On Subdifferentials Via a Generalized Conjugation Scheme: An Application to DC Problems and Optimality Conditions

This paper studies properties of a subdifferential defined using a generalized conjugation scheme. We relate this subdifferential together with the domain of an appropriate conjugate function and the ε-directional derivative. In addition, we also present necessary conditions for ε-optimality and global optimality in optimization problems involving the difference of two convex functions. These conditions will be written via this generalized notion of subdifferential studied in the first sections of the paper.


Introduction
Among the huge variety of optimization problems that can be found in real life, those whose objective function is expressed as the difference of two convex functions have attained a lot of attention since decades in the optimization community. These problems are called DC problems where DC means difference of convex functions. For an in-depth introduction as well as some applications of DC programming, we recommend the reader [1][2][3][4] and the references therein.
Given a general DC problem, there exist different approaches to study conditions for optimality. Just to mention a few, we start with the renewed paper [5], where the authors deal with optimality conditions which are necessary and sufficient for DC semi-infinite programming using subdifferentials. In [6] new global conditions for non-smooth DC optimization problems via affine support sets are developed, [7] works with DC problems under convex inequality constraints and [8] explores DC programming in reflexive Banach spaces. The works of [9] and [10] develop conditions in terms of epigraphs with infinite constraints, while [11] focuses on polynomial constraints and [12,13] on DC programs involving composite functions and canonical DC problems, respectively. We also mention [14] where a group of global optimality conditions is compared using a generalized conjugation theory as framework.
Concerning optimality conditions for global maximum of a general function in Euclidean spaces, we mention [15] and [16]. While the former paper introduces a useful characterization of global optimality using the subdifferential and the normal cone, the latter goes beyond that. More precisely, it compares the characterization coming from [15] with some other necessary and sufficient conditions named Strekalovski, Singer-Toland and Canonical-dc-programming passing through their necessary assumptions to hold and their relationships.
In the current paper we will not be motivated directly by any of these conditions, but by another from [17] written in terms of just subdifferentials: if f and g are proper convex and lower semicontinuous functions, x is a global minimizer of f = g − h if and only if This subdifferential notion is linked to Fenchel conjugation scheme, in fact it is called Fenchel subdifferential in [18]. An evidence of this connection is the following equivalence between the subdifferential of a convex and lower semicontinuous function and the subdifferential of its Fenchel conjugate Equivalence (1) holds thanks to Fenchel-Moreau theorem, which establishes the equality between a proper convex lower semicontinuous function and its Fenchel biconjugate. Nevertheless, Fenchel conjugation scheme is not suitable (in the sense that Fenchel-Moreau theorem does not hold with them) for a class of functions which generalizes the class of convex and lower semicontinuous functions, namely the evenly convex functions (see [19]). These functions have epigraphs which are evenly convex sets, i.e., intersections of arbitrary families (possibly empty) of open half-spaces. This kind of sets was initially defined by Fenchel [20] in the Euclidean space, trying to extend the polarity theory to nonclosed convex sets. Later, they were applied in linear inequality systems (see [21] and [22]), because evenly convex sets are the solution sets of linear systems containing strict inequalities. In addition, [23] contains basic properties of evenly convex sets expressed in terms of their sections and projections.
In [24] it is provided a conjugation scheme for extended real functions, called c-conjugation, and a subdifferential notion associated with it, which would allow to obtain a counterpart of (1) for proper evenly convex functions; see [25]. Another interesting application of c-conjugation developed in the last years has been the building of different dual problems for a primal convex one, in which strong duality property is related to the even convexity of the functions in the primal problem. A very recent monograph presents the state of the art in Even Convexity and Optimization; see [26].
Our commitment in this paper is to develop basic, on one hand, and interesting, on the other hand, properties of the subdifferential concept associated with c-conjugation, and to obtain optimality conditions for DC problems via this operator.
Concerning the organization, Section 2 summarizes the necessary results throughout the paper. In Section 3 we introduce the formal definition of the subdifferential of interest in this paper and its main properties showing its connection with a generalized notion of conjugate function. Section 4 is devoted to the analysis of deeper properties of this subdifferential. In particular, we will present its relationship with the domain of the conjugate function and the subdifferential of its ε-directional derivative. Section 5 develops necessary optimality conditions for DC problems when the involved functions are proper and evenly convex. Finally, Section 6 summarizes the most important achievements of this paper as well as points out related open problems for future research.

Preliminaries
We denote by X a nontrivial separated locally convex space, lcs in short, equipped with the σ(X,X * ) topology induced by X * . Here, X * represents the continuous dual space of X endowed with the σ(X * ,X) topology. Given a continuous linear functional x * ∈ X * , ⟨x, x * ⟩ represents its value at x ∈ X. If D ⊆ X , convD and clD stand for its convex hull and closure, respectively.
As we said in the previous section, Fenchel [20] defined an evenly convex set (e-convex set, in brief) as an intersection of an arbitrary family (possibly empty) of open half-spaces. The following equivalent definition provides a very useful tool in order to identify e-convex sets.

Definition 1 ([27, Def. 1])
Given C ⊆ X , we represent the smallest e-convex set in X containing C, i.e., its e-convex hull, by e − convC. If C ⊆ X is convex, the inclusions C ⊆ e − convC ⊆ clC are fulfilled. It is worthwhile adding that due to the fact that the class of e-convex sets is closed under arbitrary intersections, this operator is well defined. By hypothesis X is a separated lcs, so X * ≠{0}. Furthermore, Hahn-Banach theorem implies not only that X is e-convex, but also the fact that every closed or open convex set is e-convex, too.
If f ∶ X → ℝ , the set domf = {x ∈ X ∶ f (x) < +∞} represents its effective domain while epif = {(x, r) ∈ X × ℝ ∶ f (x) ≤ r} stands for its epigraph. The function f is proper if f (x) > −∞ for all x ∈ X and domf≠∅. With clf we mean the lower semicontinuous hull of f, i.e., the function verifying the equality epi(clf) = cl(epif). We say that f is lower semicontinuous, or lsc, if f(x) = clf(x) for all x ∈ X, and e-convex if epif is e-convex in the product space X × ℝ . It is immediate to observe that the class of lsc convex functions is contained in the class of e-convex functions. Nevertheless, this inclusion fails to be an equality between sets.

Example 1 ([28, Ex. 2.1]) Let f ∶ ℝ → ℝ be the function
It is straightforward to see that is not closed but convex. However, it is e-convex, since for any point (0, α) with nonnegative α, the hyperplane H = (x, y) ∈ ℝ 2 ∶ x = 0 passes through the point with empty intersection with epif; recall Definition 1.
We continue defining the e-convex hull of a given function f ∶ X → ℝ as the largest e-convex minorant of f and we denote it by e − convf. Using the generalized convex conjugation theory presented in Moreau [29], a conjugation scheme appropriated for e-convex functions is given in [24]. Let W ∶= X * × X * × ℝ and the coupling functions In [24] it is shown that the family of pointwise suprema of sets of c-elementary functions, that is, functions x ∈ X → c(x, (x * , u * , )) − ∈ ℝ , with (x * ,u * ,α) ∈ W and ∈ ℝ , is indeed, the family of proper e-convex functions from X to ℝ along with the function f ≡ +∞.
In [30] it is defined the notion of e ′ -convex function as that function g ∶ W → ℝ which is the pointwise supremum of sets of c ′ -elementary functions, that is, functions (x * , u * , ) ∈ W → c(x, (x * , u * , )) − ∈ ℝ with x ∈ X and ∈ ℝ . Moreover, the e ′ -convex hull of any function g ∶ W → ℝ , e � − convg , is its largest e ′ -convex minorant. The epigraphs of e ′ -convex functions are called e ′ -convex sets, and for any set D ⊂ W × ℝ , its e ′ -convex hull, denoted by e � − convD , is the smallest e ′ -convex set that contains D. We recommend the reader [31] for characterizations of e ′ -convex sets and additional properties of e ′ -convex functions.
We close this preliminary section with a result that shows the suitability of the c-conjugation scheme for e-convex functions and represents the counterpart of Fenchel-Moreau theorem for them.

First Properties of the c-subdifferential
Given a function, its subdifferentiability at a point associated with the c-conjugation scheme was considered in [24] as a particularization of the c-subdifferentiability introduced first in [33].

Definition 2 ([24, Def. 44])
The set of all c-subgradients of f at x 0 is denoted by ∂ c f(x 0 ) and is called the c-subdiffer- In [25] it was introduced the notion of c ′ -subdifferentiability, following [33] too.

Definition 3 ([25, Def. 4.2])
The set of all c ′ -subgradients of g at ( The following standard notation will be used throughout the paper. Given fix points u * ∈ X * ( (x, u, ) ∈ X × X × ℝ , resp.) and ∈ ℝ , we denote an open hyperplane in X (in W, resp.) by and respectively. According to [24], given f ∶ X → ℝ , its c-subdifferential at x 0 ∈ domf can be written as where ∂f stands for the classical (Fenchel) subdifferential. This relation between the standard subdifferential of f, ∂f, and its c-subdifferential, ∂ c f, will play a fundamental role throughout next sections.
Now we present the counterparts of some well-known results in classical subdifferential theory. The next theorem sums up some basic properties involving c-subdifferential and c ′ -subdifferential sets.

Remark 1
Statement iv) in the previous theorem establishes a relationship between subdifferential and c ′ -subdifferential sets, as it occurs in the case of c-subdifferentiability; recall (3).
The notion of ε-c-subgradient appears firstly in [30] allowing a characterization of the epigraph of the c-conjugate (see [30,Lem. 9]). Moreover, it is used to build an alternative formulation for a general regularity condition in evenly convex optimization in [25,Th. 4.9].

Definition 4 ([30, Def. 4])
Let f ∶ X → ℝ be a proper function and ε ≥ 0. A vector (x * ,u * ,α) The set of all ε-c-subgradients of f at x 0 is denoted by ∂ c,ε f(x 0 ) and is called the ε-csubdifferential set of f at x 0 . In the case f ( Regarding the relationship between the notion of ε-c-subdifferentiability and the classical ε-subdifferentiability, it is easy to see that, for all x 0 ∈ domf, Moreover, we also have that, if 0 ≤ 1 ≤ 2 < ∞ , then and In a similar way, we can define the ε-c ′ -subdifferential set of a function g ∶ W → ℝ at (x * 0 , u * 0 , 0 ) ∈ W.

Proof i) Similar to the proof of item i) in Theorem 3.
ii) (x * ,u * ,α) ∈ ∂ c,ε f(x 0 ) if and only if 〈x 0 ,u * 〉 < α and, for all x ∈ X, equivalently, 〈x 0 ,u * 〉 < α and Analogously, x 0 ∈ c � , f c (x * , u * , ) if and only if and taking into account that, according to Theorem Then ⟨x 0 , u * 1 + u * 2 ⟩ < 1 + 2 and, adding both inequalities and considering the additivity property of the coupling function in its second component, which can be applied in this case, we obtain However, it yields thus, we conclude and, consequently,

Further Properties of the c-subdifferential
We continue developing additional properties of the c-subdifferential extending them from [18]. Next proposition establishes the relationship between the c-subdifferential of a proper function and the domain of its c-conjugate.
Proposition 5 Let f ∶ X → ℝ be a proper function and x 0 ∈ domf. Then c f (x 0 ) ⊆ domf c .
Proof Take (x * ,u * ,α) ∈ ∂ c f(x 0 ). Then, by definition it holds together with ⟨x 0 , u * ⟩ < . Rearranging, this means that with ⟨x 0 , u * ⟩ < , which, after taking supremum on the right-hand-side, can be rewritten as Since ⟨x 0 , u * ⟩ < , the coupling function is finite and due to the fact that x 0 ∈ domf by hypothesis, we conclude that (x * ,u * ,α) ∈ domf c .
According to Lemma 1, for any proper function f it holds that a point (x * ,u * ,α) ∈ domf c belongs to ∂ c f(x 0 ) if and only if f c (x * ,u * ,α) = c(x 0 ,(x * ,u * ,α)) − f(x 0 ), which means that sup X {c(x, (x * , u * , )) − f (x)} must be attained at x 0 . This is not necessarily true, so the inclusion in Proposition 5 may be strict. for all x ∈ X. Next proposition shows that equality between domf c and ∂ c f(x 0 ) is a sufficient condition for the above inequality to be an equality, if f is e-convex.
Proposition 6 Let f ∶ X → ℝ be a proper e-convex function with x 0 ∈ domf and ∂ c f(x 0 ) = domf c . Then, for all x ∈ X, Proof Due to Lemma 1, for any (x * ,u * ,α) ∈ ∂ c f(x 0 ) we have that so multiplying by − 1 and adding c(x,(x * , u * , α)) we get being this equality true for all x ∈ X and (x * , u * , α) ∈ ∂ c f(x 0 ). Taking supremum over domf c = ∂ c f(x 0 ) in both sides of the previous equality, we have, for all x ∈ X, Since f is e-convex by hypothesis, according to Theorem 1, f = f cc � , and the proof ends.
The goal now is to relate the ε-directional derivative with the ε-c-subdifferential of f. We recall the definition of the ε-directional derivative of a function f at a point x along the direction u, i.e., see [18,Th. 2.1.14]. Observe that [18,Th. 2.4.4] states the relationship between f � (x 0 , ⋅)(0) and ∂ ε f(x 0 ) for a proper function and x 0 ∈ domf. The following theorem deals with this relation when c-subdifferentiability is used. First, we recall the definition of the normal cone of a convex set C ⊂ X at a point x 0 ∈ C, Theorem 7 Let f ∶ X → ℝ be a proper function, x 0 ∈ domf and ε ≥ 0. Then Proof Take (x * , u * , ) ∈ c f � (x 0 , ⋅)(0) such that ⟨x 0 , u * ⟩ < . Then α > 0 and By the definition of f ′ , Let x := x 0 + tu. Hence, for all x ∈ X and t > 0 it holds In particular, for t = 1, we get , for all u ∈ X, t > 0.
Due to Theorem 7, we pursue the counterpart of [18,Th. 2.4.11], which establishes that if f is proper convex and lsc, x 0 ∈ domf and ε > 0, then Next result comes immediately from Theorem 7.
Corollary 1 Let f ∶ X → ℝ be a proper function with x 0 ∈ domf and ε > 0. Then, for all u ∈ X it holds Remark 3 It is not an easy task to find sufficient conditions for Corollary 1 to hold with an equality, maybe a separation theorem for e-convex sets needs to be studied, in some way, besides asking the function f to be e-convex, for instance. Hence, we have decided to leave this problem to work on it in a near future.

Optimality Conditions via c-subdifferentials
In [17], Hiriart-Urruty established ε-optimality and global optimality conditions for DC programs, which are, recall, optimization problems of the type inf X {f (x) − g(x)} , where f and g are convex functions. To avoid ambiguity, we will use the usual convention +∞ − (+∞) = +∞ when minimizing DC problems. Recall that, for any ε ≥ 0, a point a ∈ X is said to be an ε-minimizer of a function h ∶ X → ℝ if h(a) is finite and for all x ∈ X. Those optimality conditions are obtained via subdifferential and ε-subdifferential sets of the involved functions in the problem.
g(a) ⊆ + f (a), for all ≥ 0. [36], where it is pointed out that the convexity and lower semicontinuity of f are not essential assumptions.

Remark 4 An alternative proof of Theorem 8 is given in
In this section, our purpose is to provide a counterpart of Theorem 8, expressed in terms of even convexity and c-subdifferentiability. We will use the following definition of the difference of two sets in W. Next lemma can be derived from [38,Th. 3.1], which is stated in the generalized conjugation theory framework. We include the proof for the sake of completeness. The following theorem is stated with equality when f and g are as in Theorem 8 and ε-subdifferential sets are used; see Theorem 1 in [36].

Remark 6
The reason why a is asked to be in domg in the Theorem 10 is that, in other case, ∂ c,λ g(a) = ∅, for all λ ≥ 0, whereas a may not be an ε-minimizer of f − g, for instance (f − g)(a) = +∞ if a∉domf.

Remark 7
Local optimality necessary or sufficient conditions for DC problems where both functions are proper and convex (although convexity for f is not essential) can be found in [17] and [39]. Again they are expressed in terms of ε-subdifferential sets.
A characterization of local optimality in the finite dimensional context given in [40,Th. 4.3], assumes that the functions f and g are convex and lsc, and it is where B(x, ν) stands for the ball of radius ν > 0 centered at x ∈ ℝ n , and η > 0 is any scalar for which a is an optimum in the ball B(a, η). It could also have its counterpart for e-convex functions and ε-c-subdifferential sets. Nevertheless, as it can be observed in [40,Th. 4.3 Proof]), some further constraint qualifications are needed to split the ε-c-subdifferential of the sum of two e-convex functions. For more information on this, we encourage the reader to check [30,Th. 11].

Conclusions and Future Research
Throughout this manuscript we have exploited the main properties that a subdifferential defined via a generalized conjugation scheme satisfies. With the purpose of generalizing some results from [18], we have investigated the role of the ε-directional derivative and we have taken an insight on how the support function of the c-subdifferential may be derived.
As a theoretical application of the c-subdifferential, we have focused on the development of global optimality and ε-optimality conditions for DC problems. For problems whose objective function reads as the difference of two e-convex functions, which can be denoted by eDC problems, we have adapted well-known results from J.B. Hiriart-Hurruty in [17], which turns out to give necessary but not sufficient conditions via c-subdifferentials.
Throughout the manuscript we have pointed out some open issues that, from our perspective, go beyond the scope of the paper and deserve to be studied thoroughly as a future research. We conclude the paper mentioning the application of the c-subdifferential in the study of Toland-Singer duality. This type of duality has become quite popular in the community of DC programming when the involved functions are proper convex and lsc, so we expect the c-subdifferential to lead the duality theory of eDC problems.

Conflict of Interests
The authors declare that they have no conflict of interest.
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. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.