DOUGLAS_RACHFORD - Douglas-rachford proximal splitting algorithm

Usage

sol = douglas_rachford(x_0,f1, f2, param);
sol = douglas_rachford(x_0,f1, f2);
[sol, info] = douglas_rachford(...);

Input parameters

`x_0`	Starting point of the algorithm
`f1`	First function to minimize
`f2`	Second function to minimize
`param`	Optional parameter

Output parameters

`sol`	Solution
`info`	Structure summarizing informations at convergence

Description

douglas_rachford algorithm solves:

\begin{equation*} sol = arg \min_x f_1(x) + f_2(x) \hspace{1cm} for \hspace{1cm} x\in R^N \end{equation*}

where \(x\) is the variable.

f1 and f2 are structures representing convex functions. Inside the structure, there have to be the prox of the function that can be called by f1.prox and the function itself that can be called by f1.eval.

param a Matlab structure containing solver paremeters. See the function solvep for more information. Additionally it contains those aditional fields:

param.lambda: is the weight of the update term. It is kind of a timestep for the proximal operators. (Warning it should not be confused with gamma, the time step for gradient descent part). By default it is set to 1. Do not change this parameter unless you know what you do.

References:

P. Combettes and J. Pesquet. A douglas--rachford splitting approach to nonsmooth convex variational signal recovery. Selected Topics in Signal Processing, IEEE Journal of, 1(4):564--574, 2007.