A Fully Parallel Distributed Algorithm for Non-Smooth Convex Optimization with Coupled Constraints: Application to Linear Algebraic Equations - Alaviani, S. Sh; Kelkar, A. G.; Vaidya, U. (Proceedings of the American Control Conference, 2022)
In this paper, collaborative optimization of sum of convex functions is considered where agents make decision using local information over networks subject to globally coupled affine equality and inequality constraints. In this problem, the globally coupled equality and inequality constraints' information is only partially accessible to each agent. The first discrete-time fully parallel distributed algorithm without diminishing step sizes, (sub)gradient, and/or solving a sub-problem at each time is derived based on monotone operator splitting approach. In the algorithm, the updates of variables happen independent of each other that results in reducing computational time per iteration significantly. The algorithm can converge to an optimal solution for any convex cost functions and any convex constraint sets of agents with arbitrary initialization over any undirected static (non-switching) networks in synchronous protocol. As an application of the problem, solving linear algebraic equations (LAEs) of the form Ax = b among m agents is considered where each agent only knows the partitioned matrix [A i , b i ] such that A = \left( {Σ _{i = 1}^m{A_i}} \right) and b = \left( {Σ _{i = 1}^m{b_i}} \right). The algorithm for LAEs is able to converge to an optimal solution for any matrices A and b.