Primal-dual extrapolation methods for monotone inclusions under local Lipschitz continuity

TL;DR

This paper introduces primal-dual extrapolation methods with backtracking line search for solving monotone inclusion problems with locally Lipschitz operators, achieving significantly improved complexity bounds over previous approaches.

Contribution

The paper develops novel primal-dual extrapolation algorithms with optimal complexity for monotone inclusions involving locally Lipschitz operators, extending to related convex optimization problems.

Findings

Operation complexity of ${ m O}(\log rac{1}{\epsilon})$ for strongly monotone cases.

Operation complexity of ${ m O}(rac{1}{\epsilon}\log rac{1}{\epsilon})$ for non-strongly cases.

Numerical results demonstrate the efficiency of the proposed methods.

Abstract

In this paper we consider a class of monotone inclusion (MI) problems of finding a zero of the sum of two monotone operators, in which one operator is maximal monotone while the other is {\it locally Lipschitz} continuous. We propose primal-dual extrapolation methods to solve them using a point and operator extrapolation technique, whose parameters are chosen by a backtracking line search scheme. The proposed methods enjoy an operation complexity of ${\cal O}(\log \epsilon^{-1})$ and ${\cal O}(\epsilon^{-1}\log \epsilon^{-1})$, measured by the number of fundamental operations consisting only of evaluations of one operator and resolvent of the other operator, for finding an $\varepsilon$-residual solution of strongly and non-strongly MI problems, respectively. The latter complexity significantly improves the previously best operation complexity ${\cal O}(\varepsilon^{-2})$. As a byproduct, complexity results of the primal-dual extrapolation methods are also obtained for finding an $\varepsilon$-KKT or $\varepsilon$-residual solution of convex conic optimization, conic constrained saddle point, and variational inequality problems under {\it local Lipschitz} continuity. We provide preliminary numerical results to demonstrate the performance of the proposed methods.