Momentum-Based Nash Set-Seeking Over Networks via Multitime Scale Hybrid Dynamic Inclusions

Multitime scale techniques, such as singular perturbations and averaging theory, have played an important role in the development of distributed Nash equilibrium seeking algorithms for network systems. Such techniques rely on the
uniform asymptotic stability
properties of the dynamics that evolve in each of the time scales of the closed-loop system. When such properties are absent, the synthesis of multitime scale Nash equilibrium-seeking algorithms is more challenging and it requires additional regularization mechanisms. In this article, we investigate the synthesis and analysis of these mechanisms in the context of
accelerated pseudogradient flows
with time-varying damping in noncooperative games. Specifically, we introduce a new class of
distributed and hybrid
Nash set-seeking algorithms that synergistically combine dynamic momentum-based flows with
coordinated
discrete-time resets. The reset mechanisms can be seen as restarting techniques that allow individual players to choose their own momentum restarting policy to potentially achieve better transient performance. The resulting closed-loop system is modeled as a hybrid dynamic inclusion, which is analyzed using tools from hybrid dynamical system's theory. Our algorithms are developed for potential games, as well as for monotone games for which a potential function does not exist. They can be implemented in games where players have access to gradient Oracles with full or partial information of the multiagent system, as well as in games where players have access only to
measurements
of their costs. In the latter case, we use tools from hybrid extremum seeking control.