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The generalized Nash equilibrium problem (GNEP) is a kind of game to find strategies for a group of players such that each player's objective function is optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs). In this paper, we propose a numerical method for finding GNEs of GNEPs of polynomials based on the polyhedral homotopy continuation and the Moment-SOS hierarchy of semidefinite relaxations. We show that our method can find all GNEs if they exist, or detect the nonexistence of GNEs, under some genericity assumptions. Some numerical experiments are made to demonstrate the efficiency of our method., Comment: 25 pages, Version to appear in Journal of Scientific Computing

This paper studies convex Generalized Nash Equilibrium Problems (GNEPs) that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing Generalized Nash Equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method., 29 pages