Strong convergence of iterative algorithms for the split equality problem

Luo Yi Shi,Rudong Chen,Yujing Wu

doi:10.1186/1029-242x-2014-478

Luo Yi Shi, Rudong Chen + Show 1 more

Open Access

https://doi.org/10.1186/1029-242x-2014-478

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Abstract

Abstract Let H 1 , H 2 , H 3 be real Hilbert spaces, C ⊆ H 1 , Q ⊆ H 2 be two nonempty closed convex sets, and let A : H 1 → H 3 , B : H 2 → H 3 be two bounded linear operators. The split equality problem (SEP) is finding x ∈ C , y ∈ Q such that A x = B y . Recently, Moudafi has presented the ACQA algorithm and the RACQA algorithm to solve SEP. However, the two algorithms are weakly convergent. It is therefore the aim of this paper to construct new algorithms for SEP so that strong convergence is guaranteed. Firstly, we define the concept of the minimal norm solution of SEP. Using Tychonov regularization, we introduce two methods to get such a minimal norm solution. And then, we introduce two algorithms which are viewed as modifications of Moudafi’s ACQA, RACQA algorithms and KM-CQ algorithm, respectively, and converge strongly to a solution of SEP. More importantly, the modifications of Moudafi’s ACQA, RACQA algorithms converge strongly to the minimal norm solution of SEP. At last, we introduce some other algorithms which converge strongly to a solution of SEP.

Highlights

Introduction and preliminaries Let C andQ be nonempty closed convex subsets of real Hilbert spaces H and H, respectively, and let A : H → H be a bounded linear operator
In Section, we introduce an algorithm which is viewed as a modification of Moudafi’s Alternating CQ-algorithm (ACQA) and Relaxed alternating CQ-algorithm (RACQA) algorithms; and we prove the strong convergence of the algorithm, more importantly, its limit is the minimum-norm solution of split equality problem (SEP) ( . )
In Section, we introduce some other iterative algorithms which converge strongly to a solution of SEP ( . )

Summary

Gwβ wβ β

It follows that wα – wβ , αwα – βwβ ≤ wα – wβ , G∗G(wβ – wα) ≤. ). wα converges strongly as α → to the minimum-norm solution wof SEP Since {wn} converges weakly to wand wn ≤ w , we can get that w ≤ lim inf n wn This shows that wis a point in which assumes a minimum norm. Due to the uniqueness of a minimum-norm element, we obtain w = w. We introduce another method to get the minimum-norm solution of SEP. Since Fix(PS) ∩ Fix(T) = = ∅, and both PS and T are averaged, from Lemma .

For α

Since α

This implies that

Proof Let Rn and R be defined by

Notice that