compressed sensing, machine learning, sensor networks.

Introduction {#Sec:intro}
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Recently, there has been a growing interest in distributed methods for data processing. Such interest was triggered both by the need of processing large quantities of data in distributed platforms, such as supercomputers, and by applications where data comes from spatially different locations and central processing is costly or impractical, as in sensor networks. On the other hand, over the last decades, convex optimization theory has proven to be an invaluable tool for deriving data processing algorithms [@Boyd:ConvexOpti]. Standard algorithms like interior-point methods are, however, unable to handle either large-scale or distributed problems. To solve these problems we thus need new algorithms.

In this paper, we aim to solve separable optimization problems in a distributed, decentralized way. In a separable optimization problem, the cost function can be decomposed as a sum of $P$ functions $f_p$ and the constraint set can be decomposed as the intersection of $P$ sets $X_p$: $$\label{Eq:IntroSeparableProb}
        \begin{array}{ll}
            \underset{x}{\text{minimize}} & f_1(x) + f_2(x) + \cdots + f_P(x) \\
            \text{subject to} & x \in X_1 \cap X_2 \cap \cdots \cap X_P\,,
        \end{array}$$ where $x \in \mathbb{R}^n$ is the global optimization variable. We will denote any solution of  with $x^\star$. 