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Inexact alternating projections on nonconvex sets

Dmitriy Drusvyatskiy(ddrusv***at***uw.edu)
Adrian S. Lewis(adrian.lewis***at***cornell.edu)

Abstract: Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex sets is typically intractable, but we show that computationally-cheap inexact projections may suffice instead. In particular, if one set is defined by sufficiently regular smooth constraints, then projecting onto the approximation obtained by linearizing those constraints around the current iterate suffices. On the other hand, if one set is a smooth manifold represented through local coordinates, then the approximate projection resulting from linearizing the coordinate system around the preceding iterate on the manifold also suffices.

Keywords: Alternating projections, Linear convergence, Variational analysis, Transversality, Approximate projection

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


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Entry Submitted: 11/03/2018
Entry Accepted: 11/04/2018
Entry Last Modified: 11/03/2018

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