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A family of variable metric methods in function space, without exact line searches

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Abstract

The purpose of this paper is to extend a family of variable metric methods, of which the BFGS algorithm (Ref. 1) is a member, into function space, in particular, for the solution of unconstrained optimal control problems. An inexact one-dimensional minimization as suggested by Fletcher (ref. 2) is used. It is shown that, with this stepsize rule and under some mild assumptions, the sequence constructed by this family of methods converges superlinearly for a strictly convex functional defined on a suitable Banach space. This result is shown to remain valid on a Hilbert space and on a Euclidean space under more relaxed assumptions. The BFGS method without line searches is used to solve several standard numerical examples, and excellent performance is observed.

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Authors and Affiliations

  1. Department of Electrical Engineering, Faculty of Engineering, University of Waterloo, Waterloo, Ontario, Canada

    R. V. Mayorga (PhD Candidate) & V. H. Quintana (Associate Professor)

Authors
  1. R. V. Mayorga
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  2. V. H. Quintana
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Communicated by H. Y. Huang

This work was supported by the Consejo Nacional de Ciencia y Tecnologia de Mexico, and by the National Research Council of Canada, Grant No. A-8835. The authors are indebted to Dr. C. Charalambous for suggesting the topic and stimulating discussions.

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Mayorga, R.V., Quintana, V.H. A family of variable metric methods in function space, without exact line searches. J Optim Theory Appl 31, 303–329 (1980). https://doi.org/10.1007/BF01262975

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