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

Abstract

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