Fixed-time Distributed Optimization: Consistent Discretization, Time-Varying Topology and Non-Convex Functions
Kunal Garg, Mayank Baranwal, Alfred O. Hero, Dimitra Panagou · arXiv (Cornell University) · 2019
This paper presents a fixed-time convergent and distributed optimization algorithm for first-order multi-agent systems over a time-varying communication topology. Each agent in the network can access its private objective function, while exchange of local information is permitted between the neighbors. The proposed optimization algorithm combines a fixed-time convergent distributed parameter estimation scheme with a fixed-time distributed consensus scheme as its solution methodology. The results are presented under the assumption that the team objective function is strongly convex, as opposed to the common assumptions in the literature requiring each of the local objective functions to be strongly convex. The results are extend to the class of possibly non-convex team objective functions satisfying only the Polyak-Łojasiewicz (PL) inequality. It is also shown that the proposed continuous-time scheme, when discretized using Euler's method, leads to consistent discretization, i.e., the fixed-time convergence behavior is preserved for the discretized dynamics. Finally, a similar scheme is presented for the case when the team objective function is strictly convex instead of strongly convex. Numerical examples presented in this paper corroborate our theoretical analysis.