A general inertial projected gradient method for variational inequality problems

Abstract

The purpose of this article is to introduce a general inertial projected gradient method with a self-adaptive stepsize for solving variational inequality problems. The proposed method incorporates two different extrapolations with respect to the previous iterates into the projected gradient method. The weak convergence for our method is proved under standard assumptions without any requirement of the knowledge of the Lipschitz constant of the mapping. Furthermore, R-linear convergence rate is established under the strong monotonicity assumption of the mapping. Finally preliminary results from numerical experiments and applications to optimal control problems are performed which show the advantage of the proposed method.

An Inertial Splitting Method for Monotone Inclusions of Three Operators

“…Notice that our proof techniques of weak convergence are more self-contained and less convoluted via introducing the characteristic operator, and the idea can be found in Eckstein (2017) and in an early version, accepted by Optimization Online in December 2018, of a very recent article Dong (2021).…”

“…where ε is any given sufficiently small positive number and t(θ k , θ k+1 , ε) is defined in (5) below.…”