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Title: Relating Lexicographic Smoothness and Directed Subdifferentiability

Abstract

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.

Authors:
 [1]
  1. Argonne National Lab. (ANL), Argonne, IL (United States)
Publication Date:
Research Org.:
Argonne National Lab. (ANL), Argonne, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1373756
Grant/Contract Number:  
AC02-06CH11357
Resource Type:
Accepted Manuscript
Journal Name:
Set-valued and Variational Analysis
Additional Journal Information:
Journal Volume: 25; Journal Issue: 2; Journal ID: ISSN 1877-0533
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 96 KNOWLEDGE MANAGEMENT AND PRESERVATION; Generalized differential calculus; Lexicographic derivative; Directed subdifferential; Directional derivative Nonsmooth analysis

Citation Formats

Khan, Kamil A. Relating Lexicographic Smoothness and Directed Subdifferentiability. United States: N. p., 2016. Web. doi:10.1007/s11228-016-0375-6.
Khan, Kamil A. Relating Lexicographic Smoothness and Directed Subdifferentiability. United States. https://doi.org/10.1007/s11228-016-0375-6
Khan, Kamil A. Fri . "Relating Lexicographic Smoothness and Directed Subdifferentiability". United States. https://doi.org/10.1007/s11228-016-0375-6. https://www.osti.gov/servlets/purl/1373756.
@article{osti_1373756,
title = {Relating Lexicographic Smoothness and Directed Subdifferentiability},
author = {Khan, Kamil A.},
abstractNote = {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.},
doi = {10.1007/s11228-016-0375-6},
journal = {Set-valued and Variational Analysis},
number = 2,
volume = 25,
place = {United States},
year = {2016},
month = {6}
}

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Works referenced in this record:

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Works referencing / citing this record:

Computationally relevant generalized derivatives: theory, evaluation and applications
journal, September 2017