Optimality conditions for bilevel programmes via Moreau envelope reformulation

For bilevel programmes with a convex lower-level programme, the classical approach replaces the lower-level programme with its Karush-Kuhn-Tucker condition and solve the resulting mathematical programme with complementarity constraint (MPCC). It is known that when the set of lower-level multipliers is not unique, MPCC may not be equivalent to the original bilevel problem, and many MPCC-tailored constraint qualifications do not hold. In this paper, we study bilevel programmes where the lower level is generalized convex. Applying the equivalent reformulation via Moreau envelope, we derive new directional optimality conditions. Even in the nondirectional case, the new optimality condition is stronger than the strong stationarity for the corresponding MPCC.