On the convergence of an active-set method for ℓ1 minimization

Abstract. We analyze an abridged version of the active-set algorithm FPC AS proposed in [18] for solving the l1-regularized problem, i.e., a weighted sum of the l1-norm ‖x‖1 and a smooth function f(x). The active set algorithm alternatively iterates between two stages. In the first “nonmonotone line search (NMLS)” stage, an iterative first-order method based on “shrinkage” is used to estimate the support at the solution. In the second “subspace optimization” stage, a smaller smooth problem is solved to recover the magnitudes of the nonzero components of x. We show that NMLS itself is globally convergent and the convergence rate is at least R-linearly. In particular, NMLS is able to identify of the zero components of a stationary point after a finite number of steps under some mild conditions. The global convergence of FPC AS is established based on the properties of NMLS.