Jsun Yui Wong
The computer program listed below seeks one integer solution or more to the following given system of two nonlinear equations:
14999
sigma [ 100 * (X(k + 1) - X(k) ^ 2) ^ 2 + (1 - X(k)) ^ 2 ] = 0,
k=1
150000
- sigma X(J22) ^ 2 = -150000,
J22=1
and each unknown = 0 or 1.
The first equation above is based on the Rosenbrock function in Schitkowski [11, pp. 118-123]. The second ccmes from Schitkowski [11, p. 194].
0 REM DEFDBL A-Z
3 DEFINT X
4 DIM X(150042), A(150042), L(150033), K(150033)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+57
91 FOR KK = 1 TO 150000
94 A(KK) = FIX(RND * 1.99)
95 NEXT KK
128 FOR I = 1 TO 9000000
129 FOR K = 1 TO 150000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
171 B = 1 + FIX(RND * 150003)
175 GOTO 190
181 B = 1 + FIX(RND * 100003)
182 IF RND < .5 THEN 183 ELSE GOTO 189
183 R = (1 - RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * R
187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
188 GOTO 191
189 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1
190 IF A(B) = 0 THEN X(B) = 1 ELSE X(B) = 0
191 NEXT IPP
193 GOTO 231
194 FOR J49 = 1 TO 50000
195 IF X(J49) < 0 THEN GOTO 1670
196 IF X(J49) > 2 THEN GOTO 1670
197 NEXT J49
201 PRODBROWN = 1
203 FOR J33 = 1 TO 50000
206 PRODBROWN = PRODBROWN * X(J33)
209 NEXT J33
222 N1 = PRODBROWN - 1
231 SUMSCHI = 0
236 FOR J22 = 1 TO 150000
239 SUMSCHI = SUMSCHI + X(J22) ^ 2
241 NEXT J22
244 N9 = -SUMSCHI + 150000
246 REM N5 = -(X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2) * (X(1) ^ 4 + X(2) ^ 4 + X(3) ^ 4 + X(4) ^ 4 + X(5) ^ 4 + X(6) ^ 4 + X(7) ^ 4 + X(8) ^ 4) + (X(1) ^ 3 + X(2) ^ 3 + X(3) ^ 3 + X(4) ^ 3 + X(5) ^ 3 + X(6) ^ 3 + X(7) ^ 3 + X(8) ^ 3) ^ 2
257 sumrose = 0
259 FOR j44 = 1 TO 149999
261 sumrose = sumrose + 100 * (X(j44 + 1) - X(j44) ^ 2) ^ 2 + (1 - X(j44)) ^ 2
263 NEXT j44
265 N6 = sumrose - 0
277 P = -ABS(N6) - ABS(N9)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 150000
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(49999), A(50000), M, JJJJ
1667 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(149996), A(149997), A(149998), A(149999), A(150000), M, JJJJ
1670 NEXT I
1888 IF M < -100 THEN 1999
1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(29996), A(29997), A(49998), A(49999), A(50000)
1912 PRINT A(1111), A(1112), A(1113), A(1114), A(1115), A(3336), A(3337), A(3338), A(3339), A(3340), A(5996), A(5997), A(25998), A(25999), A(50000), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [12]. Copied by hand from the screen, the computer program’s output is summarized below:
.
.
.
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-3454 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-3252 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-3250 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-3248 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-3046 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2844 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2642 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2440 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2238 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2036 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2034 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2032 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1830 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1828 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1626 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1424 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1222 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1020 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1018 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-816 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-614 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-412 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-210 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-208 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-206 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-204 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-202 -32000
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 150,000 A's, only the 15 A's of line 1667 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [12], the wall-clock time for reaching M=0 was 95 hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[2] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[3] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[4] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[5] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. www.SciRP.org/journal/am.
[6] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[7] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[8] 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.
[9] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[10] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
[11] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[12] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[13] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[14] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.
[15] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/
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