Extragradient Method in Optimization: Convergence and Complexity

TL;DR

This paper analyzes the convergence and complexity of the extragradient method for optimization problems involving smooth and convex functions, providing theoretical guarantees and practical implementation insights.

Contribution

It extends convergence analysis of the extragradient method under Kurdyka-Lojasiewicz assumptions and introduces an exact line search scheme for proximal decomposition methods.

Findings

Convergence to critical points with finite length under certain conditions

Sublinear convergence rate for convex functions

Numerical results show competitiveness of the method with line search

Abstract

We consider the extragradient method to minimize the sum of two functions, the first one being smooth and the second being convex. Under the Kurdyka-Lojasiewicz assumption, we prove that the sequence produced by the extragradient method converges to a critical point of the problem and has finite length. The analysis is extended to the case when both functions are convex. We provide, in this case, a sublinear convergence rate, as for gradient-based methods. Furthermore, we show that the recent small-prox complexity result can be applied to this method. Considering the extragradient method is an occasion to describe an exact line search scheme for proximal decomposition methods. We provide details for the implementation of this scheme for the one norm regularized least squares problem and demonstrate numerical results which suggest that combining nonaccelerated methods with exact line search can be a competitive choice.