Characterization of Convexity for a Piecewise C2 Function by the Limiting Second-order Subdifferential

Abstract

We prove in this paper that a piecewise $C^2$ function $\varphi: \Bbb R^n\rightarrow\Bbb R$ is convex if and only if for every $(x,y)\in{\rm gph}\partial\varphi,$ the limiting second-order subdifferential mapping $\partial^2\varphi(x,y):\Bbb R^n\rightrightarrows\Bbb R^n$ has the so-called positive semi-definiteness (PSD) - in analogy with the notion of positive semi-definiteness of symmetric real matrices. As a by-product, characterization for strong convexity of $\varphi$ is established.