  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.
