  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.
