Column Partition Based Distributed Algorithms for Coupled Convex Sparse Optimization: Dual and Exact Regularization Approaches

This paper develops column partition based distributed schemes for a class of convex sparse optimization problems, e.g., basis pursuit (BP), LASSO, denosing (BPDN), and their extensions, fused LASSO. We are particularly interested in the cases where number (scalar) decision variables is much larger than measurements, each agent has limited memory or computing capacity such that it only knows small columns measurement matrix. The problems consideration densely coupled cannot be formulated as separable programs. To overcome this difficulty, we consider dual which locally coupled. Once solution attained, shown primal can found from corresponding regularized BP-like under suitable exact regularization conditions. A wide range existing exploited to solve obtained problems. yields two-stage LASSO-like BPDN-like problems; overall convergence these established. Numerical results illustrate performance proposed schemes.