We consider a sparse convex optimization problem that involves a group of agents, under general affine constraints. This problem often arises in distributed model predictive control applications. A distributed Jacobi algorithm is proposed to solve this problem in a cooperative manner. In every iteration, each agent solves its local problem and exchanges information with its 'direct neighbours'. After that, the new and the old solutions are used in a convex combination to maintain feasibility at every iteration. The convex combination update can also be carried out locally.
We provide the a posteriori certification for centralized optimality of distributed solutions, based on comparing Lagrange multipliers of the local problems. Furthermore, we also prove a priori conditions that guarantee convergence to optimality in several problem settings.
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