Lipschitz continuity of the metric projection operator and convergence of gradient methods

Abstract

Various support conditions for a closed subset of a real Hilbert space $\mathcal H$ at a boundary point of this set are considered. These conditions ensure certain local Lipschitz continuity of the metric projection operator as a function of a point. The local Lipschitz continuity of the metric projection as a function of the set in the Hausdorff metric is also proved. This Lipschitz property is used to verify the linear convergence of some gradient methods (the gradient projection method and the conditional gradient method) without assuming that the function must be strongly convex (or even convex) and for not necessarily convex sets. The function is assumed to be differentiable with Lipschitz continuous gradient. Bibliography: 29 titles.