SDMM - Simultaneous-direction method of multipliers algorithm

Usage

sol = sdmm(F,param);
sol = sdmm(F);
[sol,info] = sdmm(...);

Input parameters

Output parameters

Description

sdmm, from simultaneous-direction method of multipliers solves:

where \(x\) belong to \(R^N\), \(L_i\) are linear operators and \(x_i\) are the minimization variables.

F is a cellarray of structure representing the functions. In the function F{i}, there have to be:

• F{i}.eval(x_i) : an operator to evaluate the function
• F{i}.prox(x_i, gamma) : an operator to evaluate the prox of the function
• F{i}.x0 : vector of initial value

Optionally you can define

• F{i}.L : linear operator, matrix or operator (default identity)
• F{i}.Lt : adjoint of linear operator, matrix or operator (default identity)

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

• param.tol : is stop criterion for the loop. The algorithm stops if

  \begin{equation*} \max_i \frac{ \| y_i(t) - y_i(t-1)\| }{ \|y_i(t)\|} < tol, \end{equation*}

  where \(y_i(t)\) are the dual variable of function i at itertion t by default, tol=10e-4.

  Warning! This stopping criterion is different from other solver!

• param.Qinv : Inverted Q matrix. \(Qinv=Q^{-1}\) with:

  \begin{equation*} Q = \sum_i L_i^T ( L_i x) \end{equation*}

  By default, Qinv is the identity matrix divided by the number of functions.

  This parameter can be given in a matrix form or in a linear operator form.