Directional Necessary Optimality Conditions for Bilevel Programs

The bilevel program is an optimization problem in which the constraint involves solutions to a parametric problem. It well known that value function reformulation provides equivalent single-level problem, but it results nonsmooth never satisfies usual qualification, such as Mangasarian–Fromovitz qualification (MFCQ). In this paper, we show even first order sufficient condition for metric subregularity (which is, general, weaker than MFCQ) fails at each feasible point of program. We introduce concept directional calmness and that, under condition, necessary optimality holds. Although sharper nondirectional one, classical and, hence, more likely hold. perform sensitivity analysis propose quasi-normality calmness. An example given may hold