Characterization of quasiconvexity for locally lipschitz vector valued functions

Characterizations of Quasiconvexity for Locally Lipschitz Vector-Valued Functions J. BENOIST 1 Abstract. The aim of the paper is to characterize the locally Lipschitz vector- valued functions which are K-quasiconvex with respect to a closed convex cone K in the sense that the sublevel sets are convex. Our criteria are written in terms of a K-quasimonotonicity notion of the generalized directional derivative and of Clarke's generalized Jacobian. This work could be compared to Sach's one in which the author gives necessary and sufficient conditions for a locally Lipschitz map f between to Euclidean spaces to be scalarly K-quasiconvex in the sense that, for any continuous linear form of the nonnegative polar cone K+, the composite function ? f is quasiconvex. Key Words. Quasiconvexity, quasimonotonicity, scalarization, generalized di- rectional derivatives, generalized Jacobians. 2000 Mathematics Subject Classification. Primary 26B25; Secondary 90C29, 49J52. 1 Introduction During the last decade, a number of characterizations of quasiconvex real- valued functions have been proposed by using properties of generalized direc- 1Associate Professor, LACO, UMR 6090, University of Limoges, Limoges, France. 1

• lipschitz vector


• cone-quasiconvex


• convex subset


• valued function


• then


• let z ?


• let ?


• weak-star closed