Characterizing Convexity of a Function by Its Fréchet and Limiting Second-Order Subdifferentials

Abstract

The Frechet and limiting second-order subdifferentials of a proper lower semicontinuous convex function \(\varphi: \mathbb R^n\rightarrow\bar{\mathbb R}\) have a property called the positive semi-definiteness (PSD)—in analogy with the notion of positive semi-definiteness of symmetric real matrices. In general, the PSD is insufficient for ensuring the convexity of an arbitrary lower semicontinuous function φ. However, if φ is a C1,1 function then the PSD property of one of the second-order subdifferentials is a complete characterization of the convexity of φ. The same assertion is valid for C1 functions of one variable. The limiting second-order subdifferential can recognize the convexity/nonconvexity of piecewise linear functions and of separable piecewise C2 functions, while its Frechet counterpart cannot.