Set-Valued Analysis

, Volume 16, Issue 2–3, pp 157–184 | Cite as

Second Order Cones for Maximal Monotone Operators via Representative Functions

  • A. C. EberhardEmail author
  • J. M. Borwein


It is shown that various first and second order derivatives of the Fitzpatrick and Penot representative functions for a maximal monotone operator T, in a reflexive Banach space, can be used to represent differential information associated with the tangent and normal cones to the Graph T. In particular we obtain formula for the proto-derivative, as well as its polar, the normal cone to the graph of T. First order derivatives are shown to be useful in recognising points of single-valuedness of T. We show that a strong form of proto-differentiability to the graph of T, is often associated with single valuedness of T.


Second order cones Maximal monotone operators Proto-differentiability 

Mathematics Subject Classifications (2000)

47H05 46N10 47H04 46A20 49J53 


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  1. 1.
    Attouch, H., Wets, R.J.-B.: A convergence for bivariate functions aimed at the convergence of saddle values. In: Cecconi, J.P., Zolezzi, T. (eds.) Lecture Notes in Mathematics, Mathematical Theories of Optimization, vol. 979, pp. 1–42 (1983)Google Scholar
  2. 2.
    Beer, G.: Topologies on Closed and Closed Convex Sets. Mathematics and its Applications, vol. 268. Kluwer Acad. Publ. (1993)Google Scholar
  3. 3.
    Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13(3/4), 561–586 (2006)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Borwein, J.M.: Maximality of sums of two maximal monotone operators in general Banach space. Proc. Amer. Math. Soc. 135, 3917–3924 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fitzpatrick, S.: Representing monotone operators by convex functions. Workshop/Miniconfe rence on Functional Analysis and Optimization (Camberra 1988), Proceedings Center Math. Anal. Austral. Nat. Univ., 20 Austral. Nat Univ. Canberra, pp. 59–65 (1988)Google Scholar
  6. 6.
    Bauschke, H.H., McLaren, D.A., Sendov, H.S.: Fitzpatrick functions: inequalities, examples, and remarks on a problem by S. Fitzpatrick. J. Convex Anal. 13(3–4), 499–523 (1996)MathSciNetGoogle Scholar
  7. 7.
    Holmes, R.B.: Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics, vol. 24. Springer-Verlag (1975)Google Scholar
  8. 8.
    Krauss, E.: A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions. Nonlinear Anal. 9(12), 1381–1399 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Mordukhovich, B.S.: Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings. Dokl. Akad. Nauk BSSR 28, 976–979 (1984)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Comprehensive Studies in Mathematics, vols. 330, 331. Springer (2005)Google Scholar
  11. 11.
    Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mordukhovich, B.S., Outrata, J.V.: On second-order subdifferentials and their applications. SIAM J. Optim. 12, 139–169 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chi Ngoc Do: Generalized second-order derivatives of a convex functions in a reflexive Banach space. Trans. Amer. Math. Soc. 334(1), 281–301 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Penot, J-P., Zălinescu, C.: Continuity of the Legendre–Fenchel transformation for some variational convergences. Optimization 53(5–6), 549–562 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Penot, J-P., Zălinescu, C.: On the convergence of maximal monotone operators. Proc. Amer. Math. Soc. 134(7), 1937–1946 (2005)CrossRefGoogle Scholar
  16. 16.
    Penot, J-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58(7–8), 855–871 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Penot, J-P., Zălinescu, C.: Some problems about the representation of monotone operators by convex functions. ANZIAM J. 47(1), 1–20 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Phelps, R.R.: Convex Functions Montone Operators and Differentiability. Lecture Notes in Mathematics, no. 1364. Springer-Verlag (1993)Google Scholar
  19. 19.
    Singer, I.: A Fenchel–Rockafellar type duality theorem for maximization. Bull. Austral. Math. Soc. 20(2), 193–198 (1979)zbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of Mathematical and Geospatial SciencesRMIT UniversityMelbourneAustralia
  2. 2.Faculty of Computing ScienceDalhousie UniversityHalifaxCanada

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