Contents

function [z, history] = lasso(A, b, lambda, rho, alpha)

% lasso  Solve lasso problem via ADMM
%
% [z, history] = lasso(A, b, lambda, rho, alpha);
%
% Solves the following problem via ADMM:
%
%   minimize 1/2*|| Ax - b ||_2^2 + \lambda || x ||_1
%
% The solution is returned in the vector x.
%
% history is a structure that contains the objective value, the primal and
% dual residual norms, and the tolerances for the primal and dual residual
% norms at each iteration.
%
% rho is the augmented Lagrangian parameter.
%
% alpha is the over-relaxation parameter (typical values for alpha are
% between 1.0 and 1.8).
%
%
%

t_start = tic;


Global constants and defaults

QUIET    = 0;
MAX_ITER = 1000;
ABSTOL   = 1e-4;
RELTOL   = 1e-2;


Data preprocessing

[m, n] = size(A);

% save a matrix-vector multiply
Atb = A'*b;


x = zeros(n,1);
z = zeros(n,1);
u = zeros(n,1);

% cache the factorization
[L U] = factor(A, rho);

if ~QUIET
fprintf('%3s\t%10s\t%10s\t%10s\t%10s\t%10s\n', 'iter', ...
'r norm', 'eps pri', 's norm', 'eps dual', 'objective');
end

for k = 1:MAX_ITER

% x-update
q = Atb + rho*(z - u);    % temporary value
if( m >= n )    % if skinny
x = U \ (L \ q);
else            % if fat
x = q/rho - (A'*(U \ ( L \ (A*q) )))/rho^2;
end

% z-update with relaxation
zold = z;
x_hat = alpha*x + (1 - alpha)*zold;
z = shrinkage(x_hat + u, lambda/rho);

% u-update
u = u + (x_hat - z);

% diagnostics, reporting, termination checks
history.objval(k)  = objective(A, b, lambda, x, z);

history.r_norm(k)  = norm(x - z);
history.s_norm(k)  = norm(-rho*(z - zold));

history.eps_pri(k) = sqrt(n)*ABSTOL + RELTOL*max(norm(x), norm(-z));
history.eps_dual(k)= sqrt(n)*ABSTOL + RELTOL*norm(rho*u);

if ~QUIET
fprintf('%3d\t%10.4f\t%10.4f\t%10.4f\t%10.4f\t%10.2f\n', k, ...
history.r_norm(k), history.eps_pri(k), ...
history.s_norm(k), history.eps_dual(k), history.objval(k));
end

if (history.r_norm(k) < history.eps_pri(k) && ...
history.s_norm(k) < history.eps_dual(k))
break;
end

end

if ~QUIET
toc(t_start);
end

end

function p = objective(A, b, lambda, x, z)
p = ( 1/2*sum((A*x - b).^2) + lambda*norm(z,1) );
end

function z = shrinkage(x, kappa)
z = max( 0, x - kappa ) - max( 0, -x - kappa );
end

function [L U] = factor(A, rho)
[m, n] = size(A);
if ( m >= n )    % if skinny
L = chol( A'*A + rho*speye(n), 'lower' );
else            % if fat
L = chol( speye(m) + 1/rho*(A*A'), 'lower' );
end

% force matlab to recognize the upper / lower triangular structure
L = sparse(L);
U = sparse(L');
end