Subgradient method with feasible inexact projections for constrained convex optimization problems

TL;DR

This paper introduces a new inexact projected subgradient method for solving nondifferentiable constrained convex optimization problems, providing convergence analysis and iteration complexity bounds for various stepsize strategies.

Contribution

It develops an inexact projection approach combined with epsilon-subgradients, extending the subgradient method to handle constraints more efficiently.

Findings

Convergence established for the proposed method.

Iteration complexity bounds derived for different stepsize rules.

Method effectively handles nondifferentiable convex problems with constraints.

Abstract

In this paper, we propose a new inexact version of the projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine $\epsilon$-subgradient method with a procedure to obtain a feasible inexact projection onto the constraint set. Asymptotic convergence results and iteration-complexity bounds for the sequence generated by the method employing the well known exogenous stepsizes, Polyak's stepsizes, and dynamic stepsizes are established.