Characterization of scalar quasiconvexity and convexity of locally lipschitz vector-valued maps

Abstract

This paper gives necessary and sufficient conditions for a locally Lipschitz vector-valued map f between Euclidean spaces R m and R n to be scalarly K-quasiconvex in the sense that, for any vector η from the nonnegative polar cone K + of the cone K, the function ηTƒ is uasiconvex where T denotes the transpose. Our criteria are written in terms of the quasimonotonicity notions of the generalized Jacobian of ƒ and the set-valued maps constructed from the Bouligand tangent cone, the intermediate (adjacent) tangent cone and the Clarke tangent cone of the graph of ƒ(·) + K. The case of K-convexity of ƒ is also considered by using suitable notions of K-monotonicity.