Subgradient Ellipsoid Method for Nonsmooth Convex Problems

TL;DR

This paper introduces a novel ellipsoid-type algorithm that combines subgradient and ellipsoid methods, providing efficient convergence for large-scale nonsmooth convex problems such as minimization, saddle-point, and variational inequality problems.

Contribution

The paper proposes a new subgradient ellipsoid algorithm with improved convergence rates and an efficient technique for generating accuracy certificates in high-dimensional nonsmooth convex optimization.

Findings

Algorithm achieves reasonable convergence rates in high dimensions.

Efficient technique for accuracy certificates enhances algorithm performance.

Applicable to a range of nonsmooth convex problems.

Abstract

In this paper, we present a new ellipsoid-type algorithm for solving nonsmooth problems with convex structure. Examples of such problems include nonsmooth convex minimization problems, convex-concave saddle-point problems and variational inequalities with monotone operator. Our algorithm can be seen as a combination of the standard Subgradient and Ellipsoid methods. However, in contrast to the latter one, the proposed method has a reasonable convergence rate even when the dimensionality of the problem is sufficiently large. For generating accuracy certificates in our algorithm, we propose an efficient technique, which ameliorates the previously known recipes.