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On the Local and Superlinear Convergence of Quasi-Newton Methods
Abstract
This paper presents a local convergence analysis for several well-known quasi-Newton methods when used, without line searches, in an iteration of the form to solve for x* such that Fx* = 0. The basic idea behind the proofs is that under certain reasonable conditions on xo, F and xo, the errors in the sequence of approximations {Hk} to F′(x*)−1 can be shown to be of bounded deterioration in that these errors, while not ensured to decrease, can increase only in a controlled way. Despite the fact that Hk is not shown to approach F′(x*)−1, the methods considered, including those based on the single-rank Broyden and double-rank Davidon-Fletcher-Powell formulae, generate locally Q-superlinearly convergent sequences {xk}.
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Cited by
Iterative Methods for Optimization
Kelley, C. T. (1999) · Society for Industrial and Applied Mathematics eBooks · doi:10.1137/1.9781611970920 · Cited by 2,052
Sequential quadratic programming
Boggs, P. T., & Tolle, J. W. (1995) · Acta Numerica · doi:10.1017/s0962492900002518 · Cited by 1,858
First- and Second-Order Methods for Learning: Between Steepest Descent and Newton's Method
Battiti, R. (1992) · Neural Computation · doi:10.1162/neco.1992.4.2.141 · Cited by 1,193
Superlinearly convergent variable metric algorithms for general nonlinear programming problems
Han, S.-P. (1976) · Mathematical Programming · doi:10.1007/bf01580395 · Cited by 477
Metadata
- Authors
- BROYDEN, C. G., DENNIS, J. E., & MORÉ, J. J.
- Venue
- IMA Journal of Applied Mathematics
- Publisher
- Oxford University Press (OUP)
- Published
- 1973
- Volume
- 12
- Issue
- 3
- Pages
- 223-245
- ISSN
- 0272-4960 (ISSN-L)
- 1464-3634
- OpenAlex
- W2084236346
- Cited by
- 502
- References
- 0
- FWCI
- 6.88
- Type
- journal article
- Language
- en
- Open access
- closed
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- CoChat. (2026). Citation check: On the Local and Superlinear Convergence of Quasi-Newton Methods. https://app.cochat.ai/papers/10.1093/imamat/12.3.223
- The paper
- BROYDEN, C. G., DENNIS, J. E., & MORÉ, J. J. (1973). On the Local and Superlinear Convergence of Quasi-Newton Methods. IMA Journal of Applied Mathematics. https://doi.org/10.1093/imamat/12.3.223
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