Inexact and accelerated proximal point algorithms

Abstract

We present inexact accelerated proximal point algorithms for minimizing a proper lower semicontinuous and convex function. We carry on a convergence analysis under different types of errors in the evaluation of the proximity operator, and we provide corresponding convergence rates for the objective function values. The proof relies on a generalization of the strategy proposed in [14] for generating estimate sequences according to the definition of Nesterov, and is based on the concept of ε-subdifferential. We show that the convergence rate of the exact accelerated algorithm 1/k2 can be recovered by constraining the errors to be of a certain type.