Fast Primal-Dual Gradient Method for Strongly Convex Minimization Problems with Linear Constraints

TL;DR

This paper introduces a primal-dual fast gradient method for strongly convex optimization problems with linear constraints, enabling efficient solutions and near-optimal primal solutions with proven convergence rates.

Contribution

It extends the fast gradient method to a primal-dual setting for constrained problems, providing convergence guarantees for both objective and feasibility.

Findings

Achieves near-optimal solutions efficiently.

Provides convergence rate theorems for primal and dual variables.

Applicable to various problems like entropy-linear programming and optimal transport.

Abstract

In this paper we consider a class of optimization problems with a strongly convex objective function and the feasible set given by an intersection of a simple convex set with a set given by a number of linear equality and inequality constraints. A number of optimization problems in applications can be stated in this form, examples being the entropy-linear programming, the ridge regression, the elastic net, the regularized optimal transport, etc. We extend the Fast Gradient Method applied to the dual problem in order to make it primal-dual so that it allows not only to solve the dual problem, but also to construct nearly optimal and nearly feasible solution of the primal problem. We also prove a theorem about the convergence rate for the proposed algorithm in terms of the objective function and the linear constraints infeasibility.