The computer program listed below seeks to solve Schittkowski’s last test problem [16, p. 213, Test Problem 395] but with 10000 unknowns instead of 50 unknowns. The source of this Test Problem 395 is given in Schittkowski [16]. Thus, the problem here is to minimize
10000
SIGMA i*(X(i)^2+X(i)^4 )
i=1
subject to
10000
SIGMA X(i)^2 =1.
i=1
0 REM DEFDBL A-Z
1 DEFINT J, K, B
2 DIM A(10000), X(10000)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+300
110 FOR J44 = 1 TO 10000
112 A(J44) = RND * .01
114 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 10000
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 10000)
183 R = (1 - RND * 2) * A(B)
187 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 IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1
199 NEXT IPP
400 SONE = 0
401 FOR J44 = 2 TO 10000
403 SONE = SONE + X(J44) ^ 2
404 NEXT J44
405 IF (1 - SONE) < .0000001 THEN 1670
406 X(1) = (1 - SONE) ^ (1 / 2)
410 STWO = 0
411 FOR J44 = 1 TO 10000
413 STWO = STWO + J44 * (X(J44) ^ 2 + X(J44) ^ 4)
415 NEXT J44
457 PD1 = -STWO
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 10000
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M < -999999999# THEN 1999
1935 PRINT A(1), A(2), A(3), A(4), A(5)
1937 PRINT A(9996), A(9997), A(9998), A(9999), A(10000)
1939 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [17]. Copied by hand from the screen, the computer program’s completer output through JJJJ= -31994 is shown below.
1 0 0 0 0
0 0 0 0 0
-2 -32000
.911705 .4108454 0 0 0
0 0 0 0 0
-1.91668 -31999
.9999998 0 6.966768E-04 0
0
0 0 0 0 0
-2 -31998
.9231066 .384544 0 0 0
0 0 0 0 0
-1.917726 -31997
.9122139 .4097142 0 0 0
0 0 0 0 0
-1.916671 -31996
.9127767 .4084589 0 0 0
0 0 0 0 0
-1.916667 -31995
1 0 0 0 0
0 0 0 0 0
-2 -31994
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 10000 unknowns, only the ten A's of line 1935 and line 1937 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 [17], the wall-clock time for obtaining the output through JJJJ= -31994 was four hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[6] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am.
[7] 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.
[8] 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.
[9] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207. .
[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 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[12] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.
[13] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[14] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.
[15] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[18] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.
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