Distributed projected-reflected-gradient algorithms for stochastic generalized Nash equilibrium problems

We consider the stochastic generalized Nash equilibrium problem (SGNEP) with joint feasibility constraints and expected-value cost functions. We propose a distributed stochastic projected reflected gradient algorithm and show its almost sure convergence when the pseudogradient mapping is monotone and the solution is unique. The algorithm is based on monotone operator splitting methods for SGNEPs when the expected-value pseudogradient mapping is approximated at each iteration via an increasing number of samples of the random variable, an approach known as stochastic approximation with variance reduction. Finally, we show that a preconditioned variant of our proposed algorithm has convergence guarantees when the pseudogradient mapping is cocoercive.