  We address the generalized Nash equilibrium seeking problem in a
partial-decision information scenario, where each agent can only exchange
information with some neighbors, although its cost function possibly depends on
the strategies of all agents. The few existing methods build on projected
pseudo-gradient dynamics, and require either double-layer iterations or
conservative conditions on the step sizes. To overcome both these flaws and
improve efficiency, we design the first fully-distributed single-layer
algorithms based on proximal best-response. Our schemes are fixed-step and
allow for inexact updates, which is crucial for reducing the computational
complexity. Under standard assumptions on the game primitives, we establish
convergence to a variational equilibrium (with linear rate for games without
coupling constraints) by recasting our algorithms as proximal-point methods,
opportunely preconditioned to distribute the computation among the agents.
Since our analysis hinges on a restricted monotonicity property, we also
provide new general results that significantly extend the domain of
applicability of proximal-point methods. Besides, the operator-theoretic
approach favors the implementation of provably correct acceleration schemes
that can further improve the convergence speed. Finally, the potential of our
algorithms is demonstrated numerically, revealing much faster convergence with
respect to projected pseudo-gradient methods and validating our theoretical
findings.
