A Unified Approach to Convex and Convexified Generalized Differentiation of Nonsmooth Functions and Set-Valued Mappings

The primary goal of this paper is to present a unified approach and shed new light on convex and Clarke generalized differentiation theories using the concepts and techniques from Mordukhovich’s developments. We show that the concepts and techniques used by Mordukhovich are important, not only to hi...

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Published in:Vietnam journal of mathematics Vol. 42; no. 4; pp. 479 - 497
Main Authors: Nam, Nguyen Mau, Hoang, Nguyen Dinh, Rector, R. Blake
Format: Journal Article
Language:English
Published: Singapore Springer Singapore 01.12.2014
Subjects:
Mathematics
Mathematics and Statistics
90C31
Subgradient
49J53
49J52
Subdifferential
Coderivative
Generalized differentiation
ISSN:2305-221X, 2305-2228
Online Access:Get full text
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Abstract The primary goal of this paper is to present a unified approach and shed new light on convex and Clarke generalized differentiation theories using the concepts and techniques from Mordukhovich’s developments. We show that the concepts and techniques used by Mordukhovich are important, not only to his generalized differentiation theory itself, but also to many other aspects of nonsmooth analysis. In particular, they can be used to derive convex subdifferential calculus rules as well as many important calculus rules of Clarke subdifferentials.
AbstractList The primary goal of this paper is to present a unified approach and shed new light on convex and Clarke generalized differentiation theories using the concepts and techniques from Mordukhovich’s developments. We show that the concepts and techniques used by Mordukhovich are important, not only to his generalized differentiation theory itself, but also to many other aspects of nonsmooth analysis. In particular, they can be used to derive convex subdifferential calculus rules as well as many important calculus rules of Clarke subdifferentials.
Author Rector, R. Blake
Nam, Nguyen Mau
Hoang, Nguyen Dinh
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  organization: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University
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Keywords 90C31
Subgradient
49J53
49J52
Subdifferential
Coderivative
Generalized differentiation
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References Hiriart-UrrutyJ-BLemaréchalCConvex Analysis and Minimization Algorithms. I. Fundamentals1993BerlinSpringer
ClarkeFHOptimization and Nonsmooth Analysis1983New YorkWiley0582.49001
MordukhovichBSVariational Analysis and Generalized Differentiation, I: Basic Theory2006BerlinSpringer
MordukhovichBSNamNMPhanHVariational analysis of marginal functions with applications to bilevel programmingJ. Optim. Theory Appl.201215255758610.1007/s10957-011-9940-128863601268.90127
RockafellarRTDirectionally Lipschitzian functions and subdifferential calculusProc. Lond. Math. Soc.19793933135510.1112/plms/s3-39.2.3315489830413.49015
Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Morgan and Claypool Publishers (2006)
J-B Hiriart-Urruty (73_CR2) 1993
RT Rockafellar (73_CR6) 1979; 39
BS Mordukhovich (73_CR3) 2006
BS Mordukhovich (73_CR5) 2012; 152
FH Clarke (73_CR1) 1983
73_CR4
References_xml – reference: ClarkeFHOptimization and Nonsmooth Analysis1983New YorkWiley0582.49001
– reference: Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Morgan and Claypool Publishers (2006)
– reference: MordukhovichBSVariational Analysis and Generalized Differentiation, I: Basic Theory2006BerlinSpringer
– reference: MordukhovichBSNamNMPhanHVariational analysis of marginal functions with applications to bilevel programmingJ. Optim. Theory Appl.201215255758610.1007/s10957-011-9940-128863601268.90127
– reference: RockafellarRTDirectionally Lipschitzian functions and subdifferential calculusProc. Lond. Math. Soc.19793933135510.1112/plms/s3-39.2.3315489830413.49015
– reference: Hiriart-UrrutyJ-BLemaréchalCConvex Analysis and Minimization Algorithms. I. Fundamentals1993BerlinSpringer
– ident: 73_CR4
– volume-title: Convex Analysis and Minimization Algorithms. I. Fundamentals
  year: 1993
  ident: 73_CR2
  doi: 10.1007/978-3-662-02796-7
– volume: 39
  start-page: 331
  year: 1979
  ident: 73_CR6
  publication-title: Proc. Lond. Math. Soc.
  doi: 10.1112/plms/s3-39.2.331
– volume: 152
  start-page: 557
  year: 2012
  ident: 73_CR5
  publication-title: J. Optim. Theory Appl.
  doi: 10.1007/s10957-011-9940-1
– volume-title: Optimization and Nonsmooth Analysis
  year: 1983
  ident: 73_CR1
– volume-title: Variational Analysis and Generalized Differentiation, I: Basic Theory
  year: 2006
  ident: 73_CR3
  doi: 10.1007/3-540-31247-1
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Title A Unified Approach to Convex and Convexified Generalized Differentiation of Nonsmooth Functions and Set-Valued Mappings
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