Semivectorial Bilevel Optimization on Riemannian Manifolds

In this paper we deal with the semivectorial bilevel problem in the Riemannian setting. The upper level is a scalar optimization problem to be solved by the leader, and the lower level is a multiobjective optimization problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing among Pareto solutions with respect to a given ordering cone. For the so-called “optimistic problem”, when the followers choice among their best responses is the most favorable for the leader, we give optimality conditions. Also for the so-called “pessimistic problem”, when there is no cooperation between the leader and the followers, and the followers choice may be the worst for the leader, we present an existence result.