Inexact proximal $\epsilon$-subgradient methods for composite convex optimization problems

TL;DR

This paper introduces two inexact proximal $ ext{ extepsilon}$-subgradient algorithms for convex optimization, analyzing their convergence and rate, including an accelerated version for smooth functions that achieves optimal convergence rates.

Contribution

It develops new inexact proximal subgradient methods with convergence analysis and proposes an accelerated variant for smooth functions, advancing optimization techniques for composite convex problems.

Findings

Convergence is guaranteed under various error criteria.

The accelerated method attains the optimal convergence rate for smooth functions.

Different stepsize rules are analyzed for practical implementation.

Abstract

We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). The algorithms involve, at each iteration, inexact evaluations of the proximal operator and approximate subgradients of the functions (namely: the $\epsilon$-subgradients). The methods use different error criteria for approximating the proximal operators. We provide an analysis of the convergence and rate of convergence properties of these methods, considering various stepsize rules, including both, diminishing and constant stepsizes. For the case where one of the functions is smooth, we propose an inexact accelerated version of the proximal gradient method, and prove that the optimal convergence rate for the function values can be achieved.