R-Linear Convergence of Limited Memory Steepest Descent

TL;DR

This paper proves that the limited memory steepest descent method (LMSD) converges at an R-linear rate for strongly convex quadratic problems, extending known results for the Barzilai-Borwein method and supporting it with numerical experiments.

Contribution

It extends the R-linear convergence analysis of the Barzilai-Borwein method to the LMSD for strongly convex quadratics, regardless of history length.

Findings

LMSD is R-linearly convergent for any history length.

Numerical experiments confirm theoretical convergence behavior.

Theoretical analysis reveals new insights into LMSD performance.

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 paper extends this analysis for LMSD, also for strongly convex quadratics. In particular, it is shown that the method is R-linearly convergent for any choice of the history length parameter. The results of numerical experiments are provided to illustrate behaviors of the method that are revealed through the theoretical analysis.