Local Linear Convergence for Alternating and Averaged Nonconvex Projections

Authors:
A. S. Lewis;D. R. Luke;J. Malick
Affiliations:
Cornell University, ORIE, 14853, Ithaca, NY, USA;University of Delaware, Department of Mathematical Sciences, 19716, Newark, DE, USA;INRIA Rhone-Alpes, 655 avenue de l’Europe, 38334, St Ismier Cedex, France
Venue:
Foundations of Computational Mathematics
Year:
2009

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Abstract

The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness, for example), we 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 linearly 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.