Journal of Optimization Theory and Applications (2022) 193:107–117 https://doi.org/10.1007/s10957-021-01970-4 On Bregman-Type Distances and Their Associated Projection Mappings Juan Enrique Martínez-Legaz1,2 · Maryam Tamadoni Jahromi3 · Eskandar Naraghirad3 Received: 15 May 2021 / Accepted: 31 October 2021 / Published online: 29 November 2021 © The Author(s) 2021 Abstract We investigate convergence properties of Bregman distances induced by convex rep- resentations of maximally monotone operators. We also introduce and study the projection mappings associated with such distances. Keywords Bregman distance · Fixed point · Maximally monotone operator · Banach space Mathematics Subject Classification 7H10 · 37C25 1 Introduction and Preliminaries This paper is motivated by the recent article [9], which introduces a notion of Bregman- type distance associated with a convex representation of an arbitrary maximally monotone operator in such a way that, when the operator is the gradient of a dif- Dedicated to Professor Franco Giannessi on the occasion of his 85th birthday. Communicated by Boris S. Mordukhovich. B Juan Enrique Martínez-Legaz JuanEnrique.Martinez.Legaz@uab.cat Maryam Tamadoni Jahromi M.tamadoni@stu.yu.ac.ir; tamadoni_maryam@yahoo.com Eskandar Naraghirad eskandarrad@gmail.com; esnaraghirad@yu.ac.ir 1 Department d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona, Bellaterra, Spain 2 Barcelona Graduate School of Mathematics (BGSMath), Barcelona, Spain 3 Department of Mathematics, Yasouj University, Yasouj 75918, Iran 123 108 Journal of Optimization Theory and Applications (2022) 193:107–117 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 Bregman- type 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 y0 ∈ X such that X × {y0 } ⊆ 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 {xn }n∈N is a sequence in E, we denote the strong convergence of {xn }n∈N to x ∈ E as n → ∞ by xn → x, and the weak convergence by xn 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 oper- ator. 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; (ii) h(x, x ∗ ) ≥ x, x ∗ , ∀(x, x ∗ ) ∈ E × E ∗ ; (iii) h(x, x ∗ ) = x, x ∗  if and only if x ∗ ∈ Sx. 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 + δG(S) , where π : E × E ∗ → R is defined by π(E, E ∗ ) := x, x ∗ , G(S) denotes the graph of S,    G (S) := x, x ∗ ∈ E × E ∗ : x ∗ ∈ Sx , 123 Journal of Optimization Theory and Applications (2022) 193:107–117 109 and δG(S) : E × E ∗ → R∪ {+∞} denotes the indicator function of G(S),   ∗  0 if (x, x ∗ ) ∈ G(S), δG(S) x, x := +∞ otherwise. The smallest element is the Fitzpatrick function FS associated with S , introduced in [16]: FS (x, x ∗ ) := x, x ∗  − inf x − y, x ∗ − y ∗ . (y,y ∗ )∈S For more details on the family H (S), see [8,10,11,16]. We recall that the domain of an operator S : E ⇒ E ∗ is the set dom(S) := {x ∈ E : Sx = ∅} , and the domains of g : E → R ∪ {+∞} and h : E × dom(S) → R ∪ {+∞} are the sets dom(g) := {x ∈ E : g(x) < +∞} and dom (h) := {(x, y) ∈ E × dom(S) : h (x, y) < +∞} , respectively. Definition 3 [9] Let S : E ⇒ E ∗ be maximally monotone and single-valued on dom(S). The Bregman-type distance associated with h ∈ H (S) is the function Dh : E × dom(S) → R ∪ {+∞} defined by Dh (x, y) := h(x, Sy) − x, Sy. Let us recall that a function g : E → R ∪ {+∞} is said to be strictly convex on a convex set X ⊆ E if g(λx + (1 − λ)y) < λg(x) + (1 − λ)g(y), ∀x, y ∈ X , x = y, ∀λ ∈ (0, 1). 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 : dom(g) × int dom(g) → R defined by D g (x, y) = g(x) − g(y) − x − y, ∇g(y), ∀x, y ∈ E. 123 110 Journal of Optimization Theory and Applications (2022) 193:107–117 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 g ∗ : E ∗ → R ∪ {+∞} , defined by    g ∗ (x ∗ ) := sup x, x ∗ − g (x) . x∈E 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 h g (x, x ∗ ) := g(x) + g ∗ (x ∗ ). (1) Then, for every (x, y) ∈ E × dom(g), we have Dh g (x, y) = D g (x, y) = g(x) − g(y) − x − y, ∇g(y). Remark 5 It is worth noticing that the continuity assumption on g implies that dom(g) is open. Indeed, one has dom (g) = g −1 (R) , and R is open in R∪ {+∞}. 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 Bregman- type projection mappings with continuous mappings. 2 Convergence Properties of Bregman-Type Distances Let us recall that a function g : E → R ∪ {+∞} is said to be coercive if lim g (x) = +∞, x→∞ and supercoercive if it satisfies the stronger condition g (x) lim = +∞. x→∞ x We start with a generalization of [3, Lemma 2.5]. Lemma 6 Let {xn }n∈N be a sequence in E, and assume that E is reflexive, S : E ⇒ E ∗ is maximally monotone and single-valued on dom(S), and h ∈ H (S) is weakly con- tinuous in its first variable. If {xn }n∈N is bounded, then the sequence {Dh (xn , y)}n∈N is bounded for every y ∈ E such that (xn , y) ∈ dom(h) and dom(h) ∩ (E × {y}) is 123 Journal of Optimization Theory and Applications (2022) 193:107–117 111 closed. Conversely, if there exists y ∈ E such that the sequence {Dh (xn , y)}n∈N is bounded and h (·, Sy) is supercoercive, then {xn }n∈N is bounded. Proof Let y ∈ E be such that (xn , y) ∈ dom(h) and dom(h) ∩ (E × {y}) is  closed,  and suppose that {Dh (xn , y)}n∈N is unbounded. Then, there is a subsequence xn k k∈N which weakly converges to some x ∈ E and satisfies Dh (xn k , y) → +∞. Clearly, (x, y) ∈ dom(h), therefore, since h is weakly continuous in its first variable, we have Dh (xn k , y) → Dh (x, y) ∈ R, which is a contradiction. Conversely, let y ∈ E be such that the sequence {Dh (xn , y)}n∈N is bounded and h (·, Sy) is supercoercive, and assume that {xn }n∈N is unbounded. After passing to a subsequence if necessary, we may assume that xn  → ∞; since h(·, Sy) is supercoercive, so is Dh (·, y), and thus Dh (xn , y) → ∞, which is absurd.   Definition 7 (see [4, Definition 2.1]) Let S and h be as in Definition 2. A sequence {xn }n∈N is said to be Dh -convergent to x ∈ dom(S) if Dh (xn , x) → 0, in symbols Dh xn → x. Our next result is a generalization of [4, Proposition 2.2], the proof of which we partially follow. Proposition 8 (Dh -convergence) Let S and h be as in Definition 2. For x ∈ dom(S) and a sequence {xn }n∈N in E, the following statements hold: Dh (i) If h (·, Sx) is continuous and xn → x, then xn → x. Dh (ii) If E is reflexive, S is strictly monotone, h (·, Sx) is supercoercive and xn → x, then xn x. (iii) If E is finite dimensional, S is strictly monotone and h (·, Sx) is continuous and supercoercive, then Dh xn → x ⇔ xn → x ⇔ xn x. Dh Proof (i) If xn → x, then limn→∞ Dh (xn , x) = Dh (x, x) = 0, hence xn → x. Dh (ii) Since xn → x and h (·, Sx) is supercoercive, {Dh (xn , x)}n∈N is bounded; so, by Lemma 6, the sequence {xn }n∈N is bounded and, in view of the reflexivity of E, there exists a subsequence {xn k }k∈N weakly convergent to some y ∈ E. Since Dh is lower semicontinuous in its first variable, we have   0 = lim Dh (xn , x) = lim Dh xn k , x ≥ Dh (y, x) ≥ 0, n→∞ k→∞ so Dh (y, x) = 0, which, by the strict monotonicity of S, yields y = x. This proves that xn x. Statement (iii) follows from (i) and (ii), since weak and strong convergence are equivalent in finite-dimensional spaces.   123 112 Journal of Optimization Theory and Applications (2022) 193:107–117 3 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 Dh (X , ·) : E → R ∪ {+∞} and h-projection mapping PXh : E ⇒ X by Dh (X , y) := inf Dh (x, y) x∈X and PXh (y) := {x ∈ X : Dh (x, y) = Dh (X , y)} , 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 PXh ) 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 PXh (y) = ∅ whenever at least one of the following conditions is satisfied: (i) X is bounded. (ii) For every unbounded sequence {xn }n∈N in X , there exists a subsequence {xn k }k∈N h(xn k ,Sy) such that inf k xn k  > Sy (this holds, in particular, if h(·, Sy) is superco- ercive). Proof Since h(·, Sy) is weakly continuous, Dh (·, y) is weakly continuous, too. Take a sequence {xn }n∈N in X such that limn→∞ Dh (xn , x) = Dh (X , x). If condition (i) holds, then the sequence {xn }n∈N is obviously bounded. If (ii) is satisfied and {xn }n∈N were unbounded, then we would have h(xn k , Sy) Dh (xn k , x) ≥ h(xn k , Sy) − xn k Sy = xn k  − Sy ; xn k  by the assumption, we obtain limk→∞ Dh (xn k , x) = +∞, a contradiction with limn→∞ Dh (xn , x) = Dh (X , x). Therefore {xn }n∈N is bounded, and hence, since E is reflexive, it has a weakly convergent subsequence {xn k }k∈N . Let x ∗ be the weak limit of xn k . Since X is weakly closed, x ∗ ∈ X . This, together with the equalities Dh (x ∗ , x) = lim Dh (xn k , x) = lim Dh (xn , x) = Dh (X , x), k→∞ n→∞ implies that x ∗ ∈ PXh (y), and the proof is complete.   123 Journal of Optimization Theory and Applications (2022) 193:107–117 113 We next give a sufficient condition for PXh (y) to have at most one element. Proposition 11 (Nonmultiplicity of PXh ) Let S and h be as is Definition 2. If X ⊆ E is convex and y ∈ E is such that h(·, Sy) is strictly convex and X ⊆ E is convex, then PXh (y) has at most one element. Proof Clearly, the strict convexity of h(·, Sy) implies that of Dh (·, y), which in turn implies that Dh (·, y) has at most one minimizer over X .   Combining Propositions 10 and 11, one obtains sufficient conditions for an h- projection mapping to be single-valued; in particular, one gets the following simple statement. Corollary 12 Let S and h be as is Definition 2. If E is reflexive, X ⊆ E is nonempty, convex and closed and, for every y ∈ E, the function h(·, Sy) is weakly continuous and strictly convex on X , then PXh is single-valued. The following characterization of the elements of PXh (y) generalizes part of [5, Fact 2.3]. Proposition 13 (Characterization of projections) Let S and h be as is Definition 2. If X ⊆ E is convex and closed, S : E ⇒ E ∗ is single-valued on its domain, y ∈ E, the function h is finite-valued and such that h(·, Sy) is subdifferentiable on PXh (y), and x̄ ∈ X , then x̄ ∈ PXh (y) if and only if there exists x ∗ ∈ ∂h(·, Sy) (x̄) such that x − x̄, Sy − x ∗  ≤ 0 ∀x ∈ X . Proof The statement follows from the equivalence x̄ ∈ PXh (y) ⇐⇒ 0 ∈ ∂ Dh (·, y) (x̄) + N X (x̄), with N X (x̄) denoting the normal cone to X at x̄, that is,   N X (x̄) := x ∗ ∈ E ∗ : x − x̄, x ∗  ≤ 0 ∀x ∈ X ; notice that ∂ Dh (·, y) (x̄) = ∂h(·, Sy) (x̄) − Sy.   Corollary 14 Let X , S, and h be as in Proposition 13. If y ∈ E is such that h(·, Sy) is differentiable on PXh (y), then x̄ ∈ PXh (y) ⇐⇒ x − x̄, Sy − ∇h(·, Sy) (x̄) ≤ 0 ∀x ∈ X . The following proposition is a special case of [19, Example 7.2], but we give its proof in order to make the paper self-contained. 123 114 Journal of Optimization Theory and Applications (2022) 193:107–117 Proposition 15 (Continuity of S) If S : E ⇒ E ∗ is maximally monotone and single- valued 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 = Rn , and notice that the proof of this result extends to arbitrary Banach spaces in a straightforward way). Let x ∈ dom(S), and let {xn }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 {xn }n∈N is contained in U ; then the sequence {Sxn }n∈N is bounded. Let Sxn k k∈N be any weakly∗ convergent subsequence, and denote its limit by x ∗ . Then,   the sequence xn k , Sxn 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 {Sxn }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 single- valued on dom(S), and h is norm×weak∗ continuous, then, for any norm-to-norm continuous f : X → dom(S), the function X × X  (x, y) → Dh (x, f (y)) ∈ R is jointly strongly continuous. Proof Let {(xn , yn )}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 (yn )}n∈N is bounded. Then, since {xn }n∈N strongly converges to x and {S f (yn )}n∈N weakly∗ converges to S f (y) , from the inequality |xn , S f (yn ) − x, S f (y)| ≤ |xn − x, S f (yn )| + |x, S f (yn ) − S f (y)| we can easily deduce that {xn , S f (yn )}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 {Dh (xn , f (yn ))}n∈N converges to Dh (x, f (y)).   Theorem 17 (Fixed point theorem for the composition of PXh with continuous map- pings) 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 PXh ◦ f has a fixed point. Proof The set A := {(x, y) ∈ X × X : Dh (x, f (y)) ≥ Dh (y, f (y))} 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 x0 ∈ X such that X × {x0 } ⊆ A, 123 Journal of Optimization Theory and Applications (2022) 193:107–117 115 that is, Dh (x0 , f (x0 )) ≤ Dh (x, f (x0 )) ∀x ∈ X ; equivalently, Dh (x0 , f (x0 )) = min Dh (x, f (x0 )), (2) x∈X which means that x0 ∈ PXh ( f (x0 )) ; thus, x0 is a fixed point of PXh ◦ f .   To derive a (finite dimensional) result on classical Bregman distances from Theorem 17, we need the following lemma. Lemma 18 If g : Rn → R ∪ {+∞} is convex, continuous and supercoercive, then the function h g defined by (1) is continuous on Rn × Rn . Proof By [2, Theorem 3.4], the conjugate function g ∗ is finite-valued; hence, since it is convex, it is continuous (see, e.g., [21, Corollary 10.1.1]). Consequently, as h g is a separable function with both terms being continuous, it is continuous, too.   h Corollary 19 (Fixed point theorem for the composition of PX g with continuous map- pings) Let X ⊆ Rn be nonempty, convex and compact, and g : Rn → R ∪ {+∞} be proper, strictly convex, continuous, supercoercive, and differentiable on dom(g). Then, for any continuous mapping f : X → dom(g), there exists a point x0 ∈ X such that D g (x0 , f (x0 )) = min D g (x, f (x0 )). (3) x∈X 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 h that PX g ◦ f has a fixed point x0 . By Proposition 4, we obtain (3).   4 Conclusions We have obtained convergence properties for Bregman-type distances associated with convex representations of maximally monotone operators. Such Bregman-type dis- tances 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 asso- ciated 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]. 123 116 Journal of Optimization Theory and Applications (2022) 193:107–117 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 the- orem for their compositions with continuous mappings, which generalizes a classical result of Fan [15] for ordinary projections in normed spaces. Acknowledgements Juan Enrique Martínez-Legaz gratefully acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities, through Grant PGC2018-097960-B-C21 and the Severo Ochoa Program for Centers of Excellence in R&D (CEX2019-000915-S). He is affiliated with MOVE (Markets, Organizations and Votes in Economics). We are grateful to two anonymous reviewers, thanks to whose valuable remarks we have corrected some mistakes and improved the presentation. Funding Open Access Funding provided by Universitat Autonoma de Barcelona. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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