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Fixed points in the family of convex representations of a maximal monotone operator

Author: B. F. Svaiter
Journal: Proc. Amer. Math. Soc. 131 (2003), 3851-3859
MSC (2000): Primary 47H05
Published electronically: May 5, 2003
MathSciNet review: 1999934
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Abstract: Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exists a convex representation of the operator which is a fixed point of this conjugation.

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  • 1. D.P. Bertsekas and S.K. Mitter.
    A descent numerical method for optimization problems with nondifferentiable cost functionals.
    SIAM Journal on Control, 11(4):637-652, 1973. MR 48:7592
  • 2. F. Bonnans, J.Ch. Gilbert, C.L. Lemaréchal and C.A. Sagastizábal.
    Optimisation Numérique, aspects théoriques et pratiques.
    Collection ``Mathématiques et applications'', SMAI-Springer-Verlag, Berlin, 1997. MR 99f:90001
  • 3. A. Brøndsted and R. T. Rockafellar.
    On the subdifferentiability of convex functions.
    Proceedings of the American Mathematical Society, 16:605-611, 1965. MR 31:2361
  • 4. Regina S. Burachik, Alfredo N. Iusem, and B. F. Svaiter.
    Enlargement of monotone operators with applications to variational inequalities.
    Set-Valued Analysis, 5(2):159-180, 1997. MR 98h:49010
  • 5. Regina Sandra Burachik and B. F. Svaiter.
    $\varepsilon$-enlargements of maximal monotone operators in Banach spaces.
    Set-Valued Analysis, 7(2):117-132, 1999. MR 2000i:47099
  • 6. Regina S. Burachik, Claudia A. Sagastizábal, and B. F. Svaiter.
    $\epsilon$-enlargements of maximal monotone operators: theory and applications.
    In M. Fukushima and L. Qi, editors, Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods (Lausanne, 1997), volume 22 of Applied Optimization, pages 25-43. Kluwer Acad. Publ., Dordrecht, 1999. MR 2000a:49030
  • 7. Burachik, R.S. and Svaiter, B.F.:
    Maximal monotone operators, convex functions and a special family of enlargements,
    Set Valued Analysis, (to appear).
  • 8. Fitzpatrick, S.:
    Representing monotone operators by convex functions,
    Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988) 59-65,
    Proc. Centre Math. Anal. Austral. Nat. Univ., 20
    Austral. Nat. Univ., Canberra, 1988. MR 90i:47054
  • 9. J.-B. Hiriart-Urruty and C. Lemaréchal.
    Convex Analysis and Minimization Algorithms.
    Number 305-306 in Grund. der math. Wiss. Springer-Verlag, 1993.
    (two volumes). MR 95m:90001; MR 95m:90002
  • 10. K.C. Kiwiel.
    Proximity control in bundle methods for convex nondifferentiable minimization.
    Mathematical Programming, 46:105-122, 1990. MR 91c:90084
  • 11. Krauss, Eckehard:
    A representation of maximal monotone operators by saddle functions,
    Rev. Roumaine Math. Pures Appl. 30 (1985), 823-837. MR 87e:47074
  • 12. C. Lemaréchal, A. Nemirovskii, and Yu. Nesterov.
    New variants of bundle methods.
    Mathematical Programming, 69:111-148, 1995. MR 96g:90075
  • 13. Martinez-Legaz, J.-E. and Théra, M.:
    A convex representation of maximal monotone operators,
    Journal of Nonlinear and Convex Analysis 2 (2001), 243-247. MR 2002e:49035
  • 14. E.A. Nurminski.
    $\varepsilon$-subgradient mapping and the problem of convex optimization.
    Cybernetics, 21(6):796-800, 1986.
  • 15. Revalski, J.P. and M. Théra,
    Generalized sums of monotone operators,
    C. R. Acad. Sci. Paris Sér. I Math. 329(11):979-984, 1999. MR 2000m:47068
  • 16. Revalski, J.P. and M. Théra,
    Variational and extended sums of monotone operators,
    Ill-posed variational problems and regularization techniques (Trier, 1998),
    229-246, Lecture Notes in Econom. and Math. Systems, 477, Springer, Berlin, 1999. MR 2000k:47071
  • 17. Rockafellar, R. T.:
    On the maximal monotonicity of subdifferential mappings,
    Pacific Journal of Mathematics 33 (1970), 209-216. MR 41:7432
  • 18. R. Tyrrell Rockafellar, Roger J-B. Wets.
    Variational Analysis.
    Springer Verlag, Berlin Heidelberg, 1998. MR 98m:49001
  • 19. Simons, S.:
    Minimax and monotonicity.
    Lecture Notes in Mathematics, 1693. Springer-Verlag, Berlin, 1998. MR 2001h:49002
  • 20. M. V. Solodov and B. F. Svaiter.
    An inexact hybrid extragradient-proximal point algorithm using the enlargement of a maximal monotone operator.
    Set-Valued Analysis, 7(4):323-345, December 1999. MR 2001a:90084
  • 21. M. V. Solodov and B. F. Svaiter.
    Error bounds for proximal point subproblems and associated inexact proximal point algorithms.
    Mathematical Programming, 88(2):371-389, 2000. MR 2001f:90052
  • 22. H. Schramm and J. Zowe.
    A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results.
    SIAM Journal on Optimization, 2(1):121-152, 1992. MR 93b:90089
  • 23. B. F. Svaiter.
    A Family of Enlargements of Maximal Monotone Operators.
    Set-Valued Analysis, 8(4):311-328, December 2000. MR 2001i:49037

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Additional Information

B. F. Svaiter
Affiliation: IMPA Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro–RJ, CEP 22460-320 Brazil

Keywords: Maximal monotone operators, conjugation, convex functions
Received by editor(s): July 31, 2002
Published electronically: May 5, 2003
Additional Notes: This work was partially supported by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society

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