The DOI matches its Crossref record; 15 of the 15 references it cites were found in a registry or an index.
Newton-Type Minimization via the Lanczos Method
Abstract
This paper discusses the use of the linear conjugate-gradient method (developed via the Lanczos method) in the solution of large-scale unconstrained minimization problems. At each iteration of a Newton-type method, the direction of search is defined as the solution of a quadratic subproblem. When the number of variables is very large, this subproblem may be solved using the linear conjugate-gradient method of Hestenes and Stiefel. We show how the equivalent Lanczos characterization of the linear conjugate-gradient method may be exploited to define a modified Newton method which can be applied to problems that do not necessarily have positive-definite Hessian matrices at all points of the region of interest. This derivation also makes it possible to compute a negative-curvature direction at a stationary point. The idea of a truncated Newton method is to perform only a limited number of iterations of the quadratic subproblem. This effectively gives a search direction that interpolates between the steepest-descent direction and the Newton direction. We describe a preconditioned linear conjugate-gradient method that defines a search direction which interpolates between the direction defined by a nonlinear conjugate-gradient-type algorithm and a modified Newton direction.
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Citations
Cites
Methods of conjugate gradients for solving linear systems
Hestenes, M. R., & Stiefel, E. (1952) · Journal of Research of the National Bureau of Standards · doi:10.6028/jres.049.044 · Cited by 8,132
Lanczos, C. (1950) · Journal of Research of the National Bureau of Standards · doi:10.6028/jres.045.026 · Cited by 4,391
The Symmetric Eigenvalue Problem
Parlett, B. N. (1998) · Society for Industrial and Applied Mathematics eBooks · doi:10.1137/1.9781611971163 · Cited by 3,192
Updating quasi-Newton matrices with limited storage
Nocedal, J. (1980) · Mathematics of Computation · doi:10.1090/s0025-5718-1980-0572855-7 · Cited by 2,958
Introduction to linear and nonlinear programming
Polak, E. (1974) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1974.1100535 · Cited by 2,100
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Solution of Sparse Indefinite Systems of Linear Equations
Paige, C. C., & Saunders, M. A. (1975) · SIAM Journal on Numerical Analysis · doi:10.1137/0712047 · Cited by 1,620
Quasi-Newton Methods, Motivation and Theory
Dennis, Jr., & Moré, J. J. (1977) · SIAM Review · doi:10.1137/1019005 · Cited by 1,553
The Symmetric Eigenvalue Problem.
Wilkinson, J. H., & Parlett, B. N. (1981) · Mathematics of Computation · doi:10.2307/2007453 · Cited by 1,186
Large-scale linearly constrained optimization
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Iterative Methods for Optimization
Kelley, C. T. (1999) · Society for Industrial and Applied Mathematics eBooks · doi:10.1137/1.9781611970920 · Cited by 2,052
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Metadata
- Authors
- Nash, S. G.
- Venue
- SIAM Journal on Numerical Analysis
- Published
- 1984-08
- Volume
- 21
- Issue
- 4
- Pages
- 770-788
- ISSN
- 0036-1429 (ISSN-L)
- 1095-7170
- OpenAlex
- W2065000345
- Cited by
- 338
- References
- 15
- FWCI
- 3.16
- Type
- journal article
- Language
- en
- Open access
- closed
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Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
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