A Hybrid Forward–Backward Algorithm and Its Optimization Application

Liya Liu,Jen-Chih Yao,Xiaolong Qin

doi:10.3390/math8030447

Abstract

In this paper, we study a hybrid forward–backward algorithm for sparse reconstruction. Our algorithm involves descent, splitting and inertial ideas. Under suitable conditions on the algorithm parameters, we establish a strong convergence solution theorem in the framework of Hilbert spaces. Numerical experiments are also provided to illustrate the application in the field of signal processing.

Highlights

Let H be a real Hilbert space. h·, ·i denotes the associated scalar product and k · k stands for the induced norm
For any initial data p0 ∈ H, pn+1 = ( Id + γn A)−1, ∀n ∈ N, where Id stands for the identity operator and γn > 0
Where Id stands for the identity operator, G is a maximally monotone operator, the parameters

Summary

Introduction

Let H be a real Hilbert space. h·, ·i denotes the associated scalar product and k · k stands for the induced norm. In [16], Alvarez and Attouch employed the first-order accelerated method to study an initial proximal point algorithm for solving the problem of finding zero of a maximally monotone operator. This iteration can be written as the following form: for any initial data p0 , p1 ∈ H, pn+1 = ( Id + λn G )−1 ( pn + αn ( pn − pn−1 )),. From this perspective, our study is the natural extension of the convergence results obtained by Attouch and Mainǵe [26] in the case of continuous dynamical systems

Preliminaries

Main Results

Numerical Experiment

Conclusions