An enhanced BaillonHaddad theorem for convex functions on convex sets
Abstract.
In this paper, we prove the BaillonHaddad theorem for Gâteaux differentiable convex functions defined on open convex sets of arbitrary Hilbert spaces. Formally, this result establishes that the gradient of a convex function defined on an open convex set is Lipschitz if and only if it is cocoercive. An application to convex optimization through dynamical systems is given.
1. Introduction
Let be a Hilbert space endowed with a scalar product , induced norm and unit ball . Given a nonempty set and , we say that an operator is cocoercive if for all
(1) 
and is Lipschitz continuous if for all
(2) 
If , then (1) means that is firmly nonexpansive and (2) that is nonexpansive (see, e.g., [5, Chapter 4]). It is clear that (1) implies (2), while the converse, in general, is false (take for example ). Despite of this negative result, the BaillonHaddad theorem ([3, Corollaire 10]) states that if is the gradient of a convex function, then (1) and (2) are equivalent. The precise statement is the following:
Theorem 1.1 (BaillonHaddad).
Let be convex, Fréchet differentiable on , and such that is Lipschitz continuous for some . Then is cocoercive.
This prominent result provides an important link between convex optimization and fixedpoint iteration [8]. Moreover, it has many applications in optimization and numerical functional analysis (see, e.g., [2, 5, 9, 16, 17]). An improved version of Theorem 1.1 appeared in [4] (see also [8, Theorem 1.2]), where the authors relate the Lipschitzianity of the gradients of a convex function with the convexity and Moreau envelopes of associated functions (see [4, Theorem 2.1]). Furthermore, they provided the following BaillonHaddad theorem for twice continuously differentiable convex functions defined on open convex sets.
Theorem 1.2.
[4, Theorem 3.3] Let be a nonempty open convex subset of , let be convex and twice continuously Fréchet differentiable on , and let . Then is Lipschitz continuous if and only if it is cocoercive.
Finally, the authors left as an open question the validity of Theorem 1.2 for Gâteaux differentiable convex functions (see [4, Remark 3.5]).
The aim of this paper is to extend Theorem 1.2 to merely Gâteaux differentiable convex functions (see Theorem 3.1). To do that, we first establish the result in finitedimensions and then we use a finite dimensional reduction.
We emphasize that extend Theorem 1.2 is of interest because it provides an important link between the gradient of convex functions defined on convex sets and cocoercive operators defined on convex sets.
Cocoercivity arises in various areas of optimization and nonlinear analysis (see, e.g., [1, 5, 6, 11, 14]). In particular, it plays an important role in the design of algorithms to solve structured monotone inclusions (which includes fixed points of nonexpansive operators). Indeed, let us consider the structured monotone inclusion: find such that
(3) 
where is a convex lower semicontinuous function and is a monotone operator. It is well known that (see, e.g., [1]) the problem (3) is equivalent to the fixed point problem: find such that
(4) 
where and is the proximal mapping of (see, e.g., [5, Definition 12.23]) defined by
To solve the fixed point problem (4), Abbas and Attouch [1] introduces the following dynamical system
(5) 
whose equilibrium points are solutions of (4). They proved the following result (see [1, Theorem 5.2])
Proposition 1.3.
Let be a convex lower semicontinuous proper function, and a maximal monotone operator which is cocoercive. Suppose that and
Then the unique solution of (5) weakly converges to some element .
The previous result was extended by Boţ and Csetnek (see [6, Theorem 12]) to solve the monotone inclusion: find such that
(6) 
where is a maximal monotone operator and is cocoercive. These two results were extended by the authors in [14], where we proved the strong convergence of a Tikhonov regularization for the dynamical system (5). It is important to emphasize that in order to solve the problems (4) and (6), it is enough that the operator is defined in and , respectively. Therefore, it is interesting to have characterizations of cocoercive operators defined on open convex subsets of . Thus, it is important to extend Theorem 1.2 to merely Gâteaux differentiable functions (see [4, Remark 3.5]).
2. Notation and Preliminaries
Let be a Hilbert space endowed with a scalar product , induced norm and unit ball . We denote by the set of continuous linear operators from into . The norm of an operator is defined by
Given an open convex set , we denote by the class of Fréchet differentiable functions whose gradient is locally Lipschitz (see, e.g., [15, Chapter 9]).
Given and , we say that is cocoercive on if for all
Example 1.
The following list provides some examples of cocoercive operators (see refer to [5, Chapter 4] for further properties on cocoercive operators):

is nonexpansive if and only if is cocoercive.

is cocoercive if and only if is Lipschitz.

A matrix is psdplus (that is, for some positive definite) if and only if the mapping is cocoercive (see [17, Proposition 2.5]).

The Yosida approximation of a maximal monotone operator is cocoercive (see [5, Corollary 23.11]).
For a convex function we consider the convex subdifferential of at as
It is wellknown that for two functions the following equality holds (see, e.g., [5, Corollary 16.48]):
(7) 
To prove our main result, we will use finite dimensional reduction arguments, thus, some elements of generalized differentiation in finite dimensions will be needed. We refer to [15] for more details.
Let be a function. For , we define the Generalized Hessian of at (see, e.g., [15, Theorem 9.62] and [12]) as the set of matrices
where is the dense set of points where is twice differentiable (by virtue of Rademacher’s theorem the set exists). The following result (see [15, Theorem 13.52]) establishes some properties of the Generalized Hessian .
Proposition 2.1.
Let be a function, where is a open set. Then is a nonempty, compact set of symmetric matrices.
The following result gives a known characterization of convexity and Lipschitzianity of functions (see, e.g., [15, 12]). We give a proof for completeness.
Proposition 2.2.
Let be a function with convex. Then

is convex if and only if for all and all one has

is Lipschitz on if and only if for all and all the inequality holds.
Proof.
(i) follows from [12, Example 2.2]. The necessity in (ii) is direct. To prove the sufficiency in (ii), it is enough to assume that for all and all the inequality holds. Fix and consider the function . Then, is locally Lipschitz on and
Thus,
Hence, according to [13, Theorem 3.5.2], the map is Lipschitz on . Finally, by virtue of [15, Exercise 9.9], we conclude that is Lipschitz on ∎
3. An enhanced BaillonHaddad theorem
In this section, we state and prove the main result of the paper, that is, the BaillonHaddad theorem for convex functions defined on convex sets, which extends [4, Theorem 3.3] and solves the question posed in [4, Remark 3.5].
Theorem 3.1.
Let be a nonempty open convex subset of a Hilbert space , let be a convex function and . Then the following are equivalent.

is Gâteaux differentiable on and is Lipschitz continuous on .

the map is convex on .

is Gâteaux differentiable on and is cocoercive.
Moreover, if any of the above conditions hold, then .
To prove Theorem 3.1, we show first the result in finite dimension under the additional assumption that (see the next lemma). Then, we obtain Theorem 3.1 in finite dimensional spaces (see Lemma 3.3). Finally, the proof of Theorem 3.1 follows from finite dimensional reductions and Lemma 3.3.
Lemma 3.2.
Let be a nonempty open convex subset of , let be a convex function and . Then the following are equivalent.

is Lipschitz continuous on .

the map is convex on .

is cocoercive.
Proof.
Let us consider the functions and . It is clear that
(8) 
On the one hand,
which shows that is equivalent to .
On the other hand,
which proves that is equivalent to . ∎
Now we proceed to delete the hypothesis from Lemma 3.2.
Lemma 3.3.
Let be a nonempty open convex subset of , let be a convex function and . Then the following are equivalent.

is Gâteaux differentiable on and is Lipschitz continuous on .

the map is convex on .

is Gâteaux differentiable on and is cocoercive.
Proof.
According to [7, Theorem 2.2.1], for functions defined on subsets of , Gâteaux differentiability is equivalent to Fréchet differentiablity. We proceed to show that any of the above conditions imply that . Indeed, it is clear that (a) and (c) implies that . To prove that (b) implies that , we follow some ideas from [8]. Let us define . Thus,
which implies that for all . Therefore, and are nonempty and contain a single element. Hence, by virtue of [7, Corollary 4.2.5], the function is Gâteaux differentiable on and, thus, Fréchet differentiable on and continuously differentiable on (see [7, Theorem 2.2.2]). It is not difficult to prove that (b) implies the following inequality:
Fix and define . Then and for all and . Thus, we obtain
(9) 
Fix and such that such that
Let and such that . Therefore, by taking in (9) and using that , we obtain
Analogously, we get
Thus, for all
which shows that is Lipschitz on . Therefore, ∎
Now we are ready to prove Theorem 3.1.
Proof of Theorem 3.1
: Let and define . We observe that is a finite dimensional Hilbert space. Thus the restriction of to , , is Gâteaux differentiable in and for all
Hence, for all
which shows that is Lipschitz on . Therefore, according to Lemma 3.3, the map
is convex on . In particular, for all , for all
Since are arbitrary, it follows that the map is convex on .
: We first observe that is convex (with finite values) and for all
Hence, by virtue of (7), for all
which implies that and are nonempty and contain a single element. Therefore, according to [7, Corollary 4.2.5], the function and are Gâteaux differentiable on . Thus, if is finite dimensional, then is convex on . Hence, by virtue of Lemma 3.3, is Lispchitz on , i.e., for all
(10) 
Let us consider
Hence, since (10) holds for any finite dimensional, we obtain
(11)  
Therefore, by taking supremum in (10), we conclude that for all
which proves .
: It is straightforward.
: Let with . Then is a Hilbert space and for all
Hence, as a result of Lemma 3.3, for all
Since is arbitrary, by using (11), we conclude that for all
which shows the equivalence between , and . Finally, the fact that any of the above conditions imply that follows from Smulian’s theorem (see, e.g., [7, Theorem 4.2.10]).
4. Application to Convex Optimization
In this section, we present an application of Theorem 3.1 to convex optimization. Let be a Hilbert space, and be a convex function defined over an open convex set with
We study the Tikhonov regularization for the projected dynamical system (5) (see [10, 14] for more details on Tikhonov regularization). Let us consider the following assumptions:
Assumption Let be a positive function satisfying

is absolutely continuous, nonincreasing and ;

;

.
We observe that, for example, the function with satisfies the previous assumption.
Theorem 4.1.
Assume, in addition to Assumption , that is Lipschitz on and . Let and be a function satisfying Assumption . Let be the unique solution of
where . Then converges strongly to , as .
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