Monotonicity of Pairs of Operators and Generalized Inertial Proximal Method

TL;DR

This paper introduces a new algorithm, GIPPA, for solving non-monotone inclusions by leveraging the concept of monotonicity of pairs of operators, with proven convergence properties and practical numerical demonstrations.

Contribution

It develops a generalized inertial proximal point algorithm using warped resolvents to handle monotone pairs, addressing design challenges in this area.

Findings

Proves weak, strong, and linear convergence of GIPPA.

Provides numerical examples demonstrating effectiveness.

Addresses the design of mappings for monotone pairs.

Abstract

Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial Proximal Point Algorithm (GIPPA) using warped resolvents under the monotonicity of pairs. The weak, strong and linear convergence of the algorithm under some mild assumptions are established. We also provide numerical examples illustrating the implementability and effectiveness of the proposed method.