# A General Framework of Exact Primal-Dual First Order Algorithms for Distributed Optimization

## TL;DR

This paper introduces a flexible distributed primal-dual optimization framework that achieves exact convergence with fixed stepsizes, balancing speed and computational effort, and demonstrates linear convergence for strongly convex problems.

## Contribution

It proposes a general class of distributed primal-dual algorithms with multiple primal steps per iteration, enabling controlled trade-offs and exact convergence with fixed stepsizes.

## Key findings

- Achieves linear convergence for strongly convex, Lipschitz gradient functions.
- Outperforms existing methods in simulation tests.
- Allows multiple primal steps to balance performance and complexity.

## Abstract

We study the problem of minimizing a sum of local objective convex functions over a network of processors/agents. This problem naturally calls for distributed optimization algorithms, in which the agents cooperatively solve the problem through local computations and communications with neighbors. While many of the existing distributed algorithms with constant stepsize can only converge to a neighborhood of optimal solution, some recent methods based on augmented Lagrangian and method of multipliers can achieve exact convergence with a fixed stepsize. However, these methods either suffer from slow convergence speed or require minimization at each iteration. In this work, we develop a class of distributed first order primal-dual methods, which allows for multiple primal steps per iteration. This general framework makes it possible to control the trade off between the performance and the execution complexity in primal-dual algorithms. We show that for strongly convex and Lipschitz gradient objective functions, this class of algorithms converges linearly to the optimal solution under appropriate constant stepsize choices. Simulation results confirm the superior performance of our algorithm compared to existing methods.