Circumcentering approximate reflections for solving the convex feasibility problem

TL;DR

This paper introduces CARM, an efficient variation of CRM that uses approximate projections to solve convex feasibility problems with proven convergence and competitive performance.

Contribution

The paper proposes CARM, a novel method using outer-approximate projections for convex feasibility, with proven convergence and linear rates under certain conditions.

Findings

CARM converges to a solution under general conditions.

CARM achieves linear convergence with an error bound.

Numerical results show CARM's efficiency compared to CRM and MAP.

Abstract

The circumcentered-reflection method (CRM) has been applied for solving convex feasibility problems. CRM iterates by computing a circumcenter upon a composition of reflections with respect to convex sets. Since reflections are based on exact projections, their computation might be costly. In this regard, we introduce the circumcentered approximate-reflection method (CARM), whose reflections rely on outer-approximate projections. The appeal of CARM is that, in rather general situations, the approximate projections we employ are available under low computational cost. We derive convergence of CARM and linear convergence under an error bound condition. We also present successful theoretical and numerical comparisons of CARM to the original CRM, to the classical method of alternating projections (MAP) and to a correspondent outer-approximate version of MAP, referred to as MAAP. Along with our results and numerical experiments, we present a couple of illustrative examples.