Computing Locally Optimal Solutions of the Bilevel Optimization Problem Using the KKT Approach

Abstract

If the lower-level problem in a bilevel optimization problem is replaced by its Karush-Kuhn-Tucker conditions, a mathematical program with complementarity constraints is obtained. Solving this nonconvex optimization problem, locally optimal solutions are computed which do in general not correspond to locally optimal solutions of the bilevel problem. Using a relaxation of this problem in two constraints it can be shown that a sequence of locally optimal solutions of the relaxed problems converges to a point which is related to a locally optimal solution of the bilevel optimization problem. If the lower-level problem is a linear one, relaxation of only the complementarity constraint is sufficient.