Generalized Nash Equilibrium Problems with Quasi-linear Constraints

We study generalized Nash equilibrium problems (GNEPs) such that objectives are polynomial functions, and each player’s constraints are linear in their own strategy. For such GNEPs, the KKT sets can be represented as unions of simpler sets by Carathéodory’s theorem. We give a convenient representation for KKT sets using partial Lagrange multiplier expressions. This produces a set of branch polynomial optimization problems, which can be efficiently solved by Moment-SOS relaxations. By doing this, we can compute all generalized Nash equilibria or detect their nonexistence. This method may not be very scalable to large-scale GNEPs. Numerical experiments are provided to demonstrate the computational efficiency.