Abstract
We introduce a family of enlargements of maximal monotone operators. The Brønsted and Rockafellar ε-subdifferential operator can be regarded as an enlargement of the subdifferential. The family of enlargements introduced in this paper generalizes the Brønsted and Rockafellar ε-subdifferential (enlargement) and also generalize the enlargement of an arbitrary maximal monotone operator recently proposed by Burachik, Iusem and Svaiter. We characterize the biggest and the smallest enlargement belonging to this family and discuss some general properties of its members. A subfamily is also studied, namely the subfamily of those enlargements which are also additive. Members of this subfamily are formally closer to the ε-subdifferential. Existence of maximal elements is proved. In the case of the subdifferential, we prove that the ε-subdifferential is maximal in this subfamily.
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References
Brøndsted, A. and Rockafellar, R. T.: On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605–611.
Burachik, R. S., Iusem, A. N. and Svaiter, B. F.: Enlargements of maximal monotone operators with application to variational inequalities, Set-Valued Anal. 5 (1997), 159–180.
Burachik, R. S., Sagastizábal, C. A. and Svaiter, B. F.: Epsilon-enlargement of maximal monotone operators with application to variational inequalities, In: M. Fukushima and L. Qi (eds), Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Acad. Publ., Dordrecht, 1997.
Burachik, R. S., Sagastizábal, C. A. and Svaiter, B. F.: Bundle methods for maximal monotone operators, In: R. Tichatschke and M. Thera (eds), Ill-posed Variational Problems and Regularization Techniques, Lecture Notes Control and Inform. Sci., Springer, New York, 1999, pp. 49–64.
Burachik, R. S. and Svaiter, B. F.: ε-enlargements of maximal monotone operators in Banach spaces, Set-Valued Anal. 7(2) (1999), 117–132.
Hiriart-Urruty, J.-B.: Lipschitz r-continuity of the approximate subdifferential of a convex function, Math. Scand. 47 (1980), 123–134.
Kato, T.: Demicontinuity, hemicontinuity and monotonicity, Bull. Amer. Math. Soc. 70 (1964), 548–550.
Lemaréchal, C.: Extensions diverses des méthodes de gradient et applications, Thèse d'Etat, Université de Paris IX, 1980.
Moreau, J.-J.: Functionelles convexes, Lectures Notes, Collège de France, 1967.
Nurminskii, E. A.: The continuity of ε-subgradient mapppings, Cybernetics 13(5) (1977).
Rockafellar, R. T.: Local boundedness of nonlinear monotone operators, Michigan Math. J. 16 (1969), 397–407.
Rockafellar, R. T.: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216.
Solodov, M. V. and Svaiter, B. F.: Error bounds for proximal point subproblems and associated inexact proximal point algorithms, Math. Program., to appear.
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Svaiter, B.F. A Family of Enlargements of Maximal Monotone Operators. Set-Valued Analysis 8, 311–328 (2000). https://doi.org/10.1023/A:1026555124541
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DOI: https://doi.org/10.1023/A:1026555124541