In this paper, we first investigate an abstract subdifferential for which (using Ekeland’s variational principle) we can prove an analog of the Brøndsted–Rockafellar property. We introduce the “ r L –density” of a subset of the product of a Banach space with its dual. A closed r L –dense monotone set is maximally monotone, but we will also consider the case of nonmonotone closed r L –dense sets. As a special case of our results, we can prove Rockafellar’s result that the subdifferential of a proper convex lower semicontinuous function is maximally monotone.
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Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011)
Borwein, J.M., Zhu, Q.: Techniques of Variational Analysis. Springer-Verlag, New York (2005)
Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Amer. Math. Soc. 16, 605–611 (1965)
Clarke, F.H.: Optimization and Nonsmooth Analysis. In: Classics in Applied Mathematics, Society for Industrial and Applied Mathematics. 2nd. SIAM, Philadelphia, PA (1990)
Lassonde, M.: Characterization of the monotone polar of subdifferentials. Optim. Lett. 8(5), 1735–1740 (2014)
Martínez-Legaz, J–E., Svaiter, B.F.: Monotone operators representable by l.s.c. convex functions. Set–Valued Anal. 13, 21–46 (2005)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer-Verlag (2006)
Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348, 1235–1280 (1996)
Phelps, R.R.: Lectures on maximal monotone operators. Extracta Mathematicae 12, 193–230 (1997)
Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. In: Lecture Notes in Mathematics. 2nd, vol. 1364. Springer-Verlag, Berlin (1993)
Rockafellar, R.T.: On the maximal monotonicity of subdifferentialmappings. Pac. J. Math. 33, 209–216 (1970)
Rockafellar, R.T.: Directionally Lipschitzian functions and subdifferential calculus. Proc. London Math. Soc. 39, 331–355 (1979)
Rockafellar, R.T.: Generalized directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32, 157–180 (1980)
Simons, S.: The range of a monotone operator. J. Math. Anal. Appl. 199, 176–201 (1996)
Simons, S.: Minimax and Monotonicity, Lecture Notes in Mathematics 1693. Springer–Verlag (1998)
Simons, S.: Maximal monotone multifunctions of Brøndsted–Rockafellar type. Set–Valued Anal. 7, 255–294 (1999)
Simons, S.: r L –density and maximal monotonicity, arXiv:1407.1100v3
Simons, S., Wang, X.: Weak subdifferentials, r L -density and maximal monotonicity, arXiv:1412.4386v2
Simons, S., Zalinescu, C.: Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal. 6(1), 1–22 (2005)
Thibault, L., Zagrodny, D.: Integration of subdifferentials of lower semicontinuous functions on Banach spaces. J. Math. Anal. Appl. 189(1), 33–58 (1995)
Zagrodny, D.: The convexity of the closure of the domain and the range of a maximal monotone multifunction of Type NI. Set–Valued Anal. 16, 759–783 (2008). doi:10.1007/s11228-008-0087-7
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co. Inc., River Edge, NJ (2002)
This paper is dedicated to Lionel Thibault, in recognition of his contribution to convex analysis and related fields
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Simons, S., Wang, X. Ubiquitous Subdifferentials, r L –density and Maximal Monotonicity. Set-Valued Var. Anal 23, 631–642 (2015). https://doi.org/10.1007/s11228-015-0326-7
- Abstract subdifferential
- Brøndsted–Rockafellar property
- Monotone polar
- r L –density
Mathematics Subject Classification (2000)
- Primary 49J52; Secondary 47H04