A Perturbed Projection Algorithm with Inertial Technique for Split Feasibility Problem

Yazheng Dang,Yanli Han,Yan Gao

doi:10.1155/2012/207323

Abstract

This paper deals with the split feasibility problem that requires to find a point closest to a closed convex set in one space such that its image under a linear transformation will be closest to another closed convex set in the image space. By combining perturbed strategy with inertial technique, we construct an inertial perturbed projection algorithm for solving the split feasibility problem. Under some suitable conditions, we show the asymptotic convergence. The results improve and extend the algorithms presented in Byrne (2002) and in Zhao and Yang (2005) and the related convergence theorem.

Highlights

Let C ⊂ Rn and Q ⊂ Rm be nonempty closed convex sets, and let A be an m × n real matrix
The CQ algorithm proposed by Byrne in 6 has the following iterative process: xk 1 PC xk γ AT PQ − I Axk, k ≥ 0, 1.3 where γ ∈ 0, 2/L, L denotes the largest eigenvalue of the matrix AT A, and I is the identity operator
It is difficult or even impossible to compute orthogonal projection; to avoid computing projection, Zhao and Yang in 7 proposed the perturbed projections algorithm for the SFP. This development was based on results of Santos and Scheimberg 8 who suggested replacing each nonempty closed convex set of the convex feasibility problem by a convergent sequence of supersets

Summary

Introduction

Let C ⊂ Rn and Q ⊂ Rm be nonempty closed convex sets, and let A be an m × n real matrix. It is difficult or even impossible to compute orthogonal projection; to avoid computing projection, Zhao and Yang in 7 proposed the perturbed projections algorithm for the SFP This development was based on results of Santos and Scheimberg 8 who suggested replacing each nonempty closed convex set of the convex feasibility problem by a convergent sequence of supersets. If such supersets can be constructed with reasonable efforts and projecting onto them is simpler than projecting onto the original convex sets, a perturbed algorithm is favorable.

Preliminaries

The Inertial Perturbed Algorithm and the Asymptotic Convergence for the SPF