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In this paper, following the optimistic approach in bilevel programming, we investigate necessary and sufficient optimality conditions for a bilevel optimization problem. Our approach consists of using the optimal value reformulation together with an exact penalization technique. Unlike Dempe and Pilecka (Journal of Global Optimization 61: 769–788, 2015), we reach our goal without resorting to convexificators; the reason is that we do not assume that the sets of all continuity directions are convex or closed. The obtained results are given in terms of directional upper semi-regular convexificators. Some examples are given to illustrate our results.

Applying directional upper semi-regular convexificators in bilevel optimization