We consider solving minimization problems with L_1-regularization: min ||x||_1 + mu f(x) particularly for f(x) = (1/2)||Ax-b||M2, where A is m by n and m < n. Our goal is to construct efficient and robust algorithms for solving large-scale problems with dense data, and our approach is based on two powerful algorithmic ideas: operator-splitting and continuation. This paper establishes q-linear convergence rates for our algorithm applied to problems with f(x) convex, but not necessarily strictly convex. We present numerical results for several types of compressed sensing problems, and show that our algorithm compares favorably with three state-of-the-art algorithms when applied to large-scale problems with noisy data.