First-order necessary optimality conditions for general bilevel programming problems

We formulate in this paper several versions of the necessary conditions for general bilevel programming problems. The technique used is related to standard methods of nonsmooth analysis. We treat separately the following cases: Lipschitz case, differentiable case, and convex case. Many typical examples are given to show the efficiency of theoretical results. In the last section, we formulate the general multilevel programming problem and give necessary conditions of optimality in the general case. We illustrate then the application of these conditions by an example.