Journal of Convex Analysis 19 (2012), No. 4, 1167--1192
Copyright Heldermann Verlag 2012
Inexact and Accelerated Proximal Point Algorithms
Saverio Salzo
DIBRIS, Università di Genova, Via Opera Pia 13, 16145 Genova, Italy
saverio.salzo@unige.it
Silvia Villa
Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy
silvia.villa@ut.it
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/k2 can be recovered by constraining the errors to be of a certain type.
Keywords: Accelerated proximal point algorithms, global convergence rates, approximation criteria.
MSC: 90C25, 49D37, 65K10
[ Fulltext-pdf (203 KB)] for subscribers only.