\begin{algorithm}[htpb]
	\caption{Randomized $r$-sets-Douglas-Rachford (RrDR) method \label{r-RDRK}}
	\begin{algorithmic}
		\Require
		$A\in \mathbb{R}^{m\times n}$, $b\in \mathbb{R}^m$, $r\in\mathbb{Z}_{+}$, $k=0$,  extrapolation/relaxation parameter $\alpha\in(0,1)$ and an initial $x^0\in \mathbb{R}^{n}$.
		\begin{enumerate}
			\item[1:] Set $z^{k}_0:=x^k$.
			\item[2:] {\bf for $\ell=1,\ldots,r$ do}
			\item[3:] \ \ \ Select $j_{k_{\ell}}\in\{1,\ldots,m\}$ with probability $\mbox{Pr}(\mbox{row}=j_{k_{\ell}})=\frac{\|a_{j_{k_{\ell}}}\|^2_2}{\|A\|_{F}^2}$.
			\item[4:] \ \ \ Compute
			$$
			z_{\ell}^{k}:=z_{\ell-1}^k-2\frac{\langle a_{j_{k_{\ell}}},z_{\ell-1}^k\rangle-b_{j_{k_{\ell}}}}{\|a_{j_{k_\ell}}\|^2_2}a_{j_{k_{\ell}}}.
			$$
			\item[5:] {\bf end for}
			\item[6:] Update
			$$
			x^{k+1}:=(1-\alpha) x^k+\alpha z_{r}^k.
			$$
			\item[7:] If the stopping rule is satisfied, stop and go to output. Otherwise, set $k=k+1$ and return to Step $1$.
		\end{enumerate}
		
		\Ensure
		The approximate solution $ x^k $.
	\end{algorithmic}
\end{algorithm}

\begin{algorithm}
[htpb]
\caption{Randomized $r$-sets-Douglas-Rachford (RrDR) method }
\begin{algorithmic}
\Require
$A\in \mathbb{R}^{m\times n}$, $b\in \mathbb{R}^m$, $r\in\mathbb{Z}_{+}$, $k=0$, extrapolation/relaxation parameter $\alpha\in(0,1)$ and an initial $x^0\in \mathbb{R}^{n}$.
\begin{enumerate}
\item[1:] Set $z^{k}_0:=x^k$.
\item[2:] {\bf for $\ell=1,\ldots,r$ do}
\item[3:] \ \ \ Select $j_{k_{\ell}}\in\{1,\ldots,m\}$ with probability $\mbox{Pr}(\mbox{row}=j_{k_{\ell}})=\frac{\|a_{j_{k_{\ell}}}\|^2_2}{\|A\|_{F}^2}$.
\item[4:] \ \ \ Compute
$$
z_{\ell}^{k}:=z_{\ell-1}^k-2\frac{\langle a_{j_{k_{\ell}}},z_{\ell-1}^k\rangle-b_{j_{k_{\ell}}}}{\|a_{j_{k_\ell}}\|^2_2}a_{j_{k_{\ell}}}.
$$
\item[5:] {\bf end for}
\item[6:] Update
$$
x^{k+1}:=(1-\alpha) x^k+\alpha z_{r}^k.
$$
\item[7:] If the stopping rule is satisfied, stop and go to output. Otherwise, set $k=k+1$ and return to Step $1$.
\end{enumerate}
\Ensure
The approximate solution $ x^k $.
\end{algorithmic}
\end{algorithm}
{
    "title": "randomized douglas-rachford methods for linear systems: improved   accuracy and efficiency",
    "id": "2207.04291",
    "abstract": "the douglas-rachford (dr) method is a widely used method for finding a point in the intersection of two closed convex sets (feasibility problem). however, the method converges weakly and the associated rate of convergence is hard to analyze in general. in addition, the direct extension of the dr method for solving more-than-two-sets feasibility problems, called the $r$-sets-dr method, is not necessarily convergent. to improve the efficiency of the optimization algorithms, the introduction of randomization and the momentum technique has attracted increasing attention. in this paper, we propose the randomized $r$-sets-dr (rrdr) method for solving the feasibility problem derived from linear systems, showing the benefit of the randomization as it brings linear convergence in expectation to the otherwise divergent $r$-sets-dr method. furthermore, the convergence rate does not depend on the dimension of the coefficient matrix. we also study rrdr with heavy ball momentum and establish its accelerated rate. numerical experiments are provided to confirm our results and demonstrate the notable improvements in accuracy and efficiency of the dr method, brought by the randomization and the momentum technique.",
    "categories": "math.oc",
    "doi": "",
    "created": "2022-07-09",
    "updated": "2024-01-09",
    "authors": [
        "deren han",
        "yansheng su",
        "jiaxin xie"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2207.04291"
}
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