Inexact and Accelerated Proximal Point Algorithms

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 by O. Güler ["New proximal point algorithms for convex minimization", SIAM J. Optimization 2(4) (1992) 649–664] 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 $1/k^2$ can be recovered by constraining the errors to be of a certain type.