On finite termination of an inexact Proximal Point algorithm

Abstract

The presence of sharp minima in nondifferentiable optimization models has been exploited, in the last decades, in the benefit of various subgradient or proximal methods. One of the long-lasting general proximal schemes of choice used to minimize nonsmooth functions is the Proximal Point Algorithm (PPA). Regarding the basic PPA, several well-known works proved finite convergence towards weak sharp minima, when supposedly each iteration is computed exactly. However, in this letter we show finite convergence of a common Inexact version of PPA (IPPA), under sufficiently low but persistent perturbations of the proximal operator. Moreover, when a simple Subgradient Method is recurrently called as an inner routine for computing each IPPA iterate, a suboptimal minimizer of the original problem lying at ϵ distance from the optimal set is obtained after a total Olog(1/ϵ) subgradient evaluations. Our preliminary numerical tests show improvements over existing restartation versions of Subgradient Method.