A partially proximal linearized alternating minimization method for finding Dantzig selectors

The Dantzig selector (DS) is an efficient estimator designed for high-dimensional linear regression problems, especially for the case where the number of samples n is much less than the dimension of features (or variables) p . In this paper, we first reformulate the underlying DS model as an unconstrained minimization problem of the sum of two nonsmooth convex functions and a smooth coupled function. Then by exploiting the structure of the resulting model, we propose a partially proximal linearized alternating minimization method (P-PLAM), whose two subproblems are easy enough with closed-form solutions. Another remarkable advantage is that P-PLAM requires only one starting point, which is potentially helpful for saving computing time. A series of computational experiments on synthetic and real-world data sets demonstrate that the proposed P-PLAM has promising numerical performance in the sense that P-PLAM can obtain higher quality solutions by taking less computing time than some existing state-of-the-art first-order solvers.