Total_variation_minimization.m

% Total variation denoising with random data

%% Generate problem data
rand('seed', 0);
randn('seed', 0);

n = 100;

x0 = ones(n,1);
for j = 1:3
    idx = randsample(n,1);
    k = randsample(1:10,1);
    x0(ceil(idx/2):idx) = k*x0(ceil(idx/2):idx);
end
b = x0 + randn(n,1);

lambda = 5;

e = ones(n,1);
D = spdiags([e -e], 0:1, n,n);

%% Solve problem

[x history] = total_variation(b, lambda, 1.0, 1.0);

%% Reporting
K = length(history.objval);                                                                                                        

h = figure;
plot(1:K, history.objval, 'k', 'MarkerSize', 10, 'LineWidth', 2); 
ylabel('f(x^k) + g(z^k)'); xlabel('iter (k)');

g = figure;
subplot(2,1,1);                                                                                                                    
semilogy(1:K, max(1e-8, history.r_norm), 'k', ...
    1:K, history.eps_pri, 'k--',  'LineWidth', 2); 
ylabel('||r||_2'); 

subplot(2,1,2);                                                                                                                    
semilogy(1:K, max(1e-8, history.s_norm), 'k', ...
    1:K, history.eps_dual, 'k--', 'LineWidth', 2);   
ylabel('||s||_2'); xlabel('iter (k)');