Retractions by Alternating Projections

Alternating projections and their variants are classical tools for computing points in intersections of sets. Existing analyses for smooth manifolds mainly focus on local convergence rates under transversality or related regularity conditions. In this work, we develop a unified framework for a broad class of (possibly inexact) alternating-projection-type methods on intersections of smooth manifolds. Specifically, under the assumption that two C2,1 embedded submanifolds M1, M2 subset ℝn intersect cleanly, we show that the associated alternating mapping admits a well-defined local limiting map psi on the intersection manifold M=M1cap M2, and that psi is a retraction on M. If, in addition, M1 and M2 are C3,1, then psi is a second-order retraction. Furthermore, the standard NewtonSLRA scheme, which exhibits quadratic local behavior under transversality, can be understood as inducing a second-order retraction on (M). This framework thus provides new retraction-based optimization tools for problems constrained to the intersection manifold.