Updating quasi-Newton matrices with limited storage
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- by Jorge Nocedal PDF
- Math. Comp. 35 (1980), 773-782 Request permission
Abstract:
We study how to use the BFGS quasi-Newton matrices to precondition minimization methods for problems where the storage is critical. We give an update formula which generates matrices using information from the last m iterations, where m is any number supplied by the user. The quasi-Newton matrix is updated at every iteration by dropping the oldest information and replacing it by the newest information. It is shown that the matrices generated have some desirable properties. The resulting algorithms are tested numerically and compared with several well-known methods.References
- C. G. Broyden, The convergence of a class of double-rank minimization algorithms. II. The new algorithm, J. Inst. Math. Appl. 6 (1970), 222–231. MR 433870
- A. G. Buckley, A combined conjugate-gradient quasi-Newton minimization algorithm, Math. Programming 15 (1978), no. 2, 200–210. MR 509962, DOI 10.1007/BF01609018 W. C. DAVIDON & L. NAZARETH, DRVOCR-A FORTRAN Implementation of Davidon’s Optimally Conditioned Method, TM-306, Argonne National Lab., Argonne, Ill., 1977.
- J. E. Dennis Jr. and Jorge J. Moré, Quasi-Newton methods, motivation and theory, SIAM Rev. 19 (1977), no. 1, 46–89. MR 445812, DOI 10.1137/1019005 R. FLETCHER, "A new approach to variable metric algorithms," Comput. J., v. 13, 1970, pp. 317-322.
- Hermann Matthies and Gilbert Strang, The solution of nonlinear finite element equations, Internat. J. Numer. Methods Engrg. 14 (1979), no. 11, 1613–1626. MR 551801, DOI 10.1002/nme.1620141104 L. NAZARETH, A Relationship Between the BFGS and Conjugate Gradient Algorithms, ANL-AMD Tech. Memo 282 (rev.), Argonne National Lab., Argonne, Ill., 1977. L. NAZARETH & J. NOCEDAL, A Study of Conjugate Gradient Methods, Tech. Rep. SOL 78-29, Dept. of Operations Research, Stanford University, Stanford, Calif., 1979. L. NAZARETH & J. NOCEDAL, "Convergence analysis of optimization methods that use variable storage," Manuscript, 1978. A. PERRY, A Modified Conjugate Gradient Algorithm, Discussion paper No. 229, Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, Ill., 1976. D. SHANNO, Conjugate Gradient Methods With Inexact Line Searches, MIS Tech. Report 22, University of Arizona, Tucson, Ariz., 1977. D. SHANNO, On Variable-Metric Methods for Sparse Hessians, MIS Tech. Rep. 27, University of Arizona, Tucson, Ariz., 1978. D. SHANNO & K. PHUA, A Variable Method Subroutine for Unconstrained Nonlinear Minimization, MIS Tech. Rep. No. 28, University of Arizona, Tucson, Ariz., 1978.
- Josef Stoer, On the convergence rate of imperfect minimization algorithms in Broyden’s $\beta$-class, Math. Programming 9 (1975), no. 3, 313–335. MR 413491, DOI 10.1007/BF01681353
- Ph. L. Toint, On sparse and symmetric matrix updating subject to a linear equation, Math. Comp. 31 (1977), no. 140, 954–961. MR 455338, DOI 10.1090/S0025-5718-1977-0455338-4
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Math. Comp. 35 (1980), 773-782
- MSC: Primary 65K05; Secondary 90C30
- DOI: https://doi.org/10.1090/S0025-5718-1980-0572855-7
- MathSciNet review: 572855