On the Strong Convergence of Subgradients of Convex Functions

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.


Introduction
There are several results on the strong convergence of subgradients of a sequence of convex functions defined on a Banach space. The most celebrated result is the Attouch theorem; see, for example, [1], where the equivalence of Mosco convergence of lower semicontinuous convex functions to the Painleve-Kuratowski graph convergence of their subdifferentials is established on reflexive Banach space. There are also results extending the Attouch theorem to general Banach spaces; see, for example, [2][3][4][5][6][7] and references therein. To the best of our knowledge all known results are of the form: there are sequences of points and subgradients such that the strong limits of sequences Communicated by Viorel Barbu.
B Dariusz Zagrodny d.zagrodny@uksw.edu.pl of subgradients exist (limits with respect to the norm of the space); see, for example, Theorem 3.1 in [7]. This is inconvenient. Simply we want to have subgradients with a desired property. The postulate of existence ("there are") does not allow one to guarantee that the subgradients are as good as it is needed. This disadvantage can be observed also, when the directional derivative is calculated. Namely, the difference quotients form sequences of functions with respect to directions, whenever we consider the limit over a discrete subset. It is natural to ask about the convergence of subgradients of this functions; see, for example, Giannessi's questions, which are recalled in (5). The question, about the existence of a convergent subsequence (at least) for this sequence of function, is the question on the existence of a convergent sequence of subgradients along a direction. In a finite-dimensional case, the existence of convergent subsequences is guaranteed by the continuity of the convex function under investigation. However, there can be subsequences with different limits; see [8]; see also [9][10][11]. It turns out that the set of "wrong directions" (there in no unique limit) has the Lebesgue measure equal zero; see Lemma 3.1. In infinite-dimensional setting, it is hard to expect the convergence. Thus, the basic question in this case, concerning directional convergence of subgradients, is: when does the union of subdifferentials along a given direction form a relatively compact set (with respect to the norm topology)? We should also ask about the uniqueness of the limit, which is the essence of Giannessi's questions in the finite-dimensional setting. In the infinite-dimensional case, results of this type are rather unknown, but it would be convenient to have such results at hand. For instance, when the limit exists, then the limiting subgradients inherit properties of a convergent sequence, like: size of norm, being in a specified closed set, a good behaving with respect to the weak convergence of arguments, and so on. In Sect. 3, we present a result which guarantees the relative compactness for some special classes of convex functions; see Theorem 3.1. In Lemmas 2.2 (in the Hilbert space setting) and 3.2 (in the reflexive Banach space setting) examples of functions from the class are provided too.

Preliminaries
In this section, some basic notions and their properties are gathered.
In the sequel, (X, · ) stands for a real normed space, X * for its dual space and H for a real Hilbert space (with a real inner product). The weak convergence is denoted by weak −→, and the limit from the right is denoted by t ↓ a, which means that t > a and t −→ a.
For every real r > 0 and every x ∈ X we denote by B X (x, r ) (resp. B X [x, r ]) the open (resp. closed) ball centered at x and of radius r , the sphere is denoted by S X [x, r ] := {y ∈ X : y − x = r } and S X := S X [0, 1], "cl " stands for the topological closure.
Following [12, page 2] span M stands for the linear hull of M and it is the smallest (in the sense of inclusion) linear subspace of X containing M. We call the set x ∈ X and w 0 ∈ X \ {0} are given; we refer to [13] and references therein for information on drops. For a given function f : We say that f : where lim inf For every nonempty set S ⊂ H the distance function from the set S is denoted by d S (·), that is, Let us assume that x ∈ X and {s i } i∈N is a sequence of points from a subset S ⊂ X such that lim i−→∞ x −s i = d S (x). It is valuable to have results preserving a relative compactness of the sequence, see, for example, Proposition 3.1 in [14]. Below a result of this kind is presented in the Hilbert space setting.

Lemma 2.1 Let H be a Hilbert space with a dimension greater than one, W be a closed subspace of H (thus W is a Hilbert space too), S ⊂ H be a nonempty subset and x
Then, if {s i } i∈N is a sequence of points from a subset S ⊂ X such that Proof Let assume the contrary, that is for some > 0 and for all i ∈ N there are and observe that Moreover, Hence, for n ∈ N large enough we get which contradicts the assumptions of the lemma.
The Asplund function, see [15, (1) page 234], is a good tool to investigate the distance function since its convexity, that is the function where S ⊂ H is a given subset. Below a directional behaving of this function is characterized.
, w, x − y = 0 for all w ∈ W and that for some t ∈]0, 1] and all u ∈ H such that where Y := {w ∈ W : w, h = 0 and w, x = 0} and

and f S h is the Asplund function for the set S h , that is, for all y ∈ H we put f S h (y)
Proof Fix θ > 0 and sequences {s i } i∈N , {z i } i∈N such that s i ∈ S h , for all i ∈ N, and t i ↓ 0. We have It follows from (2) that (4) holds true, whenever the inequality in (1) is taken into account.
Let f : X −→ R ∪ {+∞} be a lower semicontinuous convex function, which is finite at x ∈ X . The subdifferential of f at x ∈ X is defined by

Relative Compactness of Sets of Subgradients
Let us recall Giannessi's questions; see [8], see also [9][10][11] for examples of convex functions in two-dimensional spaces, for which the limit in (5) does not exist: Let f : R n −→ R, with n ≥ 2, be a convex function, and set x(t) := (t, 0, . . . , 0) ∈ R n , with t ∈ R. Assume that ∇ f (x(t)) exists for every t > 0, and consider the following limit: lim We conjecture that the above limit may not exist. Hence, however, the question is still open. The above question can be generalized in several ways. For instance, x(t) may represent a curve having the origin as endpoint instead of a ray; R n may be replaced with an infinite-dimensional space.
Below directions along which we have the weak * convergence of subgradients are indicated. (X, · ) be a reflexive Banach space and f : X −→ R ∪ {+∞} be a convex lower semicontinuous function, x ∈ dom f and M ≥ 0, w 0 ∈ X, x * ∈ X * be given. If f (x; w 0 ) is a finite real number (the directional derivative at x along w 0 is finite) such that

Lemma 3.1 Let
then for all sequences and consequently whenever Proof Let us assume that (6) holds true. The equality f (x; w 0 ) = x * , w 0 is easy to verify by a simple algebra. In fact for all t > 0 we have Thus, it follows from (6) that for all h ∈ X we have Again using (6) we get 0 ≤ − f (x; w 0 ) + x * , w 0 , which implies (9). If w 0 = 0, then ∂ f (x) = {x * } and we are done. Assume that w 0 = 0. Take Using the Eberlein-Shmulyan theorem; see Appendix to Chapter V, Section, page 141 in [16], we may assume that the sequence {x * i } i∈N is weakly convergent to some y * ∈ ∂ f (x), otherwise we choose a proper subsequence. Fix > 0, h ∈ S X and take t > 0 such that Notice that thus, since h ∈ S X is arbitrary, we get y * = x * , which translates (7) and the equality Notice that the weak convergence implies the strong one, whenever X = R n , thus using Lemma 3.1 we can pick directions along which the strong convergence is preserved and consequently the limit in (5) exists. Below we provide a property which ensures the strong convergence of subgradients. Roughly speaking the union of subdifferentials along the "good" direction forms a relatively compact set with respect to the norm topology. This property is unexpected in the infinite-dimensional setting. Let us distinguish a family of auxiliary functions. Namely, for a given sequence {t i } i∈N such that t i > 0 for all i ∈ N, and t i ↓ 0, a given closed subspace Y ⊂ X and a given positive number ρ > 0 let us put Our first example of function from the class define above is the function p defined in Lemma 2.2, see (4). Observe that the continuity of the Asplund function ensures that (11) is fulfilled for all ρ > 0 and all sequences {t i } i∈N such that t i > 0 for all i ∈ N, and t i ↓ 0, whenever we put p(0, ·) = 0. Below it is shown that weak continuous convex functions can be also used to construct functions from the class.
Proof In order to establish the inequality let us observe that for all t ∈]0, μ 0 [ by the convexity we have Let us fix ρ ∈]0, β 0 [ and a sequence {t i } i∈N such that t i > 0 for all i ∈ N, and t i ↓ 0. We have lim inf Let us recall the notion of the direct sum of two closed subspaces of a Banach space (X, · ); see Definition 4.20 in [17], where it was given for a vector topological spaces. Suppose Y is a closed subspace of X . If there exists a closed subspace Z ⊂ X such that X = Y + Z and Y ∩ Z = {0}, then Y is said to be complemented in X . In this case, X is said to be the direct sum of Y and Z , and the notation X = Y ⊕ Z is used. It is known that, if Y has a finite codimension, then Y is complemented; see, for example, Lemma 4.21 in [17] or Definition 4.1 and Theorem 5.5 in [12].