Local convergence for alternating and averaged nonconvex projections

Adrian Lewis(aslewisorie.cornell.edu)
Russell Luke(rlukemath.udel.edu)
Jerome Malick(jerome.malickinria.fr)

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.

Keywords: alternating projections, averaged projections, linear convergence, metric regularity, distance to ill-posedness, variational analysis, nonconvex, extremal principle, prox-regularity

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 3: Applications -- Science and Engineering (Control Applications )

Citation: School of ORIE, Cornell University, 1 September 2007

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Entry Submitted: 09/01/2007
Entry Accepted: 09/01/2007
Entry Last Modified: 09/01/2007

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