Local convergence for alternating and averaged nonconvex projections
Provides theoretical guarantees for projection algorithms in nonconvex settings, benefiting optimization and variational analysis.
The paper shows that strongly regular intersection of closed sets implies local linear convergence of alternating and averaged projection methods, with rates tied to a regularity modulus. Inexact versions also converge linearly.
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.