Primal-dual algorithm for distributed optimization: A dissipativity-based perspective

TL;DR

This paper presents a dissipativity-based analysis of a continuous-time primal-dual algorithm for distributed nonconvex optimization over directed graphs, showing how network design and gains influence convergence to optimal solutions.

Contribution

It reformulates the algorithm as a Lure system, analyzes its dissipativity properties, and demonstrates convergence and optimality through gain and network design, offering new insights into distributed optimization.

Findings

Linear subsystem is dissipative with respect to a suitable supply rate.

Proper gain selection and network design ensure exponential convergence.

The algorithm achieves an optimal solution to the distributed problem.

Abstract

We study a continuous-time primal-dual algorithm for distributed optimization with nonconvex local cost functions over weight-unbalanced digraphs, and analyze its performance from a dissipativity-based perspective. We first reformulate the algorithm as a Lure type system, consisting of a linear subsystem that relies on the communication topology and the algorithm gains, and a static nonlinear gradient feedback. We then show that the linear subsystem is dissipative with respect to a suitable supply rate, while the nonlinear feedback is not passive. Finally, we establish that, by properly selecting the gains or appropriately designing the communication network, this algorithm converges to an equilibrium at an exponential rate, and thus, achieves an optimal solution to the distributed problem. This work provides new insights into the roles of the network topology, algorithm gains, and cost functions in the performance of a distributed algorithm, and complements existing results from a different viewpoint.