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

Svaiter, B.

doi:10.1090/S0002-9939-03-07083-7

Fixed points in the family of convex representations of a maximal monotone operator
HTML articles powered by AMS MathViewer

by B. F. Svaiter

Proc. Amer. Math. Soc. 131 (2003), 3851-3859

DOI: https://doi.org/10.1090/S0002-9939-03-07083-7

Published electronically: May 5, 2003

PDF | Request permission

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.

References

Dimitri P. Bertsekas and Sanjoy K. Mitter, A descent numerical method for optimization problems with nondifferentiable cost functionals, SIAM J. Control 11 (1973), 637–652. MR 0329250, DOI 10.1137/0311049
J. Frédéric Bonnans, Jean Charles Gilbert, Claude Lemaréchal, and Claudia Sagastizábal, Optimisation numérique, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 27, Springer-Verlag, Berlin, 1997 (French, with French summary). Aspects théoriques et pratiques. [Theoretical and applied aspects]. MR 1613833
A. Brøndsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605–611. MR 178103, DOI 10.1090/S0002-9939-1965-0178103-8
Regina S. Burachik, Alfredo N. Iusem, and B. F. Svaiter, Enlargement of monotone operators with applications to variational inequalities, Set-Valued Anal. 5 (1997), no. 2, 159–180. MR 1463929, DOI 10.1023/A:1008615624787
Regina Sandra Burachik and B. F. Svaiter, $\epsilon$-enlargements of maximal monotone operators in Banach spaces, Set-Valued Anal. 7 (1999), no. 2, 117–132. MR 1716028, DOI 10.1023/A:1008730230603
Regina S. Burachik, Claudia A. Sagastizábal, and B. F. Svaiter, $\epsilon$-enlargements of maximal monotone operators: theory and applications, Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods (Lausanne, 1997) Appl. Optim., vol. 22, Kluwer Acad. Publ., Dordrecht, 1999, pp. 25–43. MR 1682740, DOI 10.1007/978-1-4757-6388-1_{2}
Burachik, R.S. and Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements, Set Valued Analysis, (to appear).
Simon Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59–65. MR 1009594
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
Krzysztof C. Kiwiel, Proximity control in bundle methods for convex nondifferentiable minimization, Math. Programming 46 (1990), no. 1, (Ser. A), 105–122. MR 1045575, DOI 10.1007/BF01585731
Eckehard Krauss, A representation of maximal monotone operators by saddle functions, Rev. Roumaine Math. Pures Appl. 30 (1985), no. 10, 823–837. MR 826243
Claude Lemaréchal, Arkadii Nemirovskii, and Yurii Nesterov, New variants of bundle methods, Math. Programming 69 (1995), no. 1, Ser. B, 111–147. Nondifferentiable and large-scale optimization (Geneva, 1992). MR 1354434, DOI 10.1007/BF01585555
Juan-Enrique Martinez-Legaz and Michel Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001), no. 2, 243–247. Special issue for Professor Ky Fan. MR 1848704
E.A. Nurminski. $\varepsilon$-subgradient mapping and the problem of convex optimization. Cybernetics, 21(6):796–800, 1986.
Julian P. Revalski and Michel Théra, Generalized sums of monotone operators, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 11, 979–984 (English, with English and French summaries). MR 1733905, DOI 10.1016/S0764-4442(00)88589-4
Julian P. Revalski and Michel Théra, Variational and extended sums of monotone operators, Ill-posed variational problems and regularization techniques (Trier, 1998) Lecture Notes in Econom. and Math. Systems, vol. 477, Springer, Berlin, 1999, pp. 229–246. MR 1737322, DOI 10.1007/978-3-642-45780-7_{1}4
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. MR 262827, DOI 10.2140/pjm.1970.33.209
R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3
Stephen Simons, Minimax and monotonicity, Lecture Notes in Mathematics, vol. 1693, Springer-Verlag, Berlin, 1998. MR 1723737, DOI 10.1007/BFb0093633
M. V. Solodov and B. F. Svaiter, A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator, Set-Valued Anal. 7 (1999), no. 4, 323–345. MR 1756912, DOI 10.1023/A:1008777829180
M. V. Solodov and B. F. Svaiter, Error bounds for proximal point subproblems and associated inexact proximal point algorithms, Math. Program. 88 (2000), no. 2, Ser. B, 371–389. Error bounds in mathematical programming (Kowloon, 1998). MR 1783979, DOI 10.1007/s101070050022
Helga Schramm and Jochem Zowe, A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results, SIAM J. Optim. 2 (1992), no. 1, 121–152. MR 1147886, DOI 10.1137/0802008
B. F. Svaiter, A family of enlargements of maximal monotone operators, Set-Valued Anal. 8 (2000), no. 4, 311–328. MR 1802238, DOI 10.1023/A:1026555124541