New Necessary Optimality Conditions for Bilevel Programs by Combining the MPEC and Value Function Approaches

The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush–Kuhn–Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium constraints (MPEC). In general the classical approach is not valid for nonconvex bilevel programming problems. The value function approach uses the value function of the lower level problem to define an equivalent single level problem. But the resulting problem requires a strong assumption, such as the partial calmness condition, for the KKT condition to hold. In this paper we combine the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions. The required conditions are even weaker in the case where the classical approach or the value function approach alone is applicable.