Sparse convex optimization involves optimization problems where the decision variables are constrained to have a certain number of entries equal to zero. In this paper, we focus on the sparse version of the so-called aggregative optimization scenario, i.e., on optimization problems in which the cost reads as the sum of local functions each depending on both a local decision variable and an aggregation of all of them. In this framework, we propose a novel fully-distributed scheme to address the problem over a network of cooperating agents. Specifically, by taking advantage of a suitable problem reformulation, we define an Augmented Lagrangian function. Then, we address such an Augmented Lagrangian by suitably interlacing the so-called Projected Aggregative Tracking distributed algorithm and the Block Coordinated Descent method giving rise to a novel fully-distributed scheme. The effectiveness of the proposed algorithm is corroborated via numerical simulations in problems arising in machine learning scenarios with both synthetic and real-world data sets.
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