Strong convergence of an inertial extrapolation method for a split system of minimization problems

Rabian Wangkeeree,Anteneh Getachew Gebrie

doi:10.1515/dema-2020-0025

Rabian Wangkeeree, Anteneh Getachew Gebrie

Open Access

https://doi.org/10.1515/dema-2020-0025

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Abstract

Abstract In this article, we propose an inertial extrapolation-type algorithm for solving split system of minimization problems: finding a common minimizer point of a finite family of proper, lower semicontinuous convex functions and whose image under a linear transformation is also common minimizer point of another finite family of proper, lower semicontinuous convex functions. The strong convergence theorem is given in such a way that the step sizes of our algorithm are selected without the need for any prior information about the operator norm. The results obtained in this article improve and extend many recent ones in the literature. Finally, we give one numerical example to demonstrate the efficiency and implementation of our proposed algorithm.

Highlights

Throughout this article, unless otherwise stated, we assume that H1, H2 and H are real Hilbert spaces, A : H1 → H2 is nonzero bounded linear operator and I denotes the identity operator on a Hilbert space.Assume Ci (i = 1, ..., N ) and Qi (i = 1, ..., M) are nonempty closed convex subsets of H1 and H2, respectively
If N = M = 1, the multiple-set split feasibility problem (MSSFP) (1) is reduced to the problem known as the split feasibility problem (SFP) which was first introduced by Censor and Elfving [2] for modeling inverse problems in finitedimensional Hilbert spaces
Motivated by the above theoretical views, and inspired by results in [1,21,25], in this article we introduce the strong convergence theorem of an inertial extrapolation-type algorithm that incorporates a proximal operator, a viscosity method and an inertial term to solve the so-called split system of minimization problem (SSMP), given as a task finding a point x ∈ H1 with the property x ∈ ⋂ such that Ax ∈ ⋂, (7)

Summary

Introduction

Throughout this article, unless otherwise stated, we assume that H1, H2 and H are real Hilbert spaces, A : H1 → H2 is nonzero bounded linear operator and I denotes the identity operator on a Hilbert space. Our goal is to introduce a strong convergence iterative algorithm with inertial effect solving the MSSFP (1), where Ci and Qj are solution sets of minimization problems of the form (2) for proper, lower semicontinuous convex functions fi and gj, respectively. Motivated by the above theoretical views, and inspired by results in [1,21,25], in this article we introduce the strong convergence theorem of an inertial extrapolation-type algorithm that incorporates a proximal operator, a viscosity method and an inertial term to solve the so-called split system of minimization problem (SSMP), given as a task finding a point x ∈ H1 with the property x ∈ ⋂ (arg min fi) such that Ax ∈ ⋂ (arg min gj),.

Preliminary

Main result

Numerical results

Conclusions