Nonsmooth Convex Optimization using the Specular Gradient Method with Root-Linear Convergence

TL;DR

This paper introduces the specular gradient method for one-dimensional convex functions, achieving root-linear convergence without extra assumptions, and proposes an implementation that avoids explicit derivative calculations.

Contribution

The paper presents a novel specular gradient method that guarantees root-linear convergence for convex functions and offers a practical implementation approach.

Findings

Achieves root-linear convergence for convex functions.

Provides a derivative-free implementation method.

Extends understanding of subgradient methods in convex optimization.

Abstract

In this paper, we find the special case of the subgradient method minimizing a one-dimensional real-valued function, which we term the specular gradient method, that converges root-linearly without any additional assumptions except the convexity. Furthermore, we suggest a way to implement the specular gradient method without explicitly calculating specular derivatives.