  This paper proposes Asynchronous Triggered Gradient Tracking, i.e., a
distributed optimization algorithm to solve consensus optimization over
networks with asynchronous communication. As a building block, we devise the
continuous-time counterpart of the recently proposed (discrete-time)
distributed gradient tracking called Continuous Gradient Tracking. By using a
Lyapunov approach, we prove exponential stability of the equilibrium
corresponding to agents' estimates being consensual to the optimal solution,
with arbitrary initialization of the local estimates. Then, we propose two
triggered versions of the algorithm. In the first one, the agents continuously
integrate their local dynamics and exchange with neighbors their current local
variables in a synchronous way. In Asynchronous Triggered Gradient Tracking, we
propose a totally asynchronous scheme in which each agent sends to neighbors
its current local variables based on a triggering condition that depends on a
locally verifiable condition. The triggering protocol preserves the linear
convergence of the algorithm and avoids the Zeno behavior, i.e., an infinite
number of triggering events over a finite interval of time is excluded. By
using the stability analysis of Continuous Gradient Tracking as a preparatory
result, we show exponential stability of the equilibrium point holds for both
triggered algorithms and any estimate initialization. Finally, the simulations
validate the effectiveness of the proposed methods on a data analytics problem,
showing also improved performance in terms of inter-agent communication.
