General-purpose preconditioning for regularized interior point methods

TL;DR

This paper introduces versatile preconditioners for regularized systems in interior point methods, improving convergence and stability in convex optimization problems.

Contribution

It proposes general-purpose, positive definite preconditioners and sparsification techniques tailored for interior point methods in convex programming.

Findings

Preconditioners enhance convergence of iterative solvers.

Sparsifications prevent complex eigenvalues, improving stability.

Applicable to both linear and nonlinear convex problems.

Abstract

In this paper we present general-purpose preconditioners for regularized augmented systems arising from optimization problems, and their corresponding normal equations. We discuss positive definite preconditioners, suitable for CG and MINRES. We consider "sparsifications" which avoid situations in which eigenvalues of the preconditioned matrix may become complex. Special attention is given to systems arising from the application of regularized interior point methods to linear or nonlinear convex programming problems.