A Smoothing Newton Method For Generalized Nash Equilibrium Problems

The Generalized Nash Equilibrium Problem(GNEP),where both the players' objective functions and feasible sets depend on the other players' strategies,is obtained by extending the classical Nash Equilibrium Problem introduced by Nash.The GNEP is an important model that has its roots in economic sciences but is being widely used in many different fields. However,advancements on the algorithmic side have been rather scarce.This dissertation focuses on the GNEP with shared constraints,i.e.there may exist constraints shared by all players,and presents a smoothing Newton method for computing a variational equilibrium. With the help of the smoothing Fischer-Burmeister function,the corresponding Karush-Kuhn-Tucker condition satisfied by a variational equilibrium is transformed into a nonsmooth system of equations E(ε,y) =0.Under some conditions,the nonsingularity of the Jacobian of Ewhenε≠0 is proved,and also the Clarke generalized Jacobian of E whenε= 0.Then a smoothing Newton method is given to solve the obtained nonsmooth system.The method is global convergent with a local quadratic rate.Some numerical examples are presented to illustrate the performance of the method.