A necessary and sufficient condition for discrete-time consensus on star boundaries

TL;DR

The paper addresses discrete-time consensus when agent states are constrained to star boundaries and updates are realized via radial projections of neighbor-derived conical sums. It develops a necessary and sufficient condition for asymptotic consensus of directions under strongly connected digraphs, demonstrates linear convergence, and proves continuity of the consensus point with respect to initial conditions. The analysis leverages products of time-varying matrices and a continuous mapping v(X0) to characterize when consensus occurs, and provides practical sufficiency criteria. The results generalize traditional Euclidean consensus to non-Euclidean star-boundary domains and are supported by extensive simulations on diverse star shapes, including lp-spheres, underscoring potential applications in distributed optimization on manifolds.

Abstract

It is intuitive and well known, that if agents in a multi-agent system iteratively update their states in the Euclidean space as convex combinations of neighbors' states, all states eventually converge to the same value (consensus), provided the interaction graph is sufficiently connected. However, this seems to be also true in practice if the convex combinations of states are mapped or radially projected onto any unit $l_p$-sphere or even boundaries of star-convex sets, herein referred to as star boundaries. In this paper, we present insight into this matter by providing a necessary and sufficient condition for asymptotic consensus of the normalized states (directions) for strongly connected directed graphs, which is equivalent to asymptotic consensus of states when the star boundaries are the same for all agents. Furthermore, we show that when asymptotic consensus occurs, the states converge linearly and the point of convergence is continuous in the initial states. Assuming a directed strongly connected graph provides a more general setting than that considered, for example, in gradient-based consensus protocols, where symmetric graphs are assumed. Illustrative examples and a vast number of numerical simulations showcase the theoretical results.

Figures (8)

• Figure 1: Projection of a point $y$ onto four different star boundaries according to \ref{['eq:projection1']}, where $y_1$, $y_2$, $y_{\infty}$, and $y_{\gamma}$ are the projections onto the unit $l_1$-sphere (magenta), the unit $l_2$-sphere (green), the unit $l_{\infty}$-sphere (blue), and a more complicated star boundary (red), respectively. Here, $d=2$.
• Figure 2: Left: antipodal position of the rows of $X_0$ on the unit circle in $\mathbb{R}^d$. Right: the two columns of $X_0$ in $\mathbb{R}^n$ (black dots) and the vector $v(X_0)$ (blue arrow). Clearly $v(X_0)X_0 = \bold{0}_{1,d}$.
• Figure 3: The evolution of the minimum cone $\mathcal{C}(\tilde{A}(k,t_0,X_0))$ in $\mathbb{R}^2$. The first $5$ time steps are presented, with darker colors corresponding to later time steps. The purple region represents $\mathcal{C}^{\perp}(A)$. Here, $d=n=2$.
• Figure 4: Left: The convergence of the points representing the normalized rows $P_{\mathbb{S}(n,n)}(\tilde{A}(k,t_0,X_0))$ on the unit sphere in $\mathbb{R}^3$. Right: The corresponding geodesic hulls are shrinking with time. The first $5$ time steps are presented, with darker colors corresponding to later time steps. Here, $d=n=3$.
• Figure 5: Convergence to the consensus direction of agents moving on various $l_p$-spheres in $\mathbb{R}^2$. The chosen radius and parameter $p$ for the spheres are indicated in the legend. Left: initial positions of the agents. Right: final positions of the agents, which have converged to the consensus direction. The line segment connecting the origin and final positions of the agents illustrates the consensus direction.

Theorems & Definitions (7)

• Definition 1
• Proposition 1
• Proposition 2
• Proposition 3
• Theorem 1
• Corollary 1
• Corollary 2