Convergence Analysis of Projection Method for Variational Inequalities

TL;DR

This paper introduces and analyzes an inertial projection algorithm for variational inequalities in Hilbert spaces, proving weak convergence and an $O(1/n)$ convergence rate, supported by experimental validation.

Contribution

It proposes a new inertial projection method for variational inequalities and provides a unified convergence analysis under mild assumptions.

Findings

Weak convergence of the algorithm is established.

An $O(1/n)$ convergence rate is proven.

Experimental results demonstrate the benefits of inertial extrapolation.

Abstract

The main contributions of this paper are the proposition and the convergence analysis of a class of inertial projection-type algorithm for solving variational inequality problems in real Hilbert spaces where the underline operator is monotone and uniformly continuous. We carry out a unified analysis of the proposed method under very mild assumptions. In particular, weak convergence of the generated sequence is established and nonasymptotic $O(1/n)$ rate of convergence is established, where $n$ denotes the iteration counter. We also present some experimental results to illustrate the profits gained by introducing the inertial extrapolation steps.