Relating Lexicographic Smoothness and Directed Subdifferentiability

Khan, Kamil A.

doi:10.1007/s11228-016-0375-6

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:

Khan, Kamil A. ^[1]

Argonne National Lab. (ANL), Argonne, IL (United States)

Publication Date:: Fri Jun 03 00:00:00 EDT 2016

Research Org.:: Argonne National Laboratory (ANL), Argonne, IL (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1373756

Grant/Contract Number:: AC02-06CH11357

Resource Type:: Journal Article: 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.

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                    Khan, Kamil A. Relating Lexicographic Smoothness and Directed Subdifferentiability.  United States.  https://doi.org/10.1007/s11228-016-0375-6

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                    Khan, Kamil A. 2016.  
        "Relating Lexicographic Smoothness and Directed Subdifferentiability".  United States.  https://doi.org/10.1007/s11228-016-0375-6.  https://www.osti.gov/servlets/purl/1373756.

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@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},

  url          = {https://www.osti.gov/biblio/1373756},
  journal      = {Set-valued and Variational Analysis},

  issn         = {1877-0533},

  number       = 2,

  volume       = 25,

  place        = {United States},

  year         = {Fri Jun 03 00:00:00 EDT 2016},

  month        = {Fri Jun 03 00:00:00 EDT 2016}

}

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Works referenced in this record:

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets
journal, January 2001

Baier, Robert; Farkhi, Elza M.
Set-Valued Analysis, Vol. 9, Issue 3, p. 217-245
https://doi.org/10.1023/A:1012046027626

Directed Subdifferentiable Functions and the Directed Subdifferential Without Delta-Convex Structure
journal, September 2013

Baier, Robert; Farkhi, Elza; Roshchina, Vera
Journal of Optimization Theory and Applications, Vol. 160, Issue 2
https://doi.org/10.1007/s10957-013-0401-x

Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets
journal, January 2001

Baier, Robert; Farkhi, Elza M.
Set-Valued Analysis, Vol. 9, Issue 3, p. 247-272
https://doi.org/10.1023/A:1012009529492

On Fréchet Subdifferentials
journal, January 2003

Kruger, A. Ya.
Journal of Mathematical Sciences, Vol. 116, Issue 3, p. 3325-3358
https://doi.org/10.1023/A:1023673105317

On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions
book, January 2010

Baier, Robert; Farkhi, Elza; Roshchina, Vera
Variational Analysis and Generalized Differentiation in Optimization and Control
https://doi.org/10.1007/978-1-4419-0437-9_3

Variational Analysis and Generalized Differentiation I
book, January 2006

Mordukhovich, Boris S.
Grundlehren der mathematischen Wissenschaften
https://doi.org/10.1007/3-540-31247-1

A vector forward mode of automatic differentiation for generalized derivative evaluation
journal, April 2015

Khan, Kamil A.; Barton, Paul I.
Optimization Methods and Software, Vol. 30, Issue 6
https://doi.org/10.1080/10556788.2015.1025400

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples
journal, February 2012

Baier, Robert; Farkhi, Elza; Roshchina, Vera
Nonlinear Analysis: Theory, Methods & Applications, Vol. 75, Issue 3
https://doi.org/10.1016/j.na.2011.04.074

Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides
journal, March 2014

Khan, Kamil A.; Barton, Paul I.
Journal of Optimization Theory and Applications, Vol. 163, Issue 2
https://doi.org/10.1007/s10957-014-0539-1

A minimal set-valued strong derivative for vector-valued Lipschitz functions
journal, December 1977

Sweetser, T. H.
Journal of Optimization Theory and Applications, Vol. 23, Issue 4
https://doi.org/10.1007/BF00933296

Nonlinear Analysis and Optimization II: Optimization
book, January 2010

Leizarowitz, Arie; Mordukhovich, Boris S.; Shafrir, Itai
Contemporary Mathematics, Vol. 514
https://doi.org/10.1090/conm/514/10098

Introduction to Piecewise Differentiable Equations
book, January 2012

Scholtes, Stefan
SpringerBriefs in Optimization
https://doi.org/10.1007/978-1-4614-4340-7

Lexicographic differentiation of nonsmooth functions
journal, July 2005

Nesterov, Yu.
Mathematical Programming, Vol. 104, Issue 2-3
https://doi.org/10.1007/s10107-005-0633-0

Works referencing / citing this record:

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

Barton, Paul I.; Khan, Kamil A.; Stechlinski, Peter
Optimization Methods and Software, Vol. 33, Issue 4-6
https://doi.org/10.1080/10556788.2017.1374385

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Similar Records

Title: Relating Lexicographic Smoothness and Directed Subdifferentiability

Abstract

Citation Formats

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets journal, January 2001

Directed Subdifferentiable Functions and the Directed Subdifferential Without Delta-Convex Structure journal, September 2013

Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets journal, January 2001

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Introduction to Piecewise Differentiable Equations book, January 2012

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journal, January 2001

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journal, September 2013

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A minimal set-valued strong derivative for vector-valued Lipschitz functions
journal, December 1977

Nonlinear Analysis and Optimization II: Optimization
book, January 2010

Introduction to Piecewise Differentiable Equations
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Lexicographic differentiation of nonsmooth functions
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Computationally relevant generalized derivatives: theory, evaluation and applications
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