Generalized differentiation of piecewise linear functions in second-order variational analysis

TL;DR

This paper provides a comprehensive second-order analysis of convex piecewise linear functions, deriving explicit formulas for their generalized Hessians and establishing a new second-order sum rule in nonsmooth analysis.

Contribution

It introduces explicit formulas for second-order subdifferentials of convex piecewise linear functions and proves a novel exact sum rule in second-order variational analysis.

Findings

Derived formulas for second-order subdifferentials of convex piecewise linear functions.

Established a new exact second-order sum rule for nonsmooth convex functions.

Enhanced understanding of second-order properties in variational analysis.

Abstract

The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to optimization and stability. This class consists of lower semicontinuous functions with possibly infinite values on finite-dimensional spaces, which are labeled as piecewise linear ones and can be equivalently described via the convexity of their epigraphs. In this the paper we calculate the second-order subdifferentials (generalized Hessians) of arbitrary convex piecewise linear functions, together with the corresponding geometric objects, entirely in terms of their initial data. The obtained formulas allow us, in particular, to justify a new exact (equality-type) second-order sum rule for such functions in the general nonsmooth setting.