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/