Convergence of Two Simple Methods for Solving Monotone Inclusion Problems in Reflexive Banach Spaces

TL;DR

This paper introduces two simple, efficient methods with convergence guarantees for solving monotone inclusion problems in reflexive Banach spaces, with applications to generalized Nash equilibrium problems in gas markets.

Contribution

It presents two novel methods with constant and adaptive step sizes for monotone inclusions, including convergence analysis and application to real-world equilibrium problems.

Findings

Methods require only one operator evaluation per iteration.

Weak convergence under maximal monotonicity and Lipschitz continuity.

Strong convergence when operators are strongly monotone.

Abstract

We propose two very simple methods, the first one with constant step sizes and the second one with self-adaptive step sizes, for finding a zero of the sum of two monotone operators in real reflexive Banach spaces. Our methods require only one evaluation of the single-valued operator at each iteration. Weak convergence results are obtained when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous, and strong convergence results are obtained when either one of these two operators is required, in addition, to be strongly monotone. We also obtain the rate of convergence of our proposed methods in real reflexive Banach spaces. Finally, we apply our results to solving generalized Nash equilibrium problems for gas markets.