Journal of Convex Analysis 27 (2020), No. 1, 165--178
Copyright Heldermann Verlag 2020
A Uniform Approach to Hölder Calmness of Subdifferentials
Gerald Beer
Department of Mathematics, California State University, Los Angeles, CA 90032, U.S.A.
gbeer@cslanet.calstatela.edu
María Josefa Cánovas
Center of Operations Research, Miguel Hernández University, 03202 Elche, Spain
canovas@umh.es
Marco Antonio López
Department of Mathematics, University of Alicante, 03071 Alicante, Spain
marco.antonio@ua.es
Juan Parra
Center of Operations Research, Miguel Hernández University, 03202 Elche, Spain
parra@umh.es
[Abstract-pdf]
For finite-valued convex functions $f$ defined on the $n$-dimensional Euclidean space, we are interested in the set-valued mapping assigning to each pair $(f,x)$ the subdifferential of $f$ at $x$. Our approach is uniform with respect to $f$ in the sense that it involves pairs of functions close enough to each other, but not necessarily around a nominal function. More precisely, we provide lower and upper estimates, in terms of Hausdorff excesses, of the subdifferential of one of such functions at a nominal point in terms of the subdifferential of nearby functions in a ball centered in such a point. In particular, we obtain the (1/2)\,-\,H\"{o}lder calmness of our mapping at a nominal pair $(f,x)$ under the assumption that the subdifferential mapping viewed as a set-valued mapping from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ with $f$ fixed is calm at each point of $\{x\}\times \partial f(x)$.
Keywords: Convex functions, sudifferentials, Hausdorff excess, uniform spaces, H\"{o}lder calmness.
MSC: 49J53, 52A41, 90C25, 90C31, 90C34.
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