Generalized differentiation of piecewise linear functions in second-order variational analysis. (February 2016)

• Record Type: Journal Article
• Authors: Mordukhovich, Boris S.; Sarabi, M. Ebrahim
• Is Part Of: Nonlinear analysis. Volume 132(2016)
• Page Start: 240
• Page End: 273
• Publication Date: 2016-02
• Subjects: 49J52 -- 49J53 -- 58C20; Nonlinear and variational analysis -- Piecewise linear extended-real-valued functions -- Normal cones -- Coderivatives -- First-order and second-order subdifferentials; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.11.008
• Languages: English
• ISSNs: 0362-546X

Abstract

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 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.