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

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 semicontinu...

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Bibliographic Details
Published in:Nonlinear analysis Vol. 132; pp. 240 - 273
Main Authors: Mordukhovich, Boris S., Sarabi, M. Ebrahim
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.02.2016
Subjects:
Coderivatives
Convexity
Differentiation
Equivalence
First-order and second-order subdifferentials
Mathematical analysis
Nonlinear and variational analysis
Nonlinearity
Normal cones
Optimization
Piecewise linear extended-real-valued functions
Stability
Sum rules
Normal cones
49J53
49J52
Piecewise linear extended-real-valued functions
58C20
Nonlinear and variational analysis
First-order and second-order subdifferentials
Coderivatives
ISSN:0362-546X, 1873-5215
Online Access:Get full text
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Summary: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.
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content type line 23
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2015.11.008