1/(1/X(4)+1/X(5)+1/X(6))  + 1/(1/X(9)+1/X(10))-122, through line 411 are illustrative.  Here X(11) through X(15) of line 310 through line 350, respectively, are proxy domino one through proxy domino five, respectively.  When X(11) is optimized/pushed-down, X(12) is relatively easily optimized/pushed-down.  Similarly, X(13), X(14), and X(15) are relatively easily optimized/pushed-down.

0  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),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 10
42 A(J44)=FIX(RND*400)
43 NEXT J44
126 IMAR=10+FIX(RND*32000)
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)
185  IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
189  X(J)=A(J)+PA
190 REM  X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM    X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
192 NEXT IPP
197    FOR J44=1 TO 10
201    IF X(J44)>400 THEN 1670
202    IF X(J44)<1 THEN 1670
203    NEXT J44
310 X(11)=1/ (1/X(1)+1/X(2) )   + 1/(1/X(4)+1/X(5)+1/X(6))  + 1/(1/X(9)+1/X(10))-122
320 X(12)=1/ (1/X(1)+1/X(2) )   + 1/(1/X(5)+1/X(6)+1/X(7))  + 1/(1/X(9)+1/X(10))-112
330 X(13)=1/ (1/X(2)+1/X(3) )   + 1/(1/X(4)+1/X(5)+1/X(6))  + 1/(1/X(9)+1/X(10))-105
340 X(14)=1/ (1/X(2)+1/X(3) )   + 1/(1/X(5)+1/X(6)+1/X(7))  + 1/(1/X(9)+1/X(10))-95
350 X(15)=1/ (1/X(1)+1/X(2)+1/X(3) )   + 1/(1/X(4)+1/X(5)+1/X(6)+1/X(7))  + 1/(1/X(8)+1/X(9)+1/X(10))-80
411 POBA= -ABS(X(11))-ABS(X(12))-ABS(X(13))-ABS(X(14))- ABS(X(15))
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 15
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-1E-09 THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),A(5)
1901 PRINT A(6),A(7),A(8),A(9),A(10)
1903 PRINT A(11),A(12),A(13),A(14),A(15)
1905 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31993 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

308.0525920714426   106.1138329576789   148.7127150354939
114.8101109168384   325.5764279450349   
17.37446729712957   6.04225184664533   154.9598328175128   
92.12151121457207   41.58507056766254  
-3.552713678800501D-14   1.522337811366015D-12   -8.046896482483135D-13
7.51398943066306D-13   -1.412203687323199D-12
-4.526157226791838D-12   -31997

77.57292448449295   96.15627009813874   35.51428817811416
397.8358477378793   49.75476332781962   
277.8037481413037   84.56761194556611   211.1392693013211   
58.53767634163486   135.8968557621608  
-2.401634446869139D-12   -1.392663762089796D-12   0
1.0089770684779343D-12   9.013234603116871D-12
-1.381650349685515D-11   -31993

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

References

[1] Dana H. Ballard, C. O. Jelinek, R. Schinzinger, An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3 (1974), pp. 261-266.

[2] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[3] William Conley, Computer Optimization Techniques.  New York/Princeton: Petrocelli Books, Inc., Copyright 1980.

[4] William Conley, An Electrical Resistance Network.  International Journal of Mathematical Education in Science and Technology, Volume 14, Number 3 (1983), pp. 287-291.

[5] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[6] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[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] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[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, Florida 33432, 1981.

[12] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge, UK: Cambridge University Press, 2000.
 
[13] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
 
[14] Jsun Yui Wong (2011, July 2).  A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Transformer Design Model with Integer Variables and Continuous Variables, Fourth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/02/

Testing the Domino Method of General Integer Nonlinear Programming with a Problem Involving a Nonlinear Objective Function, One Nonlinear Inequality Constraint, Five Linear Inequality Constraints, and Three Big Integer Variables

Jsun Yui Wong

The following computer program tries to solve Example 5.1 on pages 119-120 of Conley [2].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [10].  Line 193 through line 459 are illustrative.  Line 193 and line 195, which are  193 X(4)  = 8000  -X(1) -17*X(2) and 195 X(5)  = 4000  -X(2) -5*X(3), are two expressions of two of the inequality constraints.  Line 193's X(4), a slack variable, is proxy domino one; line 195's X(5), a slack variable, is proxy domino two.

0  DEFDBL A-Z
2 DEFINT I,J,K,X
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 3
42 A(J44)=FIX(RND*300)
43 NEXT J44
126 IMAR=10+FIX(RND*2000)
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*3)
183 REM R=(1-RND*2)*A(J)
185 REM IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
189 REM X(J)=A(J)+PA
190 REM  X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191  X(J)=A(J)+FIX(RND*51)-FIX(RND*51)
192 NEXT IPP
193 X(4)  = 8000  -X(1) -17*X(2)
195 X(5)  = 4000  -X(2) -5*X(3)
201    FOR J44=1 TO 3
202    IF X(J44)<0 THEN 1670
203    NEXT J44
206    FOR J44=1 TO 3
207    IF X(J44)>999 THEN 1670
208    NEXT J44
211    FOR J44=4 TO 5
212    IF X(J44)<0 THEN QS(J44)=ABS(X(J44)) ELSE QS(J44)=0
213    NEXT J44
351  PS(1)=X(1)+X(2)+ 2*X(3)     -2000
353  PS(2)=-  X(1)-7*X(2)-2*X(3)   +200
354  PS(3)=-  X(1)-X(2) -X(3)   +200
355  PS(4)=-  X(1)^2-X(2)*X(3)   +900
356   FOR J44=1 TO 4
357   IF  PS(J44)>0 THEN PS(J44)=PS(J44) ELSE PS(J44)=0
358   NEXT J44
411 POBA= -6*X(1)^2-18*X(2)^2-7*X(3)^2+2*X(1)+16*X(2)+31*X(3)+12*X(1)*X(2)*X(3)
459 POB1=POBA    -999999999#*(+QS(4)+QS(5)+PS(1)+PS(2)+PS(3)+PS(4)  )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 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
1889 IF M<1573000000# THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-28406 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

775   425   400   0   1575
1573045632   -31854

758   426   408   0   1534
1573099264   -31346

758   426   408   0   1534
1573099264   -30016

758   426   408   0   1534
1573099264   -29117

775   425   400   0   1575
1573045632   -28406

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-28406 was 13 minutes.

References

[1] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[2] William Conley, Computer Optimization Techniques.  New York/Princeton: Petrocelli Books, Inc., Copyright 1980.

[3] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[4] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[5] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[6] 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.
 
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[8] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[9] 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.

[10] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/

Testing the Domino Method of General Integer Nonlinear Programming with a Minimization Problem Involving a Nonlinear Objective Function, One Nonlinear Equality Constraint, One Linear Equality Constraint, and Three Continuous Variables

Jsun Yui Wong

The following computer program tries to solve Example 3.9 of Conley [2, p. 55].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [10].  Line 193 through line 459 are illustrative.  Line 193, which is 193 X(2)=100-3*X(1)-6*X(3), is an expression of the given linear equality constraint.  X(2) is proxy domino one.

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),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 3
42 A(J44)=FIX(RND*100)
43 NEXT J44
126 IMAR=10+FIX(RND*2000)
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*3)
183 R=(1-RND*2)*A(J)
185 IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
189 X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*51)-FIX(RND*51)
192 NEXT IPP
193 X(2)  = 100-3*X(1)       -6*  X(3)
211 FOR J44=1 TO 4
212 IF X(J44)<0 THEN 1670
213 NEXT J44
351 PS(2)=X(1)+2*X(2)^2+10*X(3)     -80
356 FOR J44=2 TO 2
357 IF  ABS(PS(J44))   >.00005 THEN PS(J44)=ABS(PS(J44))  ELSE PS(J44)=0
358 NEXT J44
411 POBA= -2*  X(1)   - 9*X(2)-  14*X(3)   -   2*X(3) *X(1)+X(3)^2
459 POB1=POBA+    -999999999#*( +PS(2)      )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 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<-117 THEN 1999
1900 PRINT A(1),A(2),A(3)
1905 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31186 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

31.69516   4.914513   0
-107.6209   -31922

31.69516   4.914513   0
-107.6209   -31597

31.69516   4.914513   0
-107.6209   -31186

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-31186 was ten minutes.

References

[1] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[2] William Conley, Optimization: A Simplified Approach.  New York/Princeton: Petrocelli Books, Inc., Copyright 1981.

[3] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[4] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[5] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[6] 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.
 
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[8] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[9] 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.

[10] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/

Testing the Domino Method of General Integer Nonlinear Programming with a Maximization Problem Involving a Nonlinear Objective Function, One Nonlinear Inequality Constraint, Three Linear Inequality Constraints, and Four General Integer Variables

Jsun Yui Wong

The following computer program tries to solve Example 3.35 of Conley [2, pp. 65-66].  Line 181 through line 1111 of the following computer program partly describe the problem.

The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [10].  Line 193 through line 459 are illustrative; line 193's X(5) is proxy domino one.

0 REM DEFDBL A-Z
2 DEFINT I,J,K,X
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
41 FOR J44=1 TO 4
42 A(J44)=FIX(RND*100)
43 NEXT J44
126 IMAR=10+FIX(RND*2000)
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 REM R=(1-RND*2)*A(J)
185 REM IF RND<.25 THEN PA=(RND)*R ELSE IF RND<.33 THEN PA=(RND^3)*R ELSE IF RND<.5 THEN PA=(RND^5)*R ELSE PA=(RND^7)*R
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*51)-FIX(RND*51)
192 NEXT IPP
193 X(5)  = 38    -X(1)-3*X(4)
211 FOR J44=1 TO 4
212 IF X(J44)<0 THEN 1670
213 NEXT J44
221 FOR J44=1 TO 4
222 IF X(J44)>30 THEN 1670
223 NEXT J44
331 IF X(5)<  0 THEN PS(1)=ABS(X(5)) ELSE PS(1)=0
351 PS(2)=X(1)^2+X(1)*X(4)     -45
353 PS(3)=   +X(2) +X(4)   -52
355 PS(4)=   X(1)+X(2) +3*X(3)   -55
356 FOR J44=2 TO 4
357 IF  PS(J44)>0 THEN PS(J44)=PS(J44) ELSE PS(J44)=0
358 NEXT J44
411 POBA= +X(1)   - X(2)+   X(3)   -X(4) +X(1)*X(2)-X(1)*X(4)+X(4)^2-X(1)^2
459 POB1=POBA+    -999999999#*(PS(1) +PS(2)  +PS(3)+PS(4)    )
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 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
1889 IF M<140 THEN 1999
1900 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The complete output through JJJJ=-31115 is shown below.  What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.

2   26   9   12   0
141   -31932

0   0   18   12   2
150   -31868

0   0   18   12   2
150   -31798

2   30   7   12   0
143   -31115

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11,  the throughput time from JJJJ=-32000 through JJJJ=-31115 was forty seconds.

References

[1] Clarence L. Barnhart, Robert K. Barnhart (Editors).  The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.

[2] William Conley, Computer Optimization Techniques, Revised Edition.  New York/Princeton: Petrocelli Books, Inc., Copyright 1984.

[3] George B. Dantzig,  Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[4] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[5] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.     

[6] 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.
 
[7] Duan Li, Xiaoling Sun, Nonlinear Integer Programming.  Boston: Springer, 2006.

[8] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems.  Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.     

[9] 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.

[10] Jsun Yui Wong (2011, April 17).  The Domino Method of Solving Nonlinear Systems of Equations.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/