A Flexible Algorithmic Framework for Strictly Convex Quadratic Minimization

TL;DR

This paper introduces a versatile algorithmic framework for minimizing strictly convex quadratic functions, unifying and extending existing methods with guarantees of linear convergence even with preconditioning.

Contribution

The paper develops a generic, flexible framework that encompasses various algorithms like steepest descent and conjugate gradients, with proven linear convergence guarantees.

Findings

Framework includes many existing algorithms as special cases

Guarantees linear convergence under broad conditions

Works with relaxation and preconditioning

Abstract

This paper presents an algorithmic framework for the minimization of strictly convex quadratic functions. The framework is flexible and generic. At every iteration the search direction is a linear combination of the negative gradient, as well as (possibly) several other `sub-search' directions, where the user determines which, and how many, sub-search directions to include. Then, a step size along each sub-direction is generated in such a way that the gradient is minimized (with respect to a matrix norm), over the hyperplane specified by the user chosen search directions. Theoretical machinery is developed, which shows that any algorithm that fits into the generic framework is guaranteed to converge at a linear rate. Moreover, these theoretical results hold even when relaxation and/or symmetric preconditioning is employed. Several state-of-the-art algorithms fit into this scheme, including steepest descent and conjugate gradients.