A uniform approach to Hölder calmness of subdifferentials

M. A. López Cerdá, G. Beer, M. J. Cánovas Cánovas, J. Parra López

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 deduce the (1/2)-Hölder calmness of our mapping at a nominal pair (f,x) under the assumption that the subdifferential mapping viewed as a multifunction from Rⁿ to Rⁿ with f fixed is calm at each point of {x}×∂f(x).

Keywords: Sudifferentials · Hausdorff excess · Uniform spaces · Hölder calmness

R. Lucchetti, G. Bernardi, S. Moretti

E. Vercher González, J. D. Bermúdez

M. J. Cánovas Cánovas, G. Beer, M. A. López Cerdá, J. Parra López