Randomized Nystr\"om Preconditioned Interior Point-Proximal Method of Multipliers

TL;DR

This paper introduces Nys-IP-PMM, a novel matrix-free convex quadratic programming solver that combines interior-point methods with randomized Nystr"om preconditioning to efficiently handle dense constraint matrices.

Contribution

The paper presents a new regularized interior-point solver that leverages low-rank structure and randomized preconditioning, improving solution speed for separable QP problems.

Findings

Demonstrates superior wallclock time performance over previous methods.

Establishes convergence of the proposed Nys-IP-PMM algorithm.

Effective for dense constraint matrices in separable QP instances.

Abstract

We present a new algorithm for convex separable quadratic programming (QP) called Nys-IP-PMM, a regularized interior-point solver that uses low-rank structure to accelerate solution of the Newton system. The algorithm combines the interior point proximal method of multipliers (IP-PMM) with the randomized Nystr\"om preconditioned conjugate gradient method as the inner linear system solver. Our algorithm is matrix-free: it accesses the input matrices solely through matrix-vector products, as opposed to methods involving matrix factorization. It works particularly well for separable QP instances with dense constraint matrices. We establish convergence of Nys-IP-PMM. Numerical experiments demonstrate its superior performance in terms of wallclock time compared to previous matrix-free IPM-based approaches.