function [sol, info] = ppxa(x_0, F, param)
%PPXA Parallel Proximal algorithm
% Usage: sol = ppxa(x_0, F, param);
% sol = ppxa(x_0, F);
% [sol, infos] = ppxa(...);
%
% Input parameters:
% x_0 : Starting point of the algorithm
% F : Array of function to minimize
% param : Optional parameter
% Output parameters:
% sol : Solution
% info : Structure summarizing informations at convergence
%
% PPXA, derived from the Douglas-Rachford algorithm, solves
%
% sol = argmin sum(W_i*f_i(x))
%
% for x in R^N, where x is the variable and x_0 is the starting point.
%
% F is a cellarray of structures representing functions. All of them
% should contains at least two fields. F{ii}.eval to evaluate the
% function and F{ii}.prox to compute the proximal operator of the
% function.
%
% param a Matlab structure containing solver paremeters. See the
% function SOLVEP for more information. Additionally it contains those
% aditional fields:
%
% param.W : the weight (all equal by default)
%
% 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 0.99. Do not change this parameter unless you know what you
% do.
%
% See also: solvep sdmm admm generalized_forward_backward
%
% References:
% P. Combettes and J. Pesquet. Proximal splitting methods in signal
% processing. Fixed-Point Algorithms for Inverse Problems in Science and
% Engineering, pages 185--212, 2011.
%
%
% Url: https://epfl-lts2.github.io/unlocbox-html/doc/solver/ppxa.html
% Copyright (C) 2012-2016 Nathanael Perraudin.
% This file is part of UNLOCBOX version 1.7.4
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% Author: Nathanael Perraudin
% Date: Oct 19 2012
% Testing : test_solver
param.algo = 'PPXA';
[sol, info] = solvep(x_0,F,param);
end
% function [sol, info, objective] = ppxa(x_0, F, param)
% %PPXA Parallel Proximal algorithm
% % Usage: sol = ppxa(x_0, F, param);
% % sol = ppxa(x_0, F);
% % [sol, infos, objective] = ppxa(...);
% %
% % Input parameters:
% % x_0 : Starting point of the algorithm
% % F : Array of function to minimize
% % param : Optional parameter
% % Output parameters:
% % sol : Solution
% % info : Structure summarizing informations at convergence
% % objective: vector (evaluation of the objective function each iteration)
% %
% % `ppxa`, derived from the Douglas-Rachford algorithm, solves
% %
% % .. sol = argmin sum(W_i*f_i(x))
% %
% % .. math:: sol = \min_x \sum_i W_i f_i(x)
% %
% % for *x* in $R^N$, where *x* is the variable and *x_0* is the starting point.
% %
% % * *F* is a cellarray of structures representing functions containing
% % operators inside and eventually the norm. The prox: *F{i}.prox* and
% % the function: *F{i}.eval* are defined in the same way as in the
% % Forward-backward and Douglas-Rachford algorithms
% %
% % * *param* a Matlab structure containing the following fields:
% %
% % General parameters:
% %
% % * *param.W* : the weight (all equal by default)
% %
% % * *param.tol* : is stop criterion for the loop. The algorithm stops if
% %
% % .. ( n(t) - n(t-1) ) / n(t) < tol,
% %
% % .. math:: \frac{ n(t) - n(t-1) }{ n(t)} < tol,
% %
% % where $n(t) = \sum W_i*f_i(x)$ is the objective function at iteration *t*
% % by default, `tol=10e-4`.
% %
% % * *param.maxit* : is the maximum number of iteration. By default, it is 200.
% %
% % * *param.verbose* : 0 no log, 1 print main steps, 2 print all steps.
% %
% % * *param.lambda* : is the weight of the update term.
% %
% % * *param.gamma* : convergence parameter (default 1)
% %
% % * *param.abs_tol* : If activated, this stopping critterion is the
% % objective function smaller than *param.tol*. By default 0.
% %
% %
% % info is a Matlab structure containing the following fields:
% %
% % * *info.algo* : Algorithm used
% %
% % * *info.iter* : Number of iteration
% %
% % * *info.time* : Time of exectution of the function in sec.
% %
% % * *info.final_eval* : Final evaluation of the objectivs functions
% %
% % * *info.crit* : Stopping critterion used
% %
% % * *info.rel_norm* : Relative norm at convergence
% %
% %
% % See also: sdmm, admm, generalized_forward_backward
% %
% % Demos: demo_ppxa
% %
% % References: combettes2011proximal
%
% % Author: Nathanael Perraudin
% % Date: Oct 19 2012
%
% % Start the time counter
% t1 = tic;
%
% % test the evaluate function
% [F] = test_eval(F);
%
% % number of functions
% m = size(F, 2);
%
% % Optional input arguments
% if nargin < 3, param = struct; end
%
% if ~isfield(param, 'gamma'), param.gamma = 1; end
% if ~isfield(param, 'tol'), param.tol = 10e-4; end
% if ~isfield(param, 'abs_tol'), param.abs_tol = 0; end
% if ~isfield(param, 'maxit'), param.maxit = 200; end
% if ~isfield(param, 'verbose'), param.verbose = 1; end
% if ~isfield(param, 'lambda'), param.lambda = 0.99; end
% if ~isfield(param, 'W'), param.W = ones(m, 1) / m; end
%
%
% W=param.W;
%
%
% % Reshape x if vector line
% if (size(W, 2) > size(W, 1))
% W = W';
% end
%
% test_gamma_ppxa(param.gamma);
% test_sum(W);
%
% % Create a table of scructure containing data
% data = [];
% for ii = 1:m
% data(ii).y = x_0; %#ok<AGROW>
% data(ii).p = zeros(size(x_0)); %#ok<AGROW>
% end
%
%
% x = w_sum(W, data, 'y');
%
%
% % outerloop
% curr_norm = 0;
% for ii = 1:m
% curr_norm = F{ii}.eval(x) + curr_norm;
% end
% [~, ~, prev_norm, iter, objective, ~] = convergence_test(curr_norm);
%
% while 1
%
% if param.verbose >= 2
% fprintf('Iteration %i:\n', iter);
% end
%
% % parallel proximal
% % compute updated prox
% for ii = 1:m
% data(ii).p = F{ii}.prox(data(ii).y, param.gamma);%#ok<AGROW>
% end
% pn = w_sum(W,data,'p');
%
%
%
% % update y
%
% for ii = 1:m
% data(ii).y = data(ii).y + param.lambda * (2*pn - x - data(ii).p); %#ok<AGROW>
% end
%
% % update x
% x = x + param.lambda * (pn - x);
%
% % update solution & relative norm
% sol = x;
% curr_norm = 0;
% for ii = 1:m
% curr_norm = F{ii}.eval(sol) + curr_norm;
% end
%
% [stop, rel_norm, prev_norm, iter, objective, crit] = convergence_test(curr_norm, prev_norm, iter, objective, param);
% [x,param] = post_process(sol, iter, curr_norm, prev_norm, objective, param);
% if stop
% break;
% end
% if param.verbose >= 2
% fprintf('Current objective function : %e \t relative norm : %e \n', curr_norm, rel_norm);
% end
%
% end
%
% if param.verbose>=2
% % Print norm
% fprintf('\n Solution found:\n');
% if param.abs_tol
% fprintf(' Final norm: %e\n', curr_norm );
% else
% fprintf(' Final relative norm: %e\n', rel_norm );
% end
%
%
% % Stopping criterion
% fprintf(' %i iterations\n', iter);
% fprintf(' Stopping criterion: %s \n\n', crit);
% elseif param.verbose>=1
% fprintf(' Solution found: ||f|| = %e, rel_norm = %e, %s\n', ...
% curr_norm, rel_norm,crit);
% end
%
% info.algo = mfilename;
% info.iter = iter;
% info.final_eval = curr_norm;
% info.crit = crit;
% info.time = toc(t1);
% info.rel_norm = rel_norm;
%
% end
%
%
%
% function test_gamma_ppxa(gamma)
% if gamma <= 0
% fprintf('Warning : gamma is not > 0\n');
% end
% end
%
%
% function test_sum(W)
% if (sum(W) > 1+eps) || (sum(W) < 1-eps)
% fprintf('Warning : sum W is not equal to 1');
% end
% end
%
% function s = w_sum(w, data, l)
% s = zeros(size(data(1)));
%
% if l == 'p'
% for ii = 1:length(data)
% s = s + w(ii) * data(ii).p;
% end
% elseif l == 'y'
% for ii = 1:length(data)
% s = s + w(ii) * data(ii).y;
% end
% else
% error('Fatal error! Unknown field in data!')
% end
% end