Abstract
Given two arbitrary closed sets in Euclidean space, to guarantee that the method of alternating projections converges locally at linear rate to a point in the intersection, a simple transversality assumption suffices. 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.
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Funding
D. Drusvyatskiy: Research supported in part by AFOSR YIP award FA9550-15-1-0237 and by National Science Foundation Grants DMS 1651851 and CCF 1740551.
A.S. Lewis: Research supported in part by National Science Foundation Grant DMS-1613996.
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Dedicated to our friend, colleague, and inspiration, Alex Ioffe, on the occasion of his 80th birthday.
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Drusvyatskiy, D., Lewis, A.S. Local Linear Convergence for Inexact Alternating Projections on Nonconvex Sets. Vietnam J. Math. 47, 669–681 (2019). https://doi.org/10.1007/s10013-019-00357-3
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DOI: https://doi.org/10.1007/s10013-019-00357-3