Extended ADMM with Paillier Cryptosystem for Nonsmooth Decentralized Optimization

This article studies a nonsmooth multiagent decentralized optimization problem where the agents aim at minimizing a sum of local strongly-convex smooth components plus a common nonsmooth term. We propose an algorithm based on ADMM framework which introduces a proximal term. We establish linear convergence of the proposed algorithm to the exact optimal solution in the presence of the nonsmooth term. Moreover, Paillier cryptosystem has been combined with our algorithm to ensure the privacy preserving in decentralized optimization in the absence of any third party or aggregator. We further provide a numerical example for a least squares problem equipped with $$L_1$$ regular term to show the linear convergence rate and how the step size influences the convergence.