Primal-dual subgradient method for constrained convex optimization problems

TL;DR

This paper analyzes a primal-dual subgradient method for convex optimization problems with constraints, demonstrating its optimality even when functions are nondifferentiable and not Lipschitz continuous.

Contribution

It establishes the optimality of the weighted dual averages method for a broad class of convex constrained problems without requiring differentiability or Lipschitz continuity.

Findings

Proves the optimality of the weighted dual averages method.

Applicable to nondifferentiable and non-Lipschitz convex functions.

Provides a simple first-order solution for constrained convex problems.

Abstract

This paper considers a general convex constrained problem setting where functions are not assumed to be differentiable nor Lipschitz continuous. Our motivation is in finding a simple first-order method for solving a wide range of convex optimization problems with minimal requirements. We study the method of weighted dual averages (Nesterov, 2009) in this setting and prove that it is an optimal method.