Relaxed inertial fixed point method for infinite family of averaged quasi-nonexpansive mapping with applications to sparse signal recovery

Yekini Shehu,Ziyue Hu,Qiao-Li Dong,Jen-Chih Yao

doi:10.1007/s00500-021-06416-7

Abstract

It is known that several optimization problems can be converted to a fixed point problem for which the underline fixed point operator is an averaged quasi-nonexpansive mapping and thus the corresponding fixed point method utilizes to solve the considered optimization problem. In this paper, we consider a fixed point method involving inertial extrapolation step with relaxation parameter to obtain a common fixed point of a countable family of averaged quasi-nonexpansive mappings in real Hilbert spaces. Our results bring a unification of several versions of fixed point methods for averaged quasi-nonexpansive mappings considered in the literature and give several implications of our results. We also give some applications to monotone inclusion problem with three-operator splitting method and composite convex and non-convex relaxed inertial proximal methods to solve both convex and nonconvex reweighted \(l_Q\) regularization for recovering a sparse signal. Finally, some numerical experiments are drawn from sparse signal recovery to illustrate our theoretical results.

Highlights

Let us consider the real Hilbert space setting: H represents a real Hilbert space with scalar product h., .i and induced norm k · k
Several fixed point iterations have been discussed in the literature cf. [8, 12, 13] and references therein for some relevant results in this direction
It is known that the standard proximal point algorithm (PPA)

Summary

Introduction

Let us consider the real Hilbert space setting: H represents a real Hilbert space with scalar product h., .i and induced norm k · k. Prominent examples for averaged quasi-nonexpansive mappings in Hilbert spaces are the projection mapping, the proximal point mapping, the resolvent operator, and several composite maps which involve at least one of these two mappings, see, e.g., [6] for more details. In [24], Maingé studied the following method to find common fixed points of infinitely many averaged quasi-nonexpansive mappings {Tn }: x0 , x1 ∈ H, xn+1 = Tn (xn + θn (xn − xn−1 )),. To establish weak convergence result of an over-relaxed inertial iterative procedure for countable families of average quasi-nonexpansive mappings in real. We show that the projection and contraction operator is β-averaged quasi-nonexpansive mappings on H. Let A : H → H be a monotone and L-Lipschitz operator on a nonempty closed and convex subset C and λ be a positive number.

Proposed Method

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