A primal dual active set with continuation algorithm for the ℓ0ℓ0-regularized optimization problem

We develop a primal dual active set with continuation algorithm for solving the ℓ0ℓ0-regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples the primal dual active set method with a continuation strategy on the regularization parameter. At each inner iteration, it first identifies the active set from both primal and dual variables, and then updates the primal variable by solving a (typically small) least-squares problem defined on the active set, from which the dual variable can be updated explicitly. Under certain conditions on the sensing matrix, i.e., mutual incoherence property or restricted isometry property, and the noise level, a finite step global convergence of the overall algorithm is established. Extensive numerical examples are presented to illustrate the efficiency and accuracy of the algorithm and its convergence behavior.