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 $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}$ is convex if and only if for every $(x,y) \in {\rm gph} \partial\varphi$, the limiting second-order subdifferential mapping $\partial^2 \varphi(x,y): \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ 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.