Proximal stabilized Interior Point Methods for quadratic programming and low-frequency-updates preconditioning techniques

TL;DR

This paper introduces a Proximal Stabilized Interior Point Method for quadratic programming that offers theoretical convergence guarantees, handles degeneracy, and employs low-frequency-updates preconditioning to improve computational efficiency.

Contribution

It develops a new Proximal Stabilized IPM framework with convergence analysis and proposes low-frequency-updates preconditioning techniques for efficient iterative linear system solutions.

Findings

The PS-IPM converges strongly and handles degenerate problems.

Preconditioners with low-frequency-updates reduce computational costs.

The method offers an alternative between first and second order approaches.

Abstract

In this work, in the context of Linear and Quadratic Programming, we interpret Primal Dual Regularized Interior Point Methods (PDR-IPMs) in the framework of the Proximal Point Method. The resulting Proximal Stabilized IPM (PS-IPM) is strongly supported by theoretical results concerning convergence and the rate of convergence, and can handle degenerate problems. Moreover, in the second part of this work, we analyse the interactions between the regularization parameters and the computational foot-print of the linear algebra routines used to solve the Newton linear systems. In particular, when these systems are solved using an iterative Krylov method, we propose general purpose preconditioners which, exploiting the regularization and a new rearrangement of the Schur complement, remain attractive for a series of subsequent IPM iterations. Therefore they need to be recomputed only in a fraction of the total IPM iterations. The resulting regularized second order methods, for which low-frequency-updates of the preconditioners are allowed, pave the path for an alternative third way in-between first and second order methods.