# Radial Subgradient Method

## TL;DR

This paper introduces a radial subgradient method for convex optimization that improves convergence rates and simplifies analysis by avoiding orthogonal projections, applicable to non-smooth, non-Lipschitz problems with known feasible points.

## Contribution

It proposes a novel radial subgradient algorithm that extends Renegar's work, offering a more natural approach with better convergence and simpler analysis for challenging convex problems.

## Key findings

- Achieves improved convergence rates over traditional subgradient methods.
- Eliminates the need for costly orthogonal projections.
- Applicable to non-smooth, non-Lipschitz convex problems with known feasible points.

## Abstract

We present a subgradient method for minimizing non-smooth, non-Lipschitz convex optimization problems. The only structure assumed is that a strictly feasible point is known. We extend the work of Renegar [5] by taking a different perspective, leading to an algorithm which is conceptually more natural, has notably improved convergence rates, and for which the analysis is surprisingly simple. At each iteration, the algorithm takes a subgradient step and then performs a line search to move radially towards (or away from) the known feasible point. Our convergence results have striking similarities to those of traditional methods that require Lipschitz continuity. Costly orthogonal projections typical of subgradient methods are entirely avoided.