Equilibrium Computation of Generalized Nash Games: A New Lagrangian-Based Approach

Abstract

This paper presents a new primal-dual method for computing an equilibrium of generalized (continuous) Nash game (referred to as generalized Nash equilibrium problem (GNEP)) where each player's feasible strategy set depends on the other players' strategies. The method is based on a new form of Lagrangian function with a quadratic approximation. First, we reformulate a GNEP as a saddle point computation problem using the new Lagrangian and establish equivalence between a saddle point of the Lagrangian and an equilibrium of the GNEP. We then propose a simple algorithm that is convergent to the saddle point. Furthermore, we establish global convergence by assuming that the Lagrangian function satisfies the Kurdyka-{\L}ojasiewicz property. A distinctive feature of our analysis is to make use of the new Lagrangian as a potential function to guide the iterate convergence. Our method has two novel features over existing approaches: (i) it requires neither boundedness assumptions on the strategy set and the set of multipliers of each player, nor any boundedness assumptions on the iterates generated by the algorithm; (ii) it leads to a Jacobi-type decomposition scheme, which, to the best of our knowledge, is the first development of a distributed algorithm to solve a general class of GNEPs. Numerical experiments are performed on benchmark test problems and the results demonstrate the effectiveness of the proposed method.