On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problems on Riemannian manifolds

Abstract

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). First, we present KKT reformulation of the bilevel optimization problems on Riemannian manifolds. Moreover, we show that global optimal solutions of theMPCCcorrespond to global optimal solutions of the bilevel problem on the Riemannian manifolds provided the lower level convex problem satisfies the Slater?s constraint qualification. But the relationship between the local solutions of the bilevel problem and its corresponding MPCC is incomplete equivalent. We then also show by examples that these correspondences can fail if the Slater?s constraint qualification fails to hold at lower-level convex problem. In addition,M- and C-type optimality conditions for the bilevel problem on Riemannian manifolds are given.