## Abdellatif Moudafi, Byrne’s extended CQ-algorithms in the light of Moreau-Yosida regularization

Full Text: PDF
DOI: 10.23952/asvao.3.2021.1.03
Volume 3, Issue 1, 30 April 2021, Pages 21-26

Abstract. Given a real $m\times n$ matrix $A$, a maximal monotone operator $S: I\!\!R^n\rightarrow 2^{I\!\!R^n}$ and a maximal monotone operator $T: I\!\!R^m\rightarrow 2^{I\!\!R^m},$ the split feasibility null-point problem is to find $\bar x$ satisfying $\bar x\in S^{-1}(0)$ with $A\bar x\in T^{-1}(0).$ Based on a regularization point of view of a Byrne’s idea, by replacing the operators by their Yosida approximates, we propose to consider the problem of finding $\bar x$ such that $(P_{\alpha^{-1}, (1-\alpha)^{-1}})$ $0=S_{\alpha^{-1}}(\bar x)+A^tT_{(1-\alpha)^{-1}}(A\bar x),$ and then to introduce the following extended CQ-Algorithm
$x_k=\frac{1}{1+\gamma\alpha}(I+\gamma\alpha J^T_{\alpha^{-1}(1+\alpha\gamma)})(I-\gamma (1-\alpha) A^t(I-J^S_{(1-\alpha)^{-1}})A)x_{k-1},$

where $\alpha\in (0,1)$ and $0\textless \gamma \textless \frac{2}{(1-\alpha)L}$ with $L=\rho(A^tA)$ being the largest eigenvalue of $A^tA$. In the context of split feasibility problems, this clearly reduces to Byrne’s extended CQ-algorithm, namely, $x_k=\frac{1}{1+\gamma\alpha}(I+\gamma\alpha P_C)(I-\gamma (1-\alpha) A^t(I-P_Q)A)x_{k-1},$ where $P_C$ and $P_Q$ are the orthogonal projections onto the nonempty closed convex sets $C$ and $Q$, respectively.