A second-order method for strongly convex ℓ 1 -regularization problems

In this paper a second-order method for solving large-scale strongly convex `1-regularized problems is developed. The proposed method is a NewtonCG (Conjugate Gradients) algorithm with backtracking line-search embedded in a doubly-continuation scheme. Worst-case iteration complexity of the proposed Newton-CG is established. Based on the analysis of Newton-CG, worstcase iteration complexity of the doubly-continuation scheme is obtained. Numerical results are presented on large-scale problems for the doublycontinuation Newton-CG algorithm, which show that the proposed secondorder method competes favourably with state-of-the-art first-order methods. In addition, `1-regularized Sparse Least-Squares problems are discussed for which a parallel block coordinate descent method stagnates.