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This paper develops column partition based distributed schemes for a class oflarge-scale 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, andeach agent has limited memory or computing capacity such that it only knows asmall number of columns of a measurement matrix. These problems inconsideration are densely coupled and cannot be formulated as separable convexprograms using column partition. To overcome this difficulty, we consider theirdual problems which are separable or locally coupled. Once a dual solution isattained, it is shown that a primal solution can be found from the dual ofcorresponding regularized BP-like problems under suitable exact regularizationconditions. A wide range of existing distributed schemes can be exploited tosolve the obtained dual problems. This yields two-stage column partition baseddistributed schemes for LASSO-like and BPDN-like problems; the overallconvergence of these schemes is established using sensitivity analysistechniques. Numerical results illustrate the effectiveness of the proposedschemes.

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