Decentralized constrained consensus optimization with primal dual splitting projection

Abstract

This paper considers the decentralized consensus optimization problem over a multi-agent network, where every agent is subject to multiple simple local constraints. In most of the existing algorithms, an agent combines its local copy of the optimization variable with their neighbors', runs local minimization or gradient descent, and then projects the estimate onto the intersection of the local constraints. To avoid the computationally-expensive projections, we propose two primal dual splitting projection algorithms such that an agent only needs to project onto individual local constraints, which significantly simplifies computation. The algorithm can be either deterministic, meaning that the local constraints are handled in parallel, or stochastic, meaning that only one local constraint is randomly sampled by every agent at every iteration. We establish convergence and rate of convergence for the deterministic case. Numerical experiments validate the effectiveness of the two proposed primal dual splitting projection algorithms.