Asymptotic Convergence Analysis of a New Class of Proximal Point Methods

Abstract

Finite dimensional local convergence results for self-adaptive proximal point methods and nonlinear functions with multiple minimizers are generalized and extended to a Hilbert space setting. The principle assumption is a local error bound condition which relates the growth in the function to the distance to the set of minimizers. A local convergence result is established for almost exact iterates. Less restrictive acceptance criteria for the proximal iterates are also analyzed. These criteria are expressed in terms of a subdifferential of the proximal function and either a subdifferential of the original function or an iteration difference. If the proximal regularization parameter $\mu({\bf x})$ is sufficiently small and bounded away from zero and f is sufficiently smooth, then there is local linear convergence to the set of minimizers. For a locally convex function, a convergence result similar to that for almost exact iterates is established. For a locally convex solution set and smooth functions, it is shown that if the proximal regularization parameter has the form $\mu({\bf x})=\beta\|f'[{\bf x}]\|^{\eta}$, where $\eta\in(0,2)$, then the convergence is at least superlinear if $\eta\in(0,1)$ and at least quadratic if $\eta\in[1,2)$.