A dual active set method for <inline-formula><tex-math id="M1">$ \ell $</tex-math></inline-formula>1-regularized problem

Abstract

We consider an $ \ell $1-regularized problem, which has many applications such as compressed sensing, MRI, and seismic images. Although there have been some methods for the problem, they cannot obtain a highly accurate solution. To overcome this drawback, we propose a dual active set method, which is an active set method for the Fenchel dual problem of the $ \ell $1-regularized problem. Our method finds a highly accurate solution quickly because of the following features of the dual problem: (i) the dimension of the decision variables is smaller; (ii) since the optimal solution of the primal problem is sparse, the active constraints of the optimal solution of the dual problem are few and can be identified quickly. Then, we show that the method outputs the solution with finite iterations when it solves certain subproblems exactly. Moreover, we give its efficient and concrete implementation for Basis Pursuit Denoising (BPDN), which is a typical $ \ell $1-regularized problem. In numerical experiments for BPDN, our method provides more accurate solutions than all baselines in comparable computing time.