Applicability of MPEC theory to the minmax complementarity reformulation

Determine whether mathematical-program-with-equilibrium-constraints theory, including constraint qualifications and S-, M-, and C-type necessary and sufficient optimality conditions, applies to the minmax complementarity reformulation (MM-CC) of pessimistic bilevel optimization.

Background

The paper reviews the reformulation (MM-CC), which transforms a pessimistic bilevel program into a mathematical program with complementarity constraints by representing the intermediate maximization problem through its optimality conditions. Although this reformulation is globally related to the pessimistic bilevel problem under the stated convexity and regularity assumptions, it contains difficult complementarity constraints and implicit variables.

The authors explicitly identify as unresolved whether the established MPEC framework can be used for (MM-CC) in the same way that it is used for KKT reformulations of optimistic bilevel programs. The open issue concerns both the applicability of MPEC-specific constraint qualifications and the derivation of S-, M-, and C-type stationarity and optimality conditions.

References

Additionally, it is unclear whether the MPEC theory w.r.t. constraint qualifications, as well as necessary and sufficient optimality conditions (in terms of S-, M-, and C-type constraint qualifications and stationarity concepts) can work as it is the case for the KKT reformulation for the optimistic bilevel optimization problem; see, e.g., [21,28,27].

— A true single-level reformulation for pessimistic bilevel optimization  (2609.38014 - Stein et al., 29 Sep 2026) in Section 2, page 6