A Framework of Conjugate Direction Methods for Symmetric Linear Systems in Optimization

TL;DR

This paper introduces a parameter-dependent class of Krylov-based methods called CD for solving symmetric linear systems, extending conjugate gradient properties and ensuring finite convergence with preconditioning.

Contribution

It proposes a new framework of conjugate direction methods that generalize CG, with proven convergence and error analysis, including preconditioning techniques.

Findings

CD methods generate conjugate directions similar to CG.

Finite convergence of CD algorithms is proven.

Preconditioning in CD retains standard error bounds.

Abstract

In this paper we introduce a parameter dependent class of Krylov-based methods, namely CD, for the solution of symmetric linear systems. We give evidence that in our proposal we generate sequences of conjugate directions, extending some properties of the standard Conjugate Gradient (CG) method, in order to preserve the conjugacy. For specific values of the parameters in our framework we obtain schemes equivalent to both the CG and the scaled-CG. We also prove the finite convergence of the algorithms in CD, and we provide some error analysis. Finally, preconditioning is introduced for CD, and we show that standard error bounds for the preconditioned CG also hold for the preconditioned CD.