Directional first order approach for a class of bilevel programs

In this paper, we study a class of bilevel optimization program, where the feasible set of the lower level program is independent of the upper level variable. For bilevel programs it is known that the first order reformulation of a bilevel program requires the convexity of the lower level program while reformulations involving the value function result in difficult optimization problems. In this paper we propose a directional first order approach which does not require convexity of the lower level program. First, we propose some conditions under which the lower level program can be equivalently characterized by its first order condition over a directional neighborhood around the local optimal condition. Next we give some conditions under which the classical first order optimality condition in the form of M-stationary condition still holds as a necessary optimality condition for the first order reformulation of the bilevel program even when the lower level program is nonconvex.