Skip to main content
SpringerLink
Log in

Variational Methods in Convex Analysis

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth subdifferential theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.M. Borwein (1981) Convex relations in analysis and optimization S. Schaible W.T. Ziemba (Eds) Generalized Concavity in Optimization and Economics Academic Press New York

    Google Scholar 

  2. J.M. Borwein (1986) ArticleTitleStability and regular points of inequality systems Jornal of Optimization Theory and Applications 48 9–52

    Google Scholar 

  3. Borwein, J.M. (2006), Maximal monotonicity via convex analysis, to appear in PAMS.

  4. Borwein, J.M. (2006), Maximal monotonicity via convex analysis, J. Convex Analysis, in press.

  5. J.M. Borwein A.S. Lewis (2000) Convex Analysis and Nonlinear Optimization: Theory and Examples CMS Springer-Verlag Books, Springer-Verlag New York

    Google Scholar 

  6. J.M. Borwein Q.J. Zhu (1999) ArticleTitleA survey of subdifferential calculus with applications Nonlinear Analysis, TMA 38 687–773 Occurrence Handle10.1016/S0362-546X(98)00142-4

    Article  Google Scholar 

  7. J.M. Borwein Q.J. Zhu (2005) Techniques of Variational Analysis CMS Springer-Verlag Books, Springer-Verlag New York

    Google Scholar 

  8. F.H Clarke Yu.S. Ledyaev (1994) ArticleTitleMean value inequalities Proc. Amer, Math. Soc 122 1075–1083 Occurrence Handle10.2307/2161175

    Article  Google Scholar 

  9. I. Ekeland (1974) ArticleTitleOn the variational principle J. Math. Anal. Appl 47 324–353 Occurrence Handle10.1016/0022-247X(74)90025-0

    Article  Google Scholar 

  10. Fitzpatrick, S. (1988), Representing monotone operators by convex functions. Work-shop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, pp. 59–65.

  11. G.J.O. Jameson (1974) Topology and Normed Spaces Chapman and Hall New York

    Google Scholar 

  12. Yu. S. Ledyaev Q.J. Zhu (1999) ArticleTitleImplicit multifunction theorems Set-Valued Analysis 7 209–238 Occurrence Handle10.1023/A:1008775413250

    Article  Google Scholar 

  13. B. Pshenichnii (1971) Necessary Conditions for an Extremum Marcel Dekker New York

    Google Scholar 

  14. Reich, S. and Simons, S. (2004), Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem, preprint.

  15. R.T. Rockafellar (1970) Convex Analysis Princeton University Press Princeton, NJ

    Google Scholar 

  16. R.T. Rockafellar (1970) ArticleTitleOn the maximality of sums of nonlinear monotone operators Trans. Amer. Math, Soc 149 75–88 Occurrence Handle10.2307/1995660

    Article  Google Scholar 

  17. S. Simons (1998) Minimax and Monotonicity Springer-Verlag Berlin

    Google Scholar 

  18. Simons, S., (2003), “A new version of the Hahn-Banach theorem,” www.math.ucsb. edu/simons/preprints/.

  19. S. Simons C.A. Zalinescu (2004) ArticleTitleNew Proof for Rockafellar’s Characterization of Maximal Monotonicity Proc Amer. Math. Soc 132 2969–2972 Occurrence Handle10.1090/S0002-9939-04-07462-3

    Article  Google Scholar 

  20. Q.J. Zhu (1998) ArticleTitleThe equivalence of several basic theorems for subdifferentials Set- Valued Analysis 6 171–185 Occurrence Handle10.1023/A:1008656207314

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computer Science, Dalhousie University, Halifax, N. S., Canada, B3H 1W5

    Jonathan M. Borwein

  2. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008, USA

    Qiji J. Zhu

Authors
  1. Jonathan M. Borwein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiji J. Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan M. Borwein.

Additional information

Research was supported by NSERC and by the Canada Research Chair Program and National Science Foundation under grant DMS 0102496.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borwein, J.M., Zhu, Q.J. Variational Methods in Convex Analysis. J Glob Optim 35, 197–213 (2006). https://doi.org/10.1007/s10898-005-3835-3

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-005-3835-3

Keywords

Search

Navigation