Advertisement

Future Challenges for Variational Analysis

  • Jonathan M. BorweinEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 47)

Abstract

Modern nonsmooth analysis is now roughly 35years old. In this chapter, I shall attempt to analyse (briefly): where the subject stands today, where it should be going, and what it will take to get there? In summary, the conclusion is that the first-order theory is rather impressive, as are many applications. The second-order theory is by comparison somewhat underdeveloped and wanting of further advance.

Keywords

Banach Space Nonexpansive Function Lipschitz Function Separable Hilbert Space Convex Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexandrov AD (1939) Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Leningrad State University, Annals, Mathematical Series 6:3–35Google Scholar
  2. 2.
    Bauschke HH, Borwein JM, and Combettes PL (2001) Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Communications in Contemporary Mathematics 3:615–647zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Borwein JM (1986) Stability and regular points of inequality systems. Journal of Optimization Theory and Applications 48:9–52zbMATHMathSciNetGoogle Scholar
  4. 4.
    Borwein JM (2005) The SIAM 100 digits challenge. Extended review in The Mathematical Intelligencer 27:40–48Google Scholar
  5. 5.
    Borwein JM (2007) Proximality and Chebyshev sets. Optimization Letters 1:21–32zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Borwein JM and Fitzpatrick S (1989) Existence of nearest points in Banach spaces. Canadian Journal of Mathematics 61:702–720MathSciNetGoogle Scholar
  7. 7.
    Borwein JM and Ioffe A (1996) Proximal analysis in smooth spaces. Set-Valued Analysis 4:1–24zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Borwein JM and Lewis AS (2005) Convex Analysis and Nonlinear Optimization. Theory and Examples, 2nd Edn. Canadian Mathematical Society – Springer, BerlinGoogle Scholar
  9. 9.
    Borwein JM and Noll D (1994) Second order differentiability of convex functions in Banach spaces. Transactions of the American Mathematical Society 342:43–82zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Borwein JM and Sciffer S (2010) An explicit non-expansive function whose subdifferential is the entire dual ball. Contemporary Mathematics 514:99–103Google Scholar
  11. 11.
    Borwein JM and Vanderwerff JM (2010) Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press, CambridgezbMATHGoogle Scholar
  12. 12.
    Borwein JM and Zhu Q (1996) Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity. SIAM Journal of Optimization 34:1568–1591zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Borwein JM and Zhu Q (2005) Techniques of Variational Analysis. Canadian Mathematical Society – Springer, BerlinGoogle Scholar
  14. 14.
    Boyd S and Vandenberghe L (2004) Convex Optimization. Cambridge University Press, CambridgezbMATHGoogle Scholar
  15. 15.
    Clarke FH (1983) Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New YorkzbMATHGoogle Scholar
  16. 16.
    Crandall M, Ishii H, and Lions P-L (1992) User’s guide to viscosity solutions of second-order partial differential equations. Bulletin of the American Mathematical Society 27:1–67zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Dontchev AL and Rockafellar RT (2009) Implicit Functions and Solution Mappings: A View from Variational Analysis. Springer Monographs in Mathematics. Springer, New YorkzbMATHCrossRefGoogle Scholar
  18. 18.
    Dutta J (2005) Generalized derivatives and nonsmooth optimization, a finite dimensional tour. With discussions and a rejoinder by the author. TOP 11:85–314Google Scholar
  19. 19.
    Eberhard AC and Wenczel R (2007) On the calculus of limiting subhessians. Set-Valued Analysis 15:377–424zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ioffe AD and Penot J-P (1997) Limiting subhessians, limiting subjets and their calculus. Transactions of the American Mathematical Society 349:789–807zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Jeyakumar V and Luc DT (2008) Nonsmooth Vector Functions and Continuous Optimization. Springer Optimization and Its Applications, vol.10. Springer, BerlinGoogle Scholar
  22. 22.
    Lewis AS and Sendov HS (2005) Nonsmooth analysis of singular values, I & II. Set-Valued Analysis 13:213–264zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Macklem M (2009) Parallel Continuous Optimization. Doctoral Thesis, Dalhousie UniversityGoogle Scholar
  24. 24.
    Mordukhovich BS (2006) Variational Analysis & Generalized Differentiation I. Basic Theory. Springer Series Fundamental Principles of Mathematical Sciences, vol.330. Springer, BerlinGoogle Scholar
  25. 25.
    Mordukhovich BS (2006) Variational Analysis & Generalized Differentiation II. Applications. Springer Series Fundamental Principles of Mathematical Sciences, vol.331. Springer, BerlinGoogle Scholar
  26. 26.
    Moussaoui M and Seeger A (1999) Second-order subgradients of convex integral functionals. Transactions of the American Mathematical Society 351:3687–3711zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Pshenichnii B (1971) Necessary Conditions for an Extremum. Marcel Dekker, New YorkGoogle Scholar
  28. 28.
    Rockafellar RT and Wets RJ-B (1998) Variational Analysis. A Series of Comprehensive Studies in Mathematics, vol.317. Springer, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.CARMA, University of NewcastleNewcastleAustralia

Personalised recommendations