부산대학교 도서관

In recent years, optimization theories and algorithms on Riemannian manifolds have been extensively studied by many scholars (see [G. de~Carvalho~Bento and J. da~Cruz~Neto, J. Optim. Theory Appl. {\bf 159} (2013), no.~1, 125--137; MR3103290; H. Bonnel, L. Todjihoundé and C.~N. Udrişte, J. Optim. Theory Appl. {\bf 167} (2015), no.~2, 464--486; MR3412446; S. Dempe and J. Dutta, Math. Program. {\bf 131} (2012), no.~1-2, Ser. A, 37--48; MR2886139; Y.~S. Ledyaev and Q.~J. Zhu, Trans. Amer. Math. Soc. {\bf 359} (2007), no.~8, 3687--3732; MR2302512] and the references therein). \par As we know, bilevel programming problems are hierarchical optimization problems combining decisions of two decision makers, the so-called leader and follower. An approach to studying bilevel optimization problems is to reformulate the lower level problem by using Karush-Kuhn-Tucker conditions, which replace the bilevel optimization problem by a so-called mathematical program with complementarity constraints (MPCC). \par In this paper, the authors present a Karush-Kuhn-Tucker reformulation of the bilevel optimization on Riemannian manifolds and show that the equivalence between the global or local optimal solutions of a bilevel problem and the corresponding MPCC under the lower level convex problem satisfies Slater's constraint qualification. In the general case, $M$- and $C$-type optimality conditions for the bilevel problem on Riemannian manifolds are given. \par The results in this paper generalize the corresponding results (see, for example, [S. Dempe and J. Dutta, op.~cit.]) from Euclidean spaces to Riemannian manifolds, and the subject is worth studying further.