Limited memory gradient methods for unconstrained optimization

TL;DR

This paper reviews and extends limited memory gradient methods for unconstrained optimization, introducing new variants and analyzing their numerical behavior for quadratic and nonlinear functions.

Contribution

It proposes novel variants of Fletcher's limited memory steepest descent method, including improved stepsize computation and secant condition modifications.

Findings

Enhanced stepsize computation using harmonic Ritz values

Existence of a secant condition with low-dimensional Hessian projection

New methods for nonlinear optimization with improved secant conditions

Abstract

The limited memory steepest descent method (Fletcher, 2012) for unconstrained optimization problems stores a few past gradients to compute multiple stepsizes at once. We review this method and propose new variants. For strictly convex quadratic objective functions, we study the numerical behavior of different techniques to compute new stepsizes. In particular, we introduce a method to improve the use of harmonic Ritz values. We also show the existence of a secant condition associated with LMSD, where the approximating Hessian is projected onto a low-dimensional space. In the general nonlinear case, we propose two new alternatives to Fletcher's method: first, the addition of symmetry constraints to the secant condition valid for the quadratic case; second, a perturbation of the last differences between consecutive gradients, to satisfy multiple secant equations simultaneously. We show that Fletcher's method can also be interpreted from this viewpoint.