Abstract

A map \(f: \mathbb{R}^m\to\mathbb{R}^n\) is called scalarly \(K\)-quasiconvex (or *-quasiconvex, in the terminology of \textit{V. Jeyakumar}, \textit{W. Oettli} and \textit{M. Natividad} [J. Math. Anal. Appl. 179, No. 2, 537-546 (1993; Zbl 0791.46002)]), \(K\subset \mathbb{R}^n\) being a closed convex cone, if \(\eta^Tf\) is quasiconvex for all \(\eta\) belonging to the nonnegative polar cone of \(K\). The author introduces two quasimonotonicity concepts for set-valued maps \(F: \mathbb{R}^m\times \mathbb{R}^m\rightrightarrows \mathbb{R}^n\) and uses them to characterize the scalar \(K\)-quasiconvexity of a locally Lipschitz map in terms of its generalized Jacobian and other generalized differentials constructed by means of various tangent cones to the graph of \(f+K\). He also defines two corresponding monotonicity notions, by means of which the \(K\)-convexity of a locally Lipschitz map is characterized in terms of its generalized differentials.