  In this paper, we study the problem of minimizing a sum of convex objective
functions, which are locally available to agents in a network. Distributed
optimization algorithms make it possible for the agents to cooperatively solve
the problem through local computations and communications with neighbors.
Lagrangian-based distributed optimization algorithms have received significant
attention in recent years, due to their exact convergence property. However,
many of these algorithms have slow convergence or are expensive to execute. In
this paper, we develop a flexible framework of first-order primal-dual
algorithms (FlexPD), which allows for multiple primal steps per iteration. This
framework includes three algorithms, FlexPD-F, FlexPD-G, and FlexPD-C that can
be used for various applications with different computation and communication
limitations. For strongly convex and Lipschitz gradient objective functions, we
establish linear convergence of our proposed framework to the optimal solution.
Simulation results confirm the superior performance of our framework compared
to the existing methods.
