Convergence of projection and contraction algorithms with outer perturbations and their applications to sparse signals recovery

TL;DR

This paper investigates the robustness of projection and contraction algorithms under perturbations for solving variational inequalities, demonstrating their convergence and rate, and proposing inertial variants.

Contribution

It establishes the bounded perturbation resilience of these algorithms and introduces inertial versions with proven convergence and convergence rate.

Findings

Algorithms are resilient to bounded perturbations.

Convergence rate of the perturbed algorithms is O(1/t).

Inertial algorithms are proposed based on resilience results.

Abstract

In this paper we study the bounded perturbation resilience of projection and contraction algorithms for solving variational inequality (VI) problems in real Hilbert spaces. Under typical and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed projection and contraction algorithms is proved. Based on the bounded perturbed resilience of projection and contraction algorithms, we present some inertial projection and contraction algorithms. In addition we show that the perturbed algorithms converges at the rate of $O(1/t)$.