Optimality results for a specific bilevel optimization problem

We develop necessary optimality conditions using techniques from variational analysis for a bilevel optimization problem without assuming the lower-level problem satisfying neither the Mangasarian Fromovitz constraint qualification nor the partial calmness condition [D. L. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems, Optimization 33 (1995), pp. 9-27; J.J. Ye and J.J. Zhu, A note on optimality conditions for bilevel programming problems, Optimization 39 (1997), pp. 361-366]. Reducing the problem into a one-level nonlinear and nonsmooth programme, we use the approximate extremal principle [B. S. Mordukhovich, Variational Analysis and Generalized Differentiation: I. Basic Theory, Grundlehren Series, Fundamental Principles of Mathematical Sciences, Vol. 330, Springer, Berlin, 2006; B. S. Mordukhovich, Variational Analysis and Generalized Differentiation: II. Applications, Grundlehren Series, Fundamental Principles of Mathematical Sciences, Vol. 331, Springer, Berlin, 2006] to get fuzzy and exact optimality conditions.