Is bilevel programming a special case of a mathematical program with complementarity constraints?
Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is "No" even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater's constraint qualification. We also show by examples that this correspondence can fail if the Slater's constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples.
bilevel programming mathematical programs with complementarity constraints optimality conditions local and global optimum