On the Strong Convergence of Subgradients of Convex Functions

Abstract

In this paper, results on the strong convergence of subgradients of convex functions along a given direction are presented; that is, the relative compactness (with respect to the norm) of the union of subdifferentials of a convex function along a given direction is investigated.

The Strong Convergence of Subgradients of Convex Functions Along Directions: Perspectives and Open Problems

“…Of course we have f (0) = 0, ∂ f (0) = {0}, so function f is Gâteaux differentiable at the origin. It follows from Lemma 3.1 that the limits lim sup t↓0 ∂ f (tw) and lim sup i→∞ ∂ f (t i w) exist for all w from a dense subset of H; see (9). Moreover, it follows from Lemma 3.1 [9] that…”

“…The convergence of subdifferentials along these directions can be provided by using Theorem 3.1 [9]. We can also predict that more sophisticated second-order conditions can be used for a specification of good directions of the convergence.…”

“…We can also predict that more sophisticated second-order conditions can be used for a specification of good directions of the convergence. Of course, if the function involved in (5) has a second-order expansion at x, then (5) An example of such function should not be looked for among antidistance functions or Asplund functions (see [9] for the definition of the function). Because it is known that Gâteaux differentiability entails the Fréchet differentiability for these functions and due Corollary 2 [16]; see also Corollary 4.2 [14], we have the directional convergence of subdifferentials for all directions.…”

“…Namely, at this moment, we know that there are convex functions such that there is the lack of convergence in some directions; B Dariusz Zagrodny d.zagrodny@uksw.edu.pl examples of such functions were provided in [6][7][8]. Moreover, when we take subgradients instead of gradients, we know that the set of such directions are negligible; the Lebesgue measure of the directions, where the convergence is not valid, amounts zero; see [9]. The infinite-dimensional case is not recognized.…”

“…The infinite-dimensional case is not recognized. There are more questions than answers; for some recent information, we refer to [9]. We do not know how large the set of good directions is.…”