Consensus seeking gradient descent flows on boundaries of convex sets

TL;DR

This paper introduces a gradient descent flow algorithm for achieving consensus on hypersurfaces, demonstrating convergence under certain conditions and linking the problem to convex set boundaries.

Contribution

It proposes a novel gradient descent flow algorithm for consensus on hypersurfaces and establishes conditions for convergence related to convex set boundaries.

Findings

Convergence occurs for almost all initial conditions if a specific inequality holds.

The algorithm on ellipsoids is equivalent to the one on the unit sphere, achieving almost global synchronization.

Strong convergence results are possible on boundaries of convex sets.

Abstract

Consensus on nonlinear spaces is of use in many control applications. This paper proposes a gradient descent flow algorithm for consensus on hypersurfaces. We show that if an inequality holds, then the system converges for almost all initial conditions and all connected graphs. The inequality involves the hypersurface Gauss map and the gradient and Hessian of the implicit equation. Moreover, for the inequality to hold, it is necessary that the manifold is the boundary of a convex set. The literature already contains an algorithm for consensus on hypersurfaces. That algorithm on any ellipsoid is equivalent to our algorithm on the unit sphere. In particular, that algorithm achieves almost global synchronization on ellipsoids. These findings suggest that strong convergence results for consensus seeking gradient descent flows may be established on manifolds that are the boundaries of convex sets.