  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.
