Local convergence for alternating and averaged nonconvex projections

TL;DR

This paper proves local linear convergence of alternating and averaged projection methods for nonconvex sets with regular intersections, extending classical results to broader nonconvex scenarios.

Contribution

It establishes local linear convergence of alternating and averaged projections for nonconvex sets with regular intersections, including inexact variants.

Findings

Alternating projections converge linearly under regularity conditions.

Averaged projections also converge linearly for multiple sets with regular intersection.

Inexact algorithms maintain linear convergence under similar conditions.

Abstract

The idea of a finite collection of closed sets having "strongly regular intersection" at a given point is crucial in variational analysis. We show that this central theoretical tool also has striking algorithmic consequences. Specifically, we consider the case of two sets, one of which we assume to be suitably "regular" (special cases being convex sets, smooth manifolds, or feasible regions satisfying the Mangasarian-Fromovitz constraint qualification). We then prove that von Neumann's method of "alternating projections" converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having strongly regular intersection at some point, the method of "averaged projections" converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly.