Circumcentering approximate reflections for solving the convex feasibility problem

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.

References in corpus (9)

• Transversality and alternating projections for nonconvex sets
• Circumcentering the Douglas--Rachford method
• On the linear convergence of the circumcentered-reflection method
• On the Circumcentered-Reflection Method for the Convex Feasibility Problem
• About intrinsic transversality of pairs of sets
• On circumcenters of finite sets in Hilbert spaces
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• On circumcenter mappings induced by nonexpansive operators
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Cited by in corpus (3)

• On the centralization of the circumcentered-reflection method
• A successive centralized circumcenter reflection method for the convex feasibility problem
• Circumcentric directions of cones