Primal-Dual Strategy (PDS) for Composite Optimization Over Directed graphs

We investigate the distributed multi-agent sharing optimization problem in a directed graph, with a composite objective function consisting of a smooth function plus a convex (possibly non-smooth) function shared by all agents. While adhering to the network connectivity structure, the goal is to minimize the sum of smooth local functions plus a non-smooth function. The proposed Primal-Dual algorithm (PD) is similar to a previous algorithm \cite{b27}, but it has additional benefits. To begin, we investigate the problem in directed graphs, where agents can only communicate in one direction and the combination matrix is not symmetric. Furthermore, the combination matrix is changing over time, and the condition coefficient weights are produced using an adaptive approach. The strong convexity assumption, adaptive coefficient weights, and a new upper bound on step-sizes are used to demonstrate that linear convergence is possible. New upper bounds on step-sizes are derived under the strong convexity assumption and adaptive coefficient weights that are time-varying in the presence of both smooth and non-smooth terms. Simulation results show the efficacy of the proposed algorithm compared to some other algorithms.