Linear convergence in (stochastic) generalized Nash equilibrium problems with coupling inequality constraints

We prove the linear convergence of a forwardbackward (FB) algorithm for solving generalized Nash equilibrium problems (GNEPs) with full rank coupling inequality constraints. The analysis relies on a contraction argument in an appropriate weighted space, and leverages partial contractivity properties for the forward and backward operators separately. This differs from existing approaches, which can only deal with equality constraints. Contractivity of the full FB algorithm then implies linear convergence of the iterates to the unique fixed point, i.e., to a generalized Nash equilibrium. The contraction perspective also allows us to extend the analysis to the case of stochastic GNEP, where we provide a novel linear convergence result for stochastic games with coupling inequality constraints