Vietnam Journal of Mathematics

, Volume 42, Issue 4, pp 479–497 | Cite as

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

  • Nguyen Mau Nam
  • Nguyen Dinh Hoang
  • R. Blake Rector


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.


Subgradient Subdifferential Coderivative Generalized differentiation 

Mathematics Subject Classifications (2010)

49J53 49J52 90C31 


  1. 1.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  2. 2.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. I. Fundamentals. Springer, Berlin (1993)Google Scholar
  3. 3.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)Google Scholar
  4. 4.
    Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Morgan and Claypool Publishers (2006)Google Scholar
  5. 5.
    Mordukhovich, B.S., Nam, N.M., Phan, H.: Variational analysis of marginal functions with applications to bilevel programming. J. Optim. Theory Appl. 152, 557–586 (2012)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Rockafellar, R.T.: Directionally Lipschitzian functions and subdifferential calculus. Proc. Lond. Math. Soc. 39, 331–355 (1979)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014

Authors and Affiliations

  • Nguyen Mau Nam
    • 1
  • Nguyen Dinh Hoang
    • 2
  • R. Blake Rector
    • 1
  1. 1.Fariborz Maseeh Department of Mathematics and StatisticsPortland State UniversityPortlandUSA
  2. 2.Tan Tao UniversityLong An ProvinceVietnam

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