The Second Order Optimalty Conditions For Nash Equilibrium Games

It is known that the non-cooperate multi-leader-follower games can be formulated as a player-convex generalized Nash equilibrium problem.Basing on the first orderoptimality conditions for Nash equilibrium problem,we proposed a second order necessary condition and a second order sufficient condition for the the generalized Nash equilibrium problemThe proposed conditions are then applied to derive the the second order necessary condition and the second order sufficient condition for the classical Nash equilibrium problem.The second-order optimality conditions on Nash equilibrium problemscan be used to demonstrate the convergence speed of the smoothing Newton methods.