Distributed constrained optimization algorithm for achieving linear convergence with less information storage

This paper studies the optimization problem where the decision variable is contained in a closed convex set. For the researched constrained optimization problem, the aim of this work is to design a distributed algorithm that exhibits a linear convergence rate and requires less information storage for the iteration variables. Towards this end, based on the implicit gradient-tracking (IGT) technique, an auxiliary variable is introduced for each agent in this work, for which the iteration at the current step does not require the information of the state variable at the previous step. But such information is necessarily involved in the previous studies where the explicit gradient-tracking technique was implemented. Thus, less information storage is needed in the proposed distributed optimization algorithm with IGT employed. Moreover, in order to handle the closed convex set constraint with IGT employed, the indirect projection method is used in this work. Consequently, the distributed constrained optimization algorithm with IGT is successfully designed over an undirected graph. Additionally, the linear convergence rate is strictly proven under several common assumptions and the exact regions of the feasible constant step-sizes are also provided. Finally, a numerical simulation on the logistic regression problem is conducted to verify the effectiveness of the established theoretical results.