Bounded perturbation resilience of extragradient-type methods and their applications

TL;DR

This paper investigates the bounded perturbation resilience of extragradient methods for variational inequalities, proving convergence under errors, and demonstrating improved solutions through superiorization with numerical results.

Contribution

It establishes the bounded perturbation resilience of extragradient methods and their inertial variants, enabling inexact solutions and superiorization techniques.

Findings

Proved convergence of perturbed extragradient methods under standard assumptions.

Demonstrated a convergence rate of O(1/t) for the algorithms.

Numerical experiments validate the effectiveness of the methods.

Abstract

In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorizion can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converges at the rate of $O(1/t)$. Numerical illustrations are given to demonstrate the performances of the algorithms.