A parallel Tseng\u2019s splitting method for solving common variational inclusion applied to signal recovery problems

Raweerote Suparatulatorn,Aviv Gibali,Thanasak Mouktonglang,Watcharaporn Cholamjiak

doi:10.1186/s13662-021-03647-8

Abstract

In this work we propose an accelerated algorithm that combines various techniques, such as inertial proximal algorithms, Tseng’s splitting algorithm, and more, for solving the common variational inclusion problem in real Hilbert spaces. We establish a strong convergence theorem of the algorithm under standard and suitable assumptions and illustrate the applicability and advantages of the new scheme for signal recovering problem arising in compressed sensing.

Highlights

Let H be a real Hilbert space such that ·, · and · are the inner product and the induced norm, respectively
We are interested in the variational inclusion problem (VIP) which is to find u ∈ H such that 0 ∈ (F + G)u, (1.1)
Inspired by the above works, we focus on the common variational inclusion problem and present a new modified Tseng’s splitting algorithm for solving it with strong converges in real Hilbert spaces

Summary

Introduction

Let H be a real Hilbert space such that ·, · and · are the inner product and the induced norm, respectively. For obtaining the strong convergence, Cholamjiak et al [4] introduced Halpern-type forward-backward splitting algorithm (HTFBSA) involving the inertial technique in a Hilbert space. Yambangwai et al [27] extended the HTFBSA to the following modified viscosity inertial forward-backward splitting algorithm (MVIFBSA): rk = uk + ξk(uk – uk–1), uk+1 = akφ(rk) + (1 – ak – bk)rk + bkJγGk rk – γkF(rk) , k ≥ 1, (1.5)

Results

Conclusion