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 meanvalue theorems. This article aims to clarify the relationship between the two generalized derivatives. In particular, for scalarvalued 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 subdifferentialthe Rubinov subdifferentialis 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:

 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:
 AC0206CH11357
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Setvalued and Variational Analysis
 Additional Journal Information:
 Journal Volume: 25; Journal Issue: 2; Journal ID: ISSN 18770533
 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/s1122801603756.
Khan, Kamil A. Relating Lexicographic Smoothness and Directed Subdifferentiability. United States. doi:10.1007/s1122801603756.
Khan, Kamil A. Fri .
"Relating Lexicographic Smoothness and Directed Subdifferentiability". United States. doi:10.1007/s1122801603756. 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 meanvalue theorems. This article aims to clarify the relationship between the two generalized derivatives. In particular, for scalarvalued 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 subdifferentialthe Rubinov subdifferentialis 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/s1122801603756},
journal = {Setvalued and Variational Analysis},
number = 2,
volume = 25,
place = {United States},
year = {2016},
month = {6}
}
Web of Science