# A subgradient method with constant step-size for $\ell_1$-composite optimization

## TL;DR

This paper introduces a subgradient method with a constant step-size for -regularized convex optimization, achieving linear convergence in strongly convex cases and demonstrating effectiveness through numerical tests.

## Contribution

It proposes a novel subgradient method with constant step-size for -regularized problems and an accelerated version with proven linear convergence.

## Key findings

- Linear convergence for strongly convex smooth terms
- Effective performance on both strongly and non-strongly convex examples
- Accelerated algorithm with adaptive restart strategy

## Abstract

Subgradient methods are the natural extension to the non-smooth case of the classical gradient descent for regular convex optimization problems. However, in general, they are characterized by slow convergence rates, and they require decreasing step-sizes to converge. In this paper we propose a subgradient method with constant step-size for composite convex objectives with $\ell_1$-regularization. If the smooth term is strongly convex, we can establish a linear convergence result for the function values. This fact relies on an accurate choice of the element of the subdifferential used for the update, and on proper actions adopted when non-differentiability regions are crossed. Then, we propose an accelerated version of the algorithm, based on conservative inertial dynamics and on an adaptive restart strategy, that is guaranteed to achieve a linear convergence rate in the strongly convex case. Finally, we test the performances of our algorithms on some strongly and non-strongly convex examples.