Proximal Projection Method for Stable Linearly Constrained Optimization

TL;DR

The paper introduces a proximal projection algorithm for efficiently solving large-scale linearly constrained convex optimization problems, with guaranteed convergence and superior empirical performance.

Contribution

It proposes a novel proximal projection method based on Douglas-Rachford splitting that ensures feasibility at each iteration and demonstrates improved results over existing methods.

Findings

PP algorithm guarantees convergence to optimal solutions.

PP outperforms or matches existing methods in various applications.

Each PP iterate is feasible, unlike many other approaches.

Abstract

Many applications using large datasets require efficient methods for minimizing a proximable convex function subject to satisfying a set of linear constraints within a specified tolerance. For this task, we present a proximal projection (PP) algorithm, which is an instance of Douglas-Rachford splitting that directly uses projections onto the set of constraints. Formal guarantees are presented to prove convergence of PP estimates to optimizers. Unlike many methods that obtain feasibility asymptotically, each PP iterate is feasible. Numerically, we show PP either matches or outperforms alternatives (e.g. linearized Bregman, primal dual hybrid gradient, proximal augmented Lagrangian, proximal gradient) on problems in basis pursuit, stable matrix completion, stable principal component pursuit, and the computation of earth mover's distances.