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.

中文翻译：

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用于耦合凸稀疏优化的基于列分区的分布式算法：对偶和精确正则化方法

本文针对一类凸稀疏优化问题开发了基于列分区的分布式方案，例如基追踪（BP）、LASSO、基追踪去噪（BPDN）及其扩展，例如融合LASSO。我们对以下情况特别感兴趣：（标量）决策变量的数量远大于（标量）测量的数量，并且每个代理的内存或计算能力有限，以至于它只知道测量的少量列矩阵。所考虑的问题是密集耦合的，不能被表述为可分离的凸程序。为了克服这个困难，我们考虑了它们可分离或局部耦合的双重问题。一旦得到双解，结果表明，在合适的精确正则化条件下，可以从相应的正则化类 BP 问题的对偶中找到原始解。可以利用广泛的现有分布式方案来解决获得的对偶问题。这为类似 LASSO 和类似 BPDN 的问题产生了基于两阶段列分区的分布式方案；建立了这些方案的整体收敛性。数值结果说明了所提出的两阶段分布式方案的性能。