
Journal of Convex Analysis 27 (2020), No. 1, 165178 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 [Abstractpdf] For finitevalued convex functions $f$ defined on the $n$dimensional Euclidean space, we are interested in the setvalued 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 setvalued 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. [ Fulltextpdf (138 KB)] for subscribers only. 