Gradient-based method with active set strategy for $\ell _1$ optimization

Abstract

In this paper, we propose an identification function and develop an active set identification technique for solving the ℓ1 optimization problem. Such a technique has a strong ability to accurately identify the zero components in a neighbourhood of an isolated stationary point without strict complementarity conditions. Based on the active set identification technique, we propose a gradient-based method for the ℓ1 optimization problem. To accelerate the algorithm, a subspace Barzilai-Borwein steplength and a subspace exact steplength are developed, respectively. Under appropriate conditions, we show that the method with the nonmonotone line search technique is globally convergent. Numerical experiments with compressive sensing problems show that our approach is competitive with several known methods for the standard ℓ2-ℓ1 problem.