We address the generalized Nash equilibrium (GNE) problem in a partial-decision information scenario, where each agent can only observe the actions of some neighbors, while its cost function possibly depends on the strategies of other agents. As main contribution, we design a fully-distributed, single-layer, fixed-step algorithm to seek a variational GNE, based on a proximal best-response augmented with consensus and constraint-violation penalization terms. Furthermore, we propose a variant of our method, specifically devised for aggregative games. We establish convergence, under strong monotonicity and Lipschitz continuity of the game mapping, by deriving our algorithms as proximal-point methods, opportunely preconditioned to distribute the computation among the agents. This operator-theoretic interpretation proves very powerful. First, it allows us to demonstrate convergence of our algorithms even if the proximal best-response is computed inexactly by the agents. Secondly, it favors the implementation of acceleration schemes that can improve the convergence speed. The potential of our algorithms is validated numerically, revealing much faster convergence with respect to known gradient-based methods.
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