Integration of the Fenchel Subdifferentials Revisited

We obtain a simple integration formula for the Fenchel subdifferentials on Euclidean spaces and analyze some of its consequences. For functions defined on locally convex spaces, we present a similar result in terms of 𝜖-subdifferentials.

the epigraph of f , even in the case when f is defined on an infinite dimensional Banach space with the Radon-Nikodym property [3,Theorem 3.7]. Another integration result for epi-pointed nonconvex functions on Banach spaces, in terms of -subdifferentials, has been recently obtained by R. Correa et al. [5,Theorem 13]. Finally, a variant of this result, valid for nonnecessarily epi-pointed functions on general normed spaces, is due to M.A. López and M. Volle [8,Theorem 4].
In this paper, we obtain a very simple integration result for convex functions on R n , in terms of the Fenchel subdifferentials, by means of a quite elementary yet powerful technique devised by P. Kocourek [7, Proof of Theorem 1], who used it to provide a simple proof of Corollary 2 below, employed also in [6] to characterize Lipschitz DC functions in terms of -subdifferentials. From our new result, the Rockafellar integration formula follows as an immediate corollary, but the two formulas are significantly different. Our formula can be expressed by means of a supremum, like in the case of the Rockafellar formula, but also as an ordinary limit, and does not require the knowledge of the whole subdifferential mapping but just of an arbitrary single valued selection of this mapping. We present a detailed analysis of the new formula and draw several of its consequences. For functions defined on general locally convex spaces, we present an alternative integration formula involving -subdifferentials. All the convex analytic notions, notations, and results that we will use are fairly standard. We refer to the classical book [12] for finite dimensional convex analysis and to [15] for an excellent treatment of its extension to the setting of locally convex spaces.
The rest of the paper consists of three sections. In Section 2, we present and analyze the new integration formula for the Fenchel subdifferentials of l.s.c. convex functions defined on R n and discuss some of its consequences. Using the new integration formula, in Section 3, we obtain a characterization of -subgradients in terms of subgradients, which is considerably simpler than that of [9,Theorem 1]. Section 4 deals with the general locally convex case, in which the difficulty caused by the absence of a suitable result on the nonemptiness of the Fenchel subdifferential is overcome by considering -subdifferentials.

Integration of the Fenchel Subdifferentials of Functions Defined on R n
In this section, we consider l.s.c. convex functions defined on R n . The main result, which is presented next, is a new integration formula for the Fenchel subdifferentials.
Proof We have Adding up these inequalities, we get and hence On the other hand, we also have and adding up these inequalities, we get therefore, since lim m→∞ f (x m,m−1 ) ≥ f (x) due to the lower semicontinuity of f , From (3) and (6), equality (1) immediately follows.
The classical Rockafellar integration formula is an immediate consequence of Theorem 1: Corollary 1 [11] Let f and x 0 be as in Theorem 1. Then, for every x ∈ cl dom f , one has Proof The inequality "≥" in (7) easily follows from the subgradient inequality. To prove the opposite inequality, use Theorem 1 after setting x i := x m+1,i and x * i := s(x m+1,i ) in (7), which yields The following result is an easy consequence of Theorem 1, too.
Then f = g + const.
Proof We have ri dom f ⊆ Dom ∂f ⊆ dom f (see [12,Theorem 23.4]), from which one can easily deduce that cl dom f = C, and hence ri dom f = ri C. In the same way, we have cl dom g = C; by [12,Theorem 6.3], from this equality, it follows that ri dom g = ri C. Then, for x 0 ∈ ri C, applying Theorem 1 with s : It is worth mentioning that Corollary 2 also follows from the classical integration result ([11, Theorem 3], [12, Theorem 24.9]) that says that ∂f = ∂g implies f = g+const. Indeed, assuming, without loss of generality, that C has a nonempty interior, condition (8) implies that the gradients of f and g coincide at every common differentiability point. Since the set of such points is dense in C [12, Theorem 25.5] and, on the other hand, the subdifferential at a point is determined by the gradients at neighboring differentiability points, it turns out that condition (8) actually implies the equality ∂f = ∂g.
Our next result states the uniqueness, up to an additive constant, of the integral of a (nonnecessarily maximal) cyclically monotone mapping under the assumption that its domain has a convex relative interior.

Corollary 3 Let
for every x ∈ R n , uniquely determined up to an additive constant.
Proof By [12, Theorem 24.8], there exists a l.s.c. convex function f : for every x ∈ R n . We can assume, without loss of generality, that cl Dom ∂f = cl Dom A. Let g be another function with the same properties as f and take a single valued selection  [10] (see also [13,Theorem 12.41]), the set Dom A is nearly convex, that is, it contains the relative interior of its convex hull, which implies that this relative interior coincides with that of Dom A. Therefore, ri Dom A is convex, and thus Corollary 3 applies.
Our next result is another consequence of Corollary 3. Using the fact that the subdifferential of a convex function f at a point in ri dom f is the sum of a compact convex set with the orthogonal subspace to the affine hull of ri dom f , and its support function is the directional derivative at that point, as a function of the direction, we obtain the following corollary of Theorem 1.

Corollary 5 Let
Corollary 6 Let f and x 0 be as in Theorem 1. Then Proof Apply Theorem 1, first with s( According to the next result, the supremum and the infimum in (9) are attained only in exceptional cases. Proof We will just consider the case of (1), since the formulas in (9) are particular cases. Assume that the supremum is attained at m. In view of (2), we have Hence, for i = 0, 1, . . . , m − 1 and t ∈ [0, 1], we obtain  (1) and (9) are not attained.
f (x m,i , −(x − x 0 ))} need not be monotone. Consider, for instance, the case when n = 1, f (x) = |x|, x 0 = −1 and s is the selection of ∂f such that s(0) = 1; in this case, for x = 1, a straightforward calculation yields However, it is easy to extract monotone subsequences, as the following proposition shows. Proof We prove the result only for the first subsequence. Setting z k := m k+1 m k , we get s(x m k+1 ,i ) .
The preceding inequality follows from the monotonicity of the subdifferential mapping; indeed, for j = iz k + 1, . . . , (i + 1)z k − 1, we have which implies If f is strictly convex, all these inequalities are strict, since then the subdifferential mapping is strictly monotone.

A Characterization of -Subgradients
Using the Rockafellar integration formula, an expression for the -subdifferential operator of a lower semicontinuous proper convex function in terms of its subdifferential was given in [9,Theorem 1]. In this section, we obtain a simpler characterization of -subgradients, thanks to our new integration formula (1).
Theorem 2 Let f and s be as in Theorem 1, x ∈ dom f , ≥ 0 and x * ∈ R n . Then Proof If x * ∈ ∂ f (x) and x 0 ∈ ri dom f , then f ( . By adding up these inequalities, simplifying, and rearranging, we easily obtain (10). Conversely, assume that x * satisfies (10). Let x 0 ∈ ri dom f and α > f (x 0 )−f (x). By (1), there exists m such that 1 Since f , being convex, is upper semicontinuous along lines [12,Theorem 10.2], and the relative interior of any segment joining x 0 and a point in dom f is contained in ri dom f , the preceding inequality also holds for an arbitrary x 0 ∈ dom f . This proves that x * ∈ ∂ f (x).
By means of the same argument used in the proof of Corollary 6, from Theorem 2, we obtain the following alternative characterization of -subgradients in terms of directional derivatives.
Corollary 8 Let f , x, ≥ 0 and x * be as in Theorem 2. Then the following statements are equivalent: Let us recall that a set is said to be nearly convex if it contains the relative interior of its convex hull.
Corollary 9 Let f : R n −→ R ∪ {+∞} be a l.s.c. proper convex function. Then ∂f is maximal in the set of mappings A : R n ⇒ R n that have a nearly convex domain and a single valued selection s on Dom A satisfying the following property: Proof Since ∂f is maximal monotone, its domain is nearly convex [10]. Moreover, property (11) is an easy consequence of the cyclic monotonicity of ∂f . Maximality immediately follows by applying Theorem 2 with = 0.

Integration of -Subdifferentials of Functions Defined on Locally Convex Spaces
In this section, we consider the more general setting of locally convex spaces. Due to the lack of a suitable result on the nonemptiness of the subdifferential in this context, our results will be expressed in terms of -subdifferentials.
Theorem 3 Let X be a real locally convex space with dual X * and duality pairing ·, · : Hence, for x ∈ dom f and x * ∈ ∂f (x 0 ), one has Proof The proof follows the same pattern as the proof of Theorem 1, with small changes. We have Adding up these inequalities, we get and hence On the other hand, we also have From (14) and (15), equality (12) immediately follows. To prove (13), we observe that the inequality "≥" is immediate and obtain the opposite inequality using (12)  From Theorem 3, one can easily derive the following analogs of Corollaries 1 and 2 by mimicking their proofs.

Corollary 10
Let X, f , and x 0 be as in Theorem 3. Then, for every x ∈ dom f , one has A formula very similar to (16) for the l.s.c. convex hull of an epi-pointed function defined on a Banach space has been obtained by R. Correa et al. [5,Theorem 13].
Then f = g + const.