$R$-linear convergence of limited memory steepest descent. (24th April 2017)

• Record Type: Journal Article
• Authors: Curtis, Frank E; Guo, Wei
• Is Part Of: IMA journal of numerical analysis. Volume 38:Number 2(2018)
• Page Start: 720
• Page End: 742
• Publication Date: 2017-04-24
• Subjects: unconstrained optimization -- steepest descent methods -- Barzilai–Borwein methods -- limited memory methods -- quadratic optimization -- R-linear rate of convergence; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drx016
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: The limited memory steepest descent method (LMSD) proposed by Fletcher is an extension of the Barzilai–Borwein 'two-point step size' strategy for steepest descent methods for solving unconstrained optimization problems. It is known that the Barzilai–Borwein strategy yields a method with an $R$ -linear rate of convergence when it is employed to minimize a strongly convex quadratic. This article extends this analysis for LMSD, also for strongly convex quadratics. In particular, it is shown that, under reasonable assumptions, the method is $R$ -linearly convergent for any choice of the history length parameter. The results of numerical experiments are also provided to illustrate behaviors of the method that are revealed through the theoretical analysis.