On Bregman-Type Distances and Their Associated Projection Mappings

We investigate convergence properties of Bregman distances induced by convex representations of maximally monotone operators. We also introduce and study the projection mappings associated with such distances.

ferentiable strictly convex function f , the Bregman-type distance associated with its convex representation coincides with the classical Bregman distance induced by f (see Proposition 4). Bregman-type distances have been further studied more recently in [7,18], the latter paper dealing with their associated farthest Voronoi cells.
The aim of this paper is to investigate the convergence properties of such Bregmantype distances as well as to study their associated projection mappings, which we introduce here.
We recall some definitions and lemmas which will be used in the sequel. We start with the following important lemma, which will play a crucial role in the proof of a fixed point result involving Bregman-type projections we will prove in the last section (see Theorem 17).
Lemma 1 [14,Lemma 4] Let X be a nonempty compact convex set in a Hausdorff topological vector space Y . Let A be a closed subset of X × X having the following properties: (i) (x, x) ∈ A for every x ∈ X.
(ii) For any fixed point y ∈ X, the set {x ∈ X : (x, y) / ∈ A} is a (possibly empty) convex set.
Then, there exists a point y 0 ∈ X such that X × {y 0 } ⊆ A.
Throughout the paper, E will be a Banach space with norm . and dual space E * . For any x ∈ E, we denote the value of x * ∈ E * at x by x, x * . When {x n } n∈N is a sequence in E, we denote the strong convergence of {x n } n∈N to x ∈ E as n → ∞ by x n → x, and the weak convergence by x n x. We say that a function h : E × E * → R∪ {+∞} is norm × weak * lower semicontinuous when every sublevel set of h is closed in the norm × weak * topology of E × E * , that is, in the product topology of the strong topology of E and the weak * topology of E * .
We first recall the notion of convex representation of a maximally monotone operator.
Definition 2 (see [17, p. 27]) Let S : E ⇒ E * be a maximally monotone operator. We say that h : E × E * → R ∪ {+∞} represents S if the following three conditions hold: (i) h is convex and norm × weak * lower semicontinuous; The set of all functions h satisfying these properties will be denoted by H (S).
For a given maximally monotone operator S : E ⇒ E * , it is well known (see, e.g., [10,Corollary 4.1]) that H (S) has a smallest element and a biggest one. The biggest element is the norm × weak * lower semicontinuous convex envelope of π + δ G(S) , where π : E × E * → R is defined by The smallest element is the Fitzpatrick function F S associated with S , introduced in [16]: For more details on the family H (S), see [8,10,11,16]. Let us recall that a function g : E → R ∪ {+∞} is said to be strictly convex on a convex set X ⊆ E if In the case when X = dom(g), we simply say that g is strictly convex.
We recall the classical definition of Bregman distance. Let g : E → R ∪ {+∞} be differentiable on int dom(g), the interior of dom(g), and strictly convex. The Bregman distance [13] (see also [6,12]) corresponding to g is the function D g : It follows from the convexity of g that D g (x, y) ≥ 0 for all (x, y) ∈ dom(g) × int dom(g) . However, D g might not be symmetric and might not satisfy the triangle inequality.
We recall that a convex function g : E → R ∪ {+∞} is said to be proper if g ≡ +∞. Its Fenchel conjugate is the lower semicontinuous proper convex function Proposition 4 [9, Proposition 3.5] Let g : E → R∪ {+∞} proper, continuous, strictly convex, and differentiable on dom (g). Take S = ∇g and h g ∈ H (S) such that (1)

Remark 5 It is worth noticing that the continuity assumption on
The rest of the paper consists of two more sections. In Sect. 2, we establish some convergence properties of Bregman-type distances. In Sect. 3, we introduce and study projections associated with Bregman-type distances; in particular, we give simple sufficient conditions for such mappings to be single-valued, a characterization of the elements of the projection sets, and a fixed point result for compositions of Bregmantype projection mappings with continuous mappings.

Convergence Properties of Bregman-Type Distances
Let us recall that a function g : and supercoercive if it satisfies the stronger condition We start with a generalization of [ Our next result is a generalization of [4, Proposition 2.2], the proof of which we partially follow.

Proposition 8 (D h -convergence) Let S and h be as in
x)} n∈N is bounded; so, by Lemma 6, the sequence {x n } n∈N is bounded and, in view of the reflexivity of E, there exists a subsequence {x n k } k∈N weakly convergent to some y ∈ E. Since D h is lower semicontinuous in its first variable, we have so D h (y, x) = 0, which, by the strict monotonicity of S, yields y = x. This proves that x n x. Statement (iii) follows from (i) and (ii), since weak and strong convergence are equivalent in finite-dimensional spaces.

Projections Associated with Bregman-Type Distances
In this section, we introduce and study the notion of h-projection, h being a convex representation of a maximally monotone operator.

Definition 9
Let S and h be as is Definition 2. For a nonempty set X ⊆ E, we define the associated h-distance function D h (X , ·) : E → R ∪ {+∞} and h-projection and respectively.
The following result establishes sufficient conditions for the nonemptiness of h-Bregman projections. It generalizes [1, Proposition 2.1], and its proof is exactly along the same lines.

Proposition 10 (The domain of P h X ) Let S and h be as is Definition 2. If E is reflexive and y ∈ E is such that h(·, Sy) is weakly continuous and X ⊆ E is nonempty and weakly closed, then P h X (y) = ∅ whenever at least one of the following conditions is satisfied:
by the assumption, we obtain lim k→∞ D h (x n k , x) = +∞, a contradiction with lim n→∞ D h (x n , x) = D h (X , x). Therefore {x n } n∈N is bounded, and hence, since E is reflexive, it has a weakly convergent subsequence {x n k } k∈N . Let x * be the weak limit of x n k . Since X is weakly closed, x * ∈ X . This, together with the equalities implies that x * ∈ P h X (y), and the proof is complete.
We next give a sufficient condition for P h

Proposition 15 (Continuity of S) If S : E ⇒ E * is maximally monotone and singlevalued on dom(S), then S is norm-to-weak * continuous.
Proof We first observe that dom(S) is open, since, for every x ∈ dom(S), the recession cone of Sx coincides with the normal cone to the closure of dom(S) at x (see [20,Theorem 1] for the case when E = R n , and notice that the proof of this result extends to arbitrary Banach spaces in a straightforward way). Let x ∈ dom(S), and let {x n } n∈N be a sequence converging to x. Since S is locally bounded at x, there exists a neighborhood U of x such that S(U ) is bounded. Without loss of generality, we assume that {x n } n∈N is contained in U ; then the sequence {Sx n } n∈N is bounded. Let Sx n k k∈N be any weakly * convergent subsequence, and denote its limit by x * . Then, the sequence x n k , Sx n k k∈N converges to (x, x * ); hence, since the graph of S is closed (because of the maximal monotonicity of S), one has x * = Sx. Thus, in view of the Banach-Alaoglu Theorem, using the well-known fact that if all the convergent subsequences of a sequence in a compact space converge to a common limit then the whole sequence converges, we conclude that {Sx n } n∈N converges to Sx, thus proving continuity of S at x.
Our next theorem will be a generalization of [15,Theorem 2]. To prove it, we will need the following lemma.

Lemma 16 Let S and h be as is Definition 2. If X ⊆ E, the operator S is singlevalued on dom(S), and h is norm×weak * continuous, then, for any norm-to-norm
Proof Let {(x n , y n )} n∈N be a sequence in X × X strongly converging to (x, y) ∈ X × X . Similarly to the proof of Proposition 15, without loss of generality we may assume that the sequence {S f (y n )} n∈N is bounded. Then, since {x n } n∈N strongly converges to x and {S f (y n )} n∈N weakly * converges to S f (y) , from the inequality | x n , S f (y n ) − x, S f (y) | ≤ | x n − x, S f (y n ) | + | x, S f (y n ) − S f (y) | we can easily deduce that { x n , S f (y n ) } n∈N converges to x, S f (y) . This, together with the norm-to-norm continuity of f , Proposition 15, and the norm×weak * continuity of h, proves that {D h (x n , f (y n ))} n∈N converges to D h (x, f (y)).
Theorem 17 (Fixed point theorem for the composition of P h X with continuous mappings) Let S and h be as is Definition 2. If X ⊆ E is nonempty, convex, and compact in the strong topology of E , the set dom(S) is convex, S is single-valued on dom(S), and h is norm×weak * continuous, then, for any norm-to-norm continuous f : X → dom(S), the mapping P h X • f has a fixed point.

Proof The set
is closed by Lemma 16, and it obviously satisfies (i) of Lemma 1. It also satisfies ii), since h is convex. Hence, by Lemma 1, there exists x 0 ∈ X such that X × {x 0 } ⊆ A, that is, equivalently, which means that To derive a (finite dimensional) result on classical Bregman distances from Theorem 17, we need the following lemma.
Proof We apply Theorem 17 with S := ∇g, followed by Proposition 4. By Proposition 15, the mapping ∇g is continuous. We set h := h g , with h g defined by (1). By Lemma 18, the function h g is continuous. Therefore, given that dom(∇g) = dom(g) is convex, all the assumptions of Theorem 17 are satisfied; hence, we conclude that P h g X • f has a fixed point x 0 . By Proposition 4, we obtain (3).

Conclusions
We have obtained convergence properties for Bregman-type distances associated with convex representations of maximally monotone operators. Such Bregman-type distances were introduced in [9] in such a way that when the maximally monotone operator is the (single-valued) subdifferential of a differentiable convex function f and its convex representation h is given by h (x, x * ) := f (x) + f * (x * ) , the associated Bregman-type distance reduces to the classical Bregman distance induced by f . When we consider this particular situation, we recover some convergence results obtained earlier by Bauschke and Combettes [4].
Our Bregman-type distances induce, in a natural way, a notion of projection onto nonempty sets. Among other results for such projections, we prove a fixed point theorem for their compositions with continuous mappings, which generalizes a classical result of Fan [15] for ordinary projections in normed spaces.