Optimality conditions for the simple convex bilevel programming problem in Banach spaces

The simple convex bilevel programming problem is a convex minimization problem whose feasible set is the solution set of another convex optimization problem. Such problems appear frequently when searching for the projection of a certain point onto the solution set of another program. Due to the nature of the problem, Slater's constraint qualification generally fails to hold at any feasible point. Hence, one has to formulate weaker constraint qualifications or stationarity notions in order to state optimality conditions. In this paper, we use two different single-level reformulations of the problem, the optimal value and the Karush-Kuhn-Tucker approach, to derive optimality conditions for the original program. Since all these considerations are carried out in Banach spaces, the results are not limited to standard optimization problems in R-n. On the road, we introduce and discuss a certain concept of M-stationarity for mathematical programs with complementarity constraints in Banach spaces.