Distributed algorithm for [formula omitted]-generalized Nash equilibria with uncertain coupled constraints

Abstract

In this paper, we design a distributed algorithm to seek generalized Nash equilibria with uncertain coupled constraints. It is hard to find the exact equilibria directly, because the parameters in the coupled constraint come from general convex sets, which may not have analytic expressions. To solve the problem, we first approximate general convex sets by inscribed polyhedrons and transform the approximate problem into a variational inequality by robust optimization. Then, with help of convex set geometry and metric spaces, we prove that the solution to the variational inequality induces an ε-generalized Nash equilibrium of the original game in the worst case. Furthermore, we propose a distributed algorithm to seek an ε-generalized Nash equilibrium, and show the convergence analysis with Lyapunov functions and variational inequalities. Finally, we illustrate the effectiveness of the distributed algorithm by a numerical example.