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Relating Lexicographic Smoothness and Directed Subdifferentiability

Journal Article · · Set-valued and Variational Analysis
 [1]
  1. Argonne National Lab. (ANL), Argonne, IL (United States)
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Lexicographic derivatives developed by Nesterov and directed subdifferentials developed by Baier, Farkhi, and Roshchina are both essentially nonconvex generalized derivatives for nonsmooth nonconvex functions and satisfy strict calculus rules and mean-value theorems. This article aims to clarify the relationship between the two generalized derivatives. In particular, for scalar-valued functions that are locally Lipschitz continuous, lexicographic smoothness and directed subdifferentiability are shown to be equivalent, along with the necessary optimality conditions corresponding to each. For such functions, the visualization of the directed subdifferential-the Rubinov subdifferential-is shown to include the lexicographic subdifferential, and is also shown to be included in its closed convex hull. As a result, various implications of these results are discussed.

Research Organization:
Argonne National Lab. (ANL), Argonne, IL (United States)
Sponsoring Organization:
USDOE Office of Science (SC)
Grant/Contract Number:
AC02-06CH11357
OSTI ID:
1373756
Journal Information:
Set-valued and Variational Analysis, Journal Name: Set-valued and Variational Analysis Journal Issue: 2 Vol. 25; ISSN 1877-0533
Publisher:
SpringerCopyright Statement
Country of Publication:
United States
Language:
English

References (14)

Variational Analysis and Generalized Differentiation I book January 2006
On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions book January 2010
Introduction to Piecewise Differentiable Equations book January 2012
A minimal set-valued strong derivative for vector-valued Lipschitz functions journal December 1977
Lexicographic differentiation of nonsmooth functions journal July 2005
Directed Subdifferentiable Functions and the Directed Subdifferential Without Delta-Convex Structure journal September 2013
Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides journal March 2014
The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples journal February 2012
Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets journal January 2001
Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets journal January 2001
On Fréchet Subdifferentials journal January 2003
A vector forward mode of automatic differentiation for generalized derivative evaluation journal April 2015
Nonlinear Analysis and Optimization II: Optimization book January 2010
Optimization and Nonsmooth Analysis book January 1990

Cited By (1)

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Related Subjects

96 KNOWLEDGE MANAGEMENT AND PRESERVATION
97 MATHEMATICS AND COMPUTING
Directed subdifferential
Directional derivative Nonsmooth analysis
Generalized differential calculus
Lexicographic derivative