Approximation approach of the stackelberg problem

Abstract

Abstract We consider the following Stackelberg Problem,two-level optimization problem, in which the player 1,called the leader, has all information about the objective function and the . constraints of player 2 whereas the later one knows nothing* but the strategy announced by player 1 : S min f 1 x, y ˜; x x ( X g 1 x, y ˜; x ≤ 0 where y ˜; x is a solution of the parametric lower level problem : min f 2 x, y y ( Y g 2 x, y ≤ 0 i,e, a solution of the stackelberg problem is an optimal strategy choosen by the leader when player 2 reacts by playing optimally. We present a general approach for approximating the Stackelberg problem by a sequence of two-level optimization problems and under appropriate assumptions we give some convergence results. Thus we particularize our approach by considering some methods using optimization technics as barrier and exterior penalty methods and we show that such methods satisfy the conditions for the convergence of our general approach. Moreover we consider the applications to control theory and to the team problems with parametric uncertainty in which the cost functionals are identical in form but explicitly dipendant on a vector parameter α and any player may have a different perception of the value of α.