Infimal convolution, c-subdifferentiability, and Fenchel duality in evenly convex optimization

M.D. Fajardo1,3, J. Vicente-Perez´ 2 and M.M.L. Rodr´ıguez3

Department of Statistics and Operations Research, University of Alicante. E-03080 Alicante, Spain.

Abstract In this paper we deal with strong Fenchel duality for infinite dimensional optimization problems where both feasible set and objective function are evenly convex. To this aim, via perturbation approach, a conjugation scheme for evenly convex functions, based on generalized convex conjugation, is used. The key is to extend some well-known results from convex analysis, involving the sum of the epigraphs of two conjugate functions, the infimal convolution and the sum formula of ε-subdifferentials for lower semicontinuous convex functions, to this more general framework.

Mathematical subject classification: 52A20, 26B25. Keywords: Evenly convex function, generalized convex conjugation, Fenchel dual problem.

1 Introduction

A subset of a locally convex real topological vector space is called evenly convex (e-convex, in brief) if it is the intersection of an arbitrary family (possibly empty) of open halfspaces. This class of sets was introduced in the finite dimensional case by Fenchel [10] in order to extend the polarity theory to nonclosed convex sets. Recently, they have been applied in linear inequality systems [12, 13] since e-convex sets are the solution sets of linear systems containing strict inequalities. Also, basic properties of this class of sets by means of their sections and projections are given in [16]. Evenly quasiconvex functions were introduced (under the name of normal quasiconvex functions) by Mart´ınez-Legazin [18] as those functions whose sublevel sets are e-convex. We can find characterizations of this type of functions in [8]. Passy and Prisman [24] showed that this class of functions has the required property for a duality framework in quasiconvex programming. In [19] Mart´ınez-Legazpresented a survey on quasiconvex programming as a

1Corresponding author. E-mail address: [email protected] 2This author has been supported by FPI Program of MICINN of Spain, Grant BES-2006-14041. 3This author has been supported by MICINN of Spain and FEDER of EU, Grant MTM2008-06695-C03-01.

1 particular case of generalized convex duality theory based on Fenchel-Moreau conjugation. This generalized conjugation pattern has also been applied in [21] to functions with e-convex epigraphs, named e-convex functions, which were introduced in [29]. This class of functions extends the important class of lower semicontinuous convex functions, which play a crucial role in optimization theory. Motivated by these results, this paper is focused on the fulfilment of strong Fenchel duality, via perturbation approach, for infinite dimensional optimization problems where both feasible set and objective function are e-convex. As we shall see, the important fact is to extend some well-known results from convex analysis, dealing with lower semicontinuous convex functions, to this more general framework. The organization is as follows. Section 2 is dedicated to the necessary preliminaries in order to make the paper self-contained. In particular, the conjugation scheme for e-convex functions will be reminded, as well as its most important properties. In Sections 3 and 4 we shall extend the fundamental results on the sum of the epigraphs of two conjugate functions, the infimal convolution and the sum formula of ε-subdifferentials for lower semicontinuous convex functions to e-convex functions and the new conjugation pattern. Finally, Section 5 will be devoted to the fulfilment of strong Fenchel duality for the problem (P ) Inf f(x) (1) s.t. x ∈ A, where A ⊂ X is a nonempty e-convex set and f : X → R is a proper e-convex function.

2 Preliminaries

We shall use the standard notation and terminology of convex analysis. Let X be a real Banach space and X∗ its topological dual space endowed with the weak* topology. For the set D ⊂ X, the closure of D is denoted by cl D. If A ⊂ X∗, then cl A stands for the weak* closure of A. As a consequence of the Hahn-Banach theorem, every open or closed convex set is e-convex. The e-convex hull of C ⊂ X, denoted by eco C, is the smallest e-convex set that contains C. This operator is well defined because X is e-convex and the class of e-convex sets is closed under intersection. Moreover, if C is convex, then C ⊂ eco C ⊂ cl C. The duality product will be denoted by ⟨·, ·⟩ : X × X∗ → R, i.e. ⟨x, x∗⟩ = x∗ (x) for all (x, x∗) ∈ X × X∗. For any function f : X → R := R ∪ {±∞}, the effective domain and the epigraph of f are denoted in the usual way, dom f := {x ∈ X | f(x) < +∞} and epi f := {(x, r) ∈ X × R | f(x) ≤ r} , respectively. The lower semicontinuous (in short, lsc) hull of f, cl f : X → R, is defined such that epi (cl f) = cl (epi f), and f is said to be lsc at x ∈ X if f(x) = (cl f)(x). Furthermore, f is said to be proper if f does not take on the value −∞ and dom f ≠ ∅. On the other hand, according to [29], we will say that f is e-convex if its epigraph is an e-convex set in X × R. Obviously, any lsc convex function is e-convex, but the reverse statement is not true (consider, for instance, the indicator function of the set ]0, +∞[ ⊂ R). The e-convex hull of f, eco f : X → R, is defined as the largest e-convex minorant of f, that is, eco f := sup {g | g is e-convex and g ≤ f} .

2 This function is e-convex since the class of e-convex functions is closed under pointwise supremum. The Fenchel conjugate of f, f ∗ : X∗ → R, is defined by

f ∗ (x∗) := sup {⟨x, x∗⟩ − f(x)} , x∈dom f

and the ε-subdifferential of f at x ∈ dom f is defined for any ε ≥ 0 as

∗ ∗ ∗ ∂εf(x) := {x ∈ X | f(y) ≥ f(x) + ⟨y − x, x ⟩ − ε, ∀y ∈ X} .

It follows easily from the definitions that if a ∈ dom f, then ∪ ∗ ∗ ∗ ∗ epi f = {(v , ⟨a, v ⟩ + ε − f(a)) | v ∈ ∂εf(a)} . (2) ε≥0

For proper convex functions f, g : X → R, the infimal convolution of f with g, denoted by f ⊕ g : X → R, is defined by

(f ⊕ g)(x) := inf {f (x1) + g (x2)} , x1+x2=x and it is said to be exact at x ∈ X if (f ⊕ g)(x) = f (a)+g (x − a) for some a ∈ X. Moreover, the infimal convolution is exact if it is exact at any x ∈ X. Also, it is easy to check that the following equality holds for every x ∈ X,

(f ⊕ g)(x) = inf {r ∈ R | (x, r) ∈ epi f + epi g} . (3)

The next theorem (see, for instance, [2, 7]) deals with the infimal convolution linked to the sum of epigraphs of conjugates and ε-subdifferentials. In particular, the equality in (ii) is known as the classical Moreau-Rockafellar formula, and an alternative proof of this result can be found in [4] as a special case in a more general context. Theorem 1. Let f, g : X → R be proper lsc convex functions such that dom f ∩ dom g ≠ ∅. Then ∗ (i) cl (epi f ∗ + epi g∗) = epi (f + g) .

∗ (ii) (f + g) = cl (f ∗ ⊕ g∗). Furthermore, the following statements are equivalent: ∗ (iii) (f + g) = f ∗ ⊕ g∗ and the infimal convolution is exact.

(iv) epi f ∗ + epi g∗ is weak* closed.

(v) For each ε ≥ 0 and for each x ∈ dom f ∩ dom g, ∪

∂ε (f + g)(x) = ∂ε1 f (x) + ∂ε2 g (x) . ε1+ε2=ε ε1,ε2≥0

3 Fenchel biconjugation theorem establishes the equivalence between a function f to be lsc convex and the equality f = f ∗∗. This theorem does not apply for e-convex functions: if we take any e-convex non lsc function f, its biconjugate f ∗∗ is lsc and f ≠ f ∗∗. Due to the fact that the classical Fenchel conjugation is not suitable for e-convex functions, a new conjugation scheme is provided for this class of convex functions in [21], based on the generalized convex conjugation theory introduced by Moreau [22]. Given a real Banach space X, we will consider the set W := X∗ × X∗ × R with the coupling function c : X × W → R given by { ⟨x, y∗⟩ if ⟨x, z∗⟩ < α, c(x, (y∗, z∗, α)) := +∞ otherwise.

For any f : X → R, its c-conjugate f c : W → R is defined by

f c((y∗, z∗, α)) := sup {c(x, (y∗, z∗, α)) − f(x)} . (4) x∈X

′ Similarly, the c′-conjugate of g : W → R is the function gc : X → R defined by

′ gc (x) := sup {c′((y∗, z∗, α), x) − g(w)} , (5) (y∗,z∗,α)∈W

where c′ : W × X → R is given by c′((y∗, z∗, α), x) = c(x, (y∗, z∗, α)). The following conven- tions are used: +∞ + (−∞) = −∞ + (+∞) = +∞ − (+∞) = −∞ − (−∞) = −∞. Functions of the form x ∈ X → c(x, (y∗, z∗, α)) − β ∈ R, with (y∗, z∗, α) ∈ W and β ∈ R are called c-elementary; in the same way, c′-elementary functions are those of the form (y∗, z∗, α) ∈ W → c(x, (y∗, z∗, α)) − β ∈ R, with x ∈ X and β ∈ R. We will denote ′ by Φc (Φc′ ) the set of c-elementary (c -elementary, respectively) functions. We will say that f : X → R is Φc-convex (g : W → R is Φc′ -convex) if it is the pointwise supremum of a subset of Φc (Φc′ , respectively). Since the class of all Φc-convex (Φc′ -convex) functions is closed under pointwise supremum, every function f : X → R (g : W → R) has a largest Φc-convex (Φc′ -convex) minorant, which is called the Φc-convex hull (Φc′ -convex hull) of f (g, respectively).

By [21, Theorem 16], the family of Φc-convex functions is precisely the family of the proper e-convex functions from X into R along with the function identically −∞. In order to use an analogous terminology, we will say that a function g : W → R is e′-convex if it is ′ ′ Φc′ -convex. Also, the e -convex hull of any function k : W → R will be denoted by e co k. The proof of the following proposition is easy (see [20, p. 243]).

Proposition 2. Let f : X → R and g : W → R. Then

′ (i) f c is e′-convex ; gc is e-convex.

′ ′ (ii) If f has a proper e-convex minorant, eco f = f cc ; e′co g = gc c.

′ (iii) If f does not take on the value −∞, then f is e-convex if and only if f = f cc ; ′ g is e′-convex if and only if g = gc c.

′ ′ (iv) f cc ≤ f ; gc c ≤ g.

4 According to (iii) in the previous proposition, g is said to be e′-convex at (y∗, z∗, α) ∈ W if ′ g(y∗, z∗, α) = gc c(y∗, z∗, α), and the e′-convex hull of any function g : W → R coincides with its second c-conjugate. Finally, the following definition and theorem from [21] will be used throughout the paper.

Definition 1. A function a : X → R is said to be e-affine if there exist y∗, z∗ ∈ X∗ and α, β ∈ R such that { ⟨x, y∗⟩ − β if ⟨x, z∗⟩ < α, a (x) = +∞ otherwise.

For any f : X → R, we denote by Ef the set of all e-affine functions minorizing f, that is, { } Ef := a : X → R | a is e-affine and a ≤ f .

Theorem 3. Let f : X → R, f not identically +∞ or −∞. Then,

f is a proper e-convex function if and only if f = sup {a | a ∈ Ef } .

The e-convex functions identically −∞ and +∞ have also a representation as in Theorem 3 if we consider Ef the empty set and the set of all e-affine functions, respectively.

3 On the sum of epigraphs of two c-conjugates

Here, our goal is to link the sets epi (f + g)c and (epi f c + epi gc) being f, g : X → R proper e-convex functions such that dom f ∩ dom g ≠ ∅, and taking Theorem 1 (i) as a reference. The first thing to think about is what sort of hull we need to make (epi f c + epi gc) to reach epi (f + g)c. Applying Proposition 2, (f + g)c is an e′-convex function, hence its epigraph is an e-convex set due to the fact that every e′-convex function k : W → R is also e-convex. Therefore, it is possible to think in taking eco (epi f c + epi gc). However, we can not assert that this set is the epigraph of an e′-convex function defined on W , as we observe in the following remark. Remark 1. Consider X = Rn. Let 0 be the zero vector of W = R2n+1. Since (0, 1) belongs to both the recession cones of epi f c and epi gc, we have that (0, 1) belongs also to the recession cone of eco (epi f c + epi gc). According to [29, Proposition 2.13], k : W → R defined for all (y∗, z∗, α) ∈ W by

k (y∗, z∗, α) := inf {a ∈ R | (y∗, z∗, α, a) ∈ eco (epi f c + epi gc)}

is an e-convex function, and epi k = eco (epi f c + epi gc) ∪ gph k, where gph k stands for the graph of k. This set will be the tightest e-convex epigraph containing eco (epi f c + epi gc), and it may not coincide with eco (epi f c + epi gc). Observe that the e-convex hull of a sum of two epigraphs is not necessarily an e-convex epigraph: consider, for instance, the functions (x − 1)2 defined on [0, 1] ⊂ R and 0 defined on ]0, +∞[ ⊂ R. Therefore, we need to define another kind of hull to deal with this problem. For this purpose, we shall use the e′-convexity of (f + g)c and we establish the following definition.

5 Definition 2. We will say that a set D ⊂ W × R is e′-convex if there exists an e′-convex function k : W → R such that D = epi k. As the intersection of an arbitrary family of e′-convex sets is an e′-convex set, then the e′-convex hull of an arbitrary set D ⊂ W × R is defined as the smallest e′-convex set containing D, and it will be denoted by e′co D.

Remark 2. Observe that e′co D is nothing else than the epigraph of the e′-convex hull of the ∗ ∗ ∗ ∗ function fD : W → R defined by fD (y , z , α) := inf {a ∈ R | (y , z , α, a) ∈ D}. Hence,

′ ′ c′c e co D = epi (e co fD) = epi fD . (6)

Definition 3. Consider two functions f, g : X → R. A function a : X → R belongs to the e set Ef,g if there exist a1 ∈ Ef , a2 ∈ Eg such that, if { { ⟨·, y∗⟩ − β if ⟨·, z∗⟩ < α , ⟨·, y∗⟩ − β if ⟨·, z∗⟩ < α , a (·) = 1 1 1 1 and a (·) = 2 2 2 2 1 +∞ otherwise, 2 +∞ otherwise,

then { ⟨·, y∗ + y∗⟩ − (β + β ) if ⟨·, z∗ + z∗⟩ < α + α , a (·) = 1 2 1 2 1 2 1 2 +∞ otherwise. e Obviously, every a ∈ Ef,g is an e-affine function and also a ≤ a1 + a2 ≤ f + g. Consequently, e e Ef,g is a new set of e-affine minorants of f + g, and the inclusion Ef,g ⊂ Ef+g holds. For the sake of brevity in most of the results of sections 3 and 4, we will deal with f, g : X → R two proper e-convex functions satisfying the condition

dom f ∩ dom g ≠ ∅, (7)

which ensures that f + g is a proper e-convex function, in virtue of [29, Proposition 3.3]4. Associated to f and g, we define the function h : X → R given by { } e h := sup a | a ∈ Ef,g . (8)

Clearly, h is a proper e-convex function and one has h ≤ f + g.

Theorem 4. Let f, g : X → R be proper e-convex functions such that (7) holds, and let h be the function defined in (8). Then

e′co (epi f c + epi gc) = epi hc.

c c ⊂ c ∗ ∗ ∈ c Proof. First of all, we shall show that epi f +epi g epi h . Consider (y1, z1, α1, β1) epi f ∗ ∗ ∈ c and (y2, z2, α2, β2) epi g . Then, ∗ ∗ − ≤ c (x, (y1, z1, α1)) β1 f(x) and ∗ ∗ − ≤ c (x, (y2, z2, α2)) β2 g(x) 4Although the results in [29] are established on functions defined on Rn, [21, Remark 12] allows to extend most of them to the framework of locally convex spaces.

6 ∈ · ∗ ∗ − · ∗ ∗ − for all x X. The e-affine functions c ( , (y1, z1, α1)) β1 and c ( , (y2, z2, α2)) β2 belong to E E · ∗ ∗ ∗ ∗ − ∈ Ee f and g, respectively, so c ( , (y1 + y2, z1 + z2, α1 + α2)) (β1 + β2) f,g. Therefore, the following inequality holds for all x ∈ X, ∗ ∗ ∗ ∗ − ≤ c (x, (y1 + y2, z1 + z2, α1 + α2)) h(x) (β1 + β2) . c ∗ ∗ ∗ ∗ ≤ In consequence, h (y1 + y2, z1 + z2, α1 + α2) β1 + β2 and ∗ ∗ ∗ ∗ ∈ c (y1 + y2, z1 + z2, α1 + α2, β1 + β2) epi h . Since hc is an e′-convex function, its epigraph will be an e′-convex set, hence e′co (epi f c + epi gc) ⊂ epi hc. (9) In order to prove the equality in (9), we consider an e′-convex function k : W → R such ′ that e′co (epi f c + epi gc) = epi k. In virtue of Proposition 2 we know that k = kc c. This ′ means that k is the c-conjugate of kc : X → R, which is an e-convex function. Therefore, we have ′ epi f c + epi gc ⊂ e′co (epi f c + epi gc) = epi kc c. (10)

Ee ⊂ E ′ ∈ Ee ∈ E ∈ E Now, we shall prove that f,g kc . Pick any a f,g, then there exist a1 f , a2 g such that { { ⟨x, y∗⟩ − β if ⟨x, z∗⟩ < α , ⟨x, y∗⟩ − β if ⟨x, z∗⟩ < α , a (x) = 1 1 1 1 a (x) = 2 2 2 2 1 +∞ otherwise, 2 +∞ otherwise, and { ⟨x, y∗ + y∗⟩ − (β + β ) if ⟨x, z∗ + z∗⟩ < α + α , a (x) = 1 2 1 2 1 2 1 2 +∞ otherwise, ∈ ∗ ∗ − ≤ ∈ c ∗ ∗ ≤ for all x X. Hence, c (x, (y1, z1, α1)) f (x) β1 for all x X and f (y1, z1, α1) β1. c ∗ ∗ ≤ ∗ ∗ ∈ c ∗ ∗ ∈ c Analogously, g (y2, z2, α2) β2. We have (y1, z1, α1, β1) epi f and (y2, z2, α2, β2) epi g , ∗ ∗ ∗ ∗ ∈ c′c therefore, (y1 + y2, z1 + z2, α1 + α2, β1 + β2) epi k according to (10). This means that ∗ ∗ ∗ ∗ − ≤ c′ c (x, (y1 + y2, z1 + z2, α1 + α2)) (β1 + β2) k (x) , ′ ∈ ≤ c ∈ E ′ for all x X, which actually means a k . Consequently, a kc . ′ ′ ′ Ee ⊂ E ′ ≤ c c c ≤ c c ⊂ c c The set containment f,g kc implies h k . Hence, k h and epi h epi k , which entails epi hc ⊂ e′co (epi f c + epi gc). Corollary 5. Let f, g : X → R be proper e-convex functions such that (7) holds, and let h be the function defined in (8). Then e′co (epi f c + epi gc) = epi (f + g)c if and only if f + g = h.

Proof. (⇐) This statement follows easily from Theorem 4. (⇒) Again by Theorem 4, we have epi (f + g)c = epi hc,

′ ′ which is equivalent to (f + g)c = hc, and this implies (f + g)cc = hcc . Since f + g and h are both proper and e-convex functions, applying Proposition 2, we obtain f + g = h.

7 4 The infimal convolution of c-conjugates

In this section, the e′-convex hull of the infimal convolution f c ⊕gc, where f and g are proper e-convex funtions, is established, and ε-subdifferential sum formulas for e-convex functions are presented under suitable conditions. The last result we show here, Corollary 15, is the counterpart, in the e-convex setting, of the classical Fenchel Duality Theorem. It will be the main tool in Section 5 in order to derive Strong Fenchel Duality for e-convex optimization problems. First of all, the next lemma is to be considered. Lemma 6. Let f : X → R. Then, f is proper if and only if f c is proper. Proof. It is a straightforward consequence of the definition of proper functions, the well- known fact that f is proper if and only if f ∗ is proper, and the following detailed formula from [21] for the c-conjugate (4) that is applied for every (y∗, z∗, α) ∈ W , { f ∗(y∗) if ⟨x, z∗⟩ < α for all x ∈ dom f, f c(y∗, z∗, α) = +∞ otherwise.

Moreover, dom f c = dom f ∗ × {(z∗, α) ∈ X∗ × R | ⟨x, z∗⟩ < α, ∀x ∈ dom f}. Theorem 7. Let f, g : X → R be proper e-convex functions such that (7) holds, and let h be the function defined in (8). Then

′ (i) hc = (f c ⊕ gc)c c.

′ (ii) h = (f c ⊕ gc)c .

′ Proof. (i) According to (6), we can write e′co (epi f c + epi gc) = epi kc c, where the function k : W → R is given by

k (y∗, z∗, α) := inf {a ∈ R | (y∗, z∗, α, a) ∈ epi f c + epi gc} .

From (3), it follows that k = f c ⊕ gc, and applying Theorem 4, we have

′ epi hc = e′co (epi f c + epi gc) = epi (f c ⊕ gc)c c .

′ Consequently, hc = (f c ⊕ gc)c c. ′ (ii) Since h is an e-convex function, then h = hc c. According to Proposition 2, we also ( ) ′ ′ c ′ know that (f c ⊕ gc)c c = (f c ⊕ gc)c . Finally, if we apply c′-conjugates to the equality in (i) we obtain (ii). The following result is a direct consequence of the above theorem and Corollary 5, and therefore its proof is omitted. Corollary 8. Let f, g : X → R be proper e-convex functions such that (7) holds, and let h be the function defined in (8). Then

(f + g)c is the e′-convex hull of f c ⊕ gc if and only if f + g = h.

8 We recall from [21] the subdifferentiability notion associated with the conjugation pattern described in this paper. Definition 4. Let f : X → R be a function and ε ≥ 0. We will say that the vector ∗ ∗ ∗ (u , v , ω) ∈ W is an ε-c-subgradient of f at x0 ∈ X if f(x0) ∈ R, ⟨x0, v ⟩ < ω and

∗ ∗ ∗ ∗ f(x) − f(x0) ≥ c (x, (u , v , ω)) − c (x0, (u , v , ω)) − ε, ∀x ∈ X. (11)

We denote by ∂c,εf (x0) the set of all the ε-c-subgradients of f at x0, and it will be called the ε-c-subdifferential of f at x0. We set ∂c,εf (x0) = ∅ if f(x0) ∈/ R. Moreover, if ε = 0 then the vectors are simply called c-subgradients of f at x0, and the c-subdifferential of f at x0, denoted by ∂cf (x0), is the 0-c-subdifferential of f at x0. The following lemma generalizes the formula given in (2).

Lemma 9. Let f : X → R be a proper function. Then, if x0 ∈ dom f, ∪ c ∗ ∗ ∗ ∗ ∗ epi f = {(u , v , ω, ⟨x0, u ⟩ + ε − f (x0)) | (u , v , ω) ∈ ∂c,εf (x0)} . ε≥0

Proof. Pick any (u∗, v∗, ω, β) ∈ epi f c. Then, c (x, (u∗, v∗, ω)) − f (x) ≤ β holds for all x ∈ X. ∗ ∗ ∗ Hence, in particular we have c (x0, (u , v , ω)) − β ≤ f (x0) < +∞, which implies ⟨x0, v ⟩ < ω ∗ ∗ ∗ and c (x0, (u , v , ω)) = ⟨x0, u ⟩. In this case, we can write

∗ ∗ c (x0, (u , v , ω)) − β = f (x0) − ε,

for a certain ε ≥ 0. As a consequence, the following inequality holds for all x ∈ X,

∗ ∗ ∗ ∗ c (x, (u , v , ω)) − f (x) ≤ c (x0, (u , v , ω)) − f (x0) + ε,

∗ ∗ ∗ and therefore, (u , v , ω) ∈ ∂c,εf (x0). Moreover, β = ⟨x0, u ⟩ + ε − f (x0). ∗ ∗ ∗ Conversely, take ε ≥ 0 and (u , v , ω) ∈ ∂c,εf (x0). By definition, ⟨x0, v ⟩ < ω and inequality (11) holds, which implies

∗ ∗ ∗ c (x, (u , v , ω)) − f (x) ≤ ⟨x0, u ⟩ − f (x0) + ε, ∀x ∈ X.

c ∗ ∗ ∗ ∗ ∗ ∗ c Hence, f (u , v , ω) ≤ ⟨x0, u ⟩ + ε − f (x0) and (u , v , ω, ⟨x0, u ⟩ + ε − f (x0)) ∈ epi f . Theorem 10. Let f, g : X → R be proper e-convex functions such that (7) holds, and let h be the function defined in (8). Then, the following statements are equivalent: (i) hc = f c ⊕ gc and the infimal convolution is exact.

(ii) epi f c + epi gc is e′-convex. Proof. (i) ⇒ (ii). We shall prove that epi f c + epi gc = epi hc. Due to the fact that hc is an e′-convex function, the proof would be finished. By Theorem 4 we know that epi f c +epi gc ⊂ epi hc. For proving the reverse inclusion, consider (y∗, z∗, α, β) ∈ epi hc. Applying (i) we have

≥ c ∗ ∗ c ⊕ c ∗ ∗ c ∗ ∗ c ∗ ∗ β h (y , z , α) = (f g )(y , z , α) = f (y1, z1, α1) + g (y2, z2, α2) ,

9 ∗ ∗ ∗ ∗ ∈ ∗ ∗ ∗ ∗ ∗ ∗ for certain (y1, z1, α1), (y2, z2, α2) W such that (y1, z1, α1) + (y2, z2, α2) = (y , z , α). Con- c ∗ ∗ − sider now the scalars β1 := f (y1, z1, α1) and β2 := β β1. Then, ∗ ∗ ∈ c ∗ ∗ ∈ c (y1, z1, α1, β1) epi f and (y2, z2, α2, β2) epi g . ∗ ∗ ∗ ∗ ∗ ∗ ∈ c c Hence, (y , z , α, β) = (y1, z1, α1, β1) + (y2, z2, α2, β2) epi f + epi g . (ii) ⇒ (i). As the set epi f c + epi gc is e′-convex, by means of Theorem 4 we conclude that epi f c + epi gc = epi hc. Hence, for all (y∗, z∗, α) ∈ W we have

hc (y∗, z∗, α) = inf {a | (y∗, z∗, α, a) ∈ epi hc} = inf {a | (y∗, z∗, α, a) ∈ epi f c + epi gc} = (f c ⊕ gc)(y∗, z∗, α) , and consequently, hc = f c ⊕ gc. Let us check that the infimal convolution is exact. Since h is a proper convex function, then f c ⊕ gc is a proper function according to Lemma 6. Suppose that (f c ⊕ gc)(y∗, z∗, α) = +∞ for some (y∗, z∗, α) ∈ W . Since gc (0, 0, 0) = +∞ and f c is proper, then we can write

(f c ⊕ gc)(y∗, z∗, α) = f c (y∗, z∗, α) + gc (0, 0, 0) , and therefore, the infimal convolution is exact at (y∗, z∗, α). Now, assume that β := (f c ⊕ gc)(y∗, z∗, α) ∈ R. Then, (y∗, z∗, α, β) ∈ epi hc = epi f c + epi gc, ∗ ∗ ∈ c ∗ ∗ ∈ c ∗ ∗ and there exist (¯y1, z¯1, α¯1, β1) epi f and (¯y2, z¯2, α¯2, β2) epi g such that (y , z , α, β) = ∗ ∗ ∗ ∗ (¯y1, z¯1, α¯1, β1) + (¯y2, z¯2, α¯2, β2) and

c c ∗ ∗ f ⊕ g (y , z , α) = β1 + β2 ≥ c ∗ ∗ c ∗ ∗ f (¯y1, z¯1, α¯1) + g (¯y2, z¯2, α¯2) ≥ { c ∗ ∗ c ∗ ∗ | ∗ ∗ ∗ ∗ ∗ ∗ } inf f (y1, z1, α1) + g (y2, z2, α2) (y1, z1, α1) + (y2, z2, α2) = (y , z , α) = f c ⊕ gc (y∗, z∗, α) .

Hence, all the inequalities above are in fact equalities, and the infimal convolution is exact. The following theorem examines how the sum of epigraphs, epi f c + epi gc, is linked to the infimal convolution of f c and gc, and to the ε-c-subdifferentials of f and g. Theorem 11. Let f, g : X → R be proper e-convex functions such that (7) holds, let h be the function defined in (8) and assume that f + g = h. Then, the following statements are equivalent: (i) (f + g)c = f c ⊕ gc and the infimal convolution is exact.

(ii) epi f c + epi gc is e′-convex.

(iii) For each ε ≥ 0 and for each x ∈ dom f ∩ dom g, ∪

∂c,ε (f + g)(x) = ∂c,ε1 f (x) + ∂c,ε2 g (x) . (12) ε1+ε2=ε ε1,ε2≥0

10 Proof. (i) ⇔ (ii). It follows directly from Theorem 10. ∗ ∗ (ii) ⇒ (iii). Consider any ε ≥ 0 and x0 ∈ dom f ∩ dom g. If (u , v , ω) ∈ ∂c,ε (f + g)(x0), combining Corollary 5, condition (ii) and Lemma 9, we can write

∗ ∗ ∗ c c c (u , v , ω, ⟨x0, u ⟩ + ε − (f + g)(x0)) ∈ epi (f + g) = epi f + epi g .

∗ ∗ ∈ c ∗ ∗ ∈ c Hence, there exist (u1, v1, ω1, β1) epi f and (u2, v2, ω2, β2) epi g such that ∗ ∗ ∗ ∗ ∗ ∗ ⟨ ∗⟩ − (u1, v1, ω1, β1) + (u2, v2, ω2, β2) = (u , v , ω, x0, u + ε (f + g)(x0)) .

Applying again Lemma 9, there exist ε1, ε2 ≥ 0 such that ∗ ∗ ∈ ⟨ ∗⟩ − (u1, v1, ω1) ∂c,ε1 f (x0) , β1 = x0, u1 + ε1 f (x0) , ∗ ∗ ∈ ⟨ ∗⟩ − (u2, v2, ω2) ∂c,ε2 g (x0) , β2 = x0, u2 + ε2 g (x0) , ∗ ∗ ∈ which implies ε1 + ε2 = ε. As a consequence, (u , v , ω) ∂c,ε1 f (x0) + ∂c,ε2 g (x0), and therefore, ∪ ⊂ ∂c,ε (f + g)(x0) ∂c,ε1 f (x0) + ∂c,ε2 g (x0) . ε1+ε2=ε ε1,ε2≥0 ∗ ∗ ∈ ∗ ∗ ∈ ≥ Conversely, consider (u1, v1, ω1) ∂c,ε1 f (x0) and (u2, v2, ω2) ∂c,ε2 g (x0), where ε1, ε2 0 are such that ε1 + ε2 = ε. Lemma 9 is used again to obtain

∗ ∗ ∗ ∗ ⟨ ∗ ∗⟩ − ∈ c c (u1 + u2, v1 + v2, ω1 + ω2, x0, u1 + u2 + ε (f + g)(x0)) epi f + epi g .

c c c ∗ ∗ ∗ ∗ ∈ Since epi f + epi g = epi (f + g) , then (u1, v1, ω1) + (u2, v2, ω2) ∂c,ε (f + g)(x0), which completes the proof of equality (12). (iii) ⇒ (ii). By Corollary 5 we know that e′co (epi f c + epi gc) = epi (f + g)c. So, for prov- ing (ii) it suffices to show that epi (f + g)c ⊂ epi f c + epi gc. Let (y∗, z∗, α, β) ∈ epi (f + g)c and take any x0 ∈ dom f ∩ dom g. By Lemma 9, there exists ε ≥ 0 such that

∗ ∗ ∗ (y , z , α) ∈ ∂c,ε (f + g)(x0) and β = ⟨x0, y ⟩ + ε − (f + g)(x0) . ≥ ∗ ∗ ∈ ∗ ∗ ∈ Since there exist ε1, ε2 0 with ε1 + ε2 = ε, and (u1, v1, ω1) ∂c,ε1 f (x0), (u2, v2, ω2) ∗ ∗ ∗ ∗ ∗ ∗ ⟨ ∗⟩ − ∂c,ε2 g (x0) verifying (y , z , α) = (u1, v1, ω1) + (u2, v2, ω2), name β1 := x0, u1 + ε1 f (x0) ⟨ ∗⟩ − and β2 := x0, u2 + ε2 g (x0). Observe that β1 + β2 = β. Again by Lemma 9, we obtain

∗ ∗ ∈ c (u1, v1, ω1, β1) epi f , ∗ ∗ ∈ c (u2, v2, ω2, β2) epi g .

Therefore, (y∗, z∗, α, β) ∈ epi f c + epi gc.

Corollary 12. Let f, g : X → R be proper e-convex functions such that (7) holds, let h be the function defined in (8) and assume that f + g = h. If epi f c + epi gc is e′-convex, then

∂c (f + g)(x) = ∂cf (x) + ∂cg (x) , ∀x ∈ dom f ∩ dom g.

11 Theorem 13. Let f, g : X → R be proper e-convex functions such that (7) holds, let h be the function defined in (8) and assume that there exists α > 0 such that hc (0, 0, α) is finite. Then, the following statements are equivalent: (i) hc (0, 0, α) = (f c ⊕ gc) (0, 0, α) and the infimal convolution is exact at (0, 0, α). (ii) (0, 0, α, hc (0, 0, α)) ∈ epi f c + epi gc. (iii) (f c ⊕ gc) is e′-convex at (0, 0, α) and the infimal convolution is exact at (0, 0, α).

∗ ∗ ∗ Proof. (i) ⇒ (ii). For some y , z ∈ X and α1, α2 ∈ R with α1 + α2 = α, we can write

c c c c ∗ ∗ c ∗ ∗ h (0, 0, α) = (f ⊕ g ) (0, 0, α) = f (y , z , α1) + g (−y , −z , α2) ,

c ∗ ∗ c ∗ ∗ with f (y , z , α1) ∈ R and g (−y , −z , α2) ∈ R. Then,

c ∗ ∗ c ∗ ∗ ∗ ∗ c ∗ ∗ (0, 0, α, h (0, 0, α)) = (y , z , α1, f (y , z , α1)) + (−y , −z , α2, g (−y , −z , α2)) , and therefore, (0, 0, α, hc (0, 0, α)) ∈ epi f c + epi gc. (ii) ⇒ (i). By assumption, we know that

c ∗ ∗ ∗ ∗ (0, 0, α, h (0, 0, α)) = (y1, z1, α1, β1) + (y2, z2, α2, β2) , ∗ ∗ ∈ c ∗ ∗ ∈ c for certain (y1, z1, α1, β1) epi f and (y2, z2, α2, β2) epi g . Consequently, we have that ∗ ∗ ∗ ∗ (0, 0, α) = (y1, z1, α1) + (y2, z2, α2) and c ≥ c ∗ ∗ c ∗ ∗ ≥ c ⊕ c h (0, 0, α) = β1 + β2 f (y1, z1, α1) + g (y2, z2, α2) (f g ) (0, 0, α) . (13) On the other hand, if we combine Theorem 7 and Proposition 2 we obtain

′ hc (0, 0, α) ≤ (f c ⊕ gc)c c (0, 0, α) ≤ (f c ⊕ gc) (0, 0, α) . (14)

Statement (i) is a straightforward consequence of (13) and (14). (i) ⇔ (iii). It follows easily from Theorem 7 and the definition of e′-convexity at a point.

Corollary 14. Let f, g : X → R be proper e-convex functions such that (7) holds and let h be the function defined in (8). Assume that f + g = h and there exists α > 0 such that (f + g)c (0, 0, α) is finite. Then, the following statements are equivalent: (i) (f + g)c (0, 0, α) = (f c ⊕ gc) (0, 0, α) and the infimal convolution is exact at (0, 0, α). (ii) (0, 0, α, (f + g)c (0, 0, α)) ∈ epi f c + epi gc. (iii) (f c ⊕ gc) is e′-convex at (0, 0, α) and the infimal convolution is exact at (0, 0, α).

Corollary 15. Let f, g : X → R be proper e-convex functions such that (7) holds, let h be the function defined in (8) and assume that f + g = h. If epi f c + epi gc is e′-convex, then

c ∗ ∗ c ∗ ∗ inf {f (x) + g (x)} = max {−f (y , z , α1) − g (−y , −z , α2)} . (15) x∈X y∗,z∗∈X∗ α1+α2>0

12 Proof. Take any α > 0. By the definition of c-conjugate, we can write

(f + g)c (0, 0, α) = sup {−f (x) − g (x)} = − inf {f (x) + g (x)} , (16) x∈X x∈X and clearly, it does not depend on α. On the other hand, since the infimal convolution is exact according to Theorem 11, then we have

c c c ∗ ∗ c ∗ ∗ (f ⊕ g ) (0, 0, α) = min {f (y , z , α1) + g (−y , −z , α2)} . (17) y∗,z∗∈X∗ α1+α2=α It suffices to combine (16), (17) and Theorem 11 (i) to complete the proof of (15). Remark 3. Observe that, if (f + g)c (0, 0, α) is finite for some α > 0, then the maximum in the right-hand side of (15) is finite. In such a case, we can replace the condition on the e′-convexity of epi f c + epi gc in the above Corollary by a weaker condition, which is (ii) in Corollary 14.

5 Fenchel Duality

Duality is a fundamental concept that plays a central role in optimization. Via a perturbation approach, different dual problems for a primal one can be obtained, depending on the chosen perturbation function. We recall some works dealing with the perturbational approach to duality theory, which are [2, 3, 9, 20, 27, 28, 30]. One of the most known dual problems is Fenchel’s one, and many works can be found in the literature dealing with the problem of finding regularity conditions which ensure strong duality. In the convex setting with finite dimensional space, background information on Fenchel duality is due to Rockafellar [26, 27]. With infinite dimensional space, we refer for instance to [5, 7, 11, 23, 17]. The main idea is to associate to a general optimization problem (GP ) Inf F (x) s.t. x ∈ X,

where F : X → R, a dual problem using a perturbation function Φ : X × Θ → R, where Θ is the perturbation variable space, and Φ must verify the relation

Φ(x, 0Θ) = F (x) ,

for all x ∈ X, 0Θ denoting the zero element of the space Θ. A dual problem for (GP ) can be defined in the following way: (GD) Sup −Φ∗ (0, z∗) s.t. z∗ ∈ Θ∗.

In connection with the perturbation function Φ, one defines the infimum value function, p :Θ → R, p (z) := inf Φ(x, z) . (18) x∈X

13 It is clear that the primal problem (GP ) can be written as follows

p (0Θ) = inf Φ(x, 0Θ) , x∈X and the dual problem (GD) Sup −p∗ (z∗) s.t. z∗ ∈ Θ∗. If we denote by v(GP ) and v(GD) the optimal values of the primal and the dual problems, respectively, it holds v(GD) ≤ v(GP ) ≤ +∞, and this inequality is called weak duality, whereas strong duality is v(GD) = v(GP ) and the dual problem is solvable when v(GP ) is finite. In a particular convex context, the following primal problem is considered:

(P ) Inf f(x) 1 (19) s.t. x ∈ A, where A ⊂ X is a nonempty closed convex set and f : X → R is a proper lsc convex function. The problem (P1) is a particular case of (GP ) with F = f + δA, where δA is the indicator function of A. We will consider the perturbation function Φ : X × X → R given by { f (x + u) if x ∈ A, Φ(x, u) := (20) +∞ otherwise.

The perturbation variable is u ∈ Θ ≡ X. It is easy to check that the dual problem

∗ ∗ (D1) Sup −Φ (0, u ) s.t. u∗ ∈ X∗ becomes the Fenchel dual problem − ∗ ∗ − ∗ − ∗ (DF ) Sup f (u ) δA ( u ) s.t. u∗ ∈ X∗. ∩ ̸ ∅ ∗ ∗ The following result (cf. [7, Corollary 3]) is well-known: if A dom f = and epi f + epi δA ∗ is weak closed, then strong duality holds for (P1)–(DF ), i.e., {− ∗ ∗ − ∗ − ∗ } inf f (x) = max f (u ) δA ( u ) . x∈A u∗∈X∗ The aim of this section is to establish strong Fenchel duality results for the primal problem (P ) in (1), being A ⊂ X a nonempty e-convex set and f : X → R a proper e-convex function, which generalizes the problem (P1) in (19). We will construct a dual problem for (P ) by means of the perturbational approach, taking the perturbation function given in (20), but applying the c-conjugation scheme instead of classical Fenchel conjugation. If we set Z := X × X, its dual space Z∗ can be identified with X∗ × X∗ (cf. [25]) using the dual product ⟨(x, u) , (x∗, u∗)⟩ = ⟨x, x∗⟩ + ⟨u, u∗⟩ ,

14 for all (x, u) ∈ Z and (x∗, u∗) ∈ Z∗. The c-conjugate of the perturbation function Φ in (20), Φc : Z∗ × Z∗ × R → R, is defined by

Φc ((x∗, u∗) , (y∗, v∗) , α) = sup {c¯((x, u) , ((x∗, u∗) , (y∗, v∗) , α)) − Φ(x, u)} , (x,u)∈Z wherec ¯ : Z × Z∗ × Z∗ × R → R is { ⟨x, x∗⟩ + ⟨u, u∗⟩ if ⟨x, y∗⟩ + ⟨u, v∗⟩ < α c ((x, u) , ((x∗, u∗) , (y∗, v∗) , α)) = +∞ otherwise.

Since for all (x, u) ∈ Z and for all ((x∗, u∗) , (y∗, v∗) , α) ∈ Z∗ × Z∗ × R we have

Φ(x, u) + Φc ((x∗, u∗) , (y∗, v∗) , α) ≥ c¯((x, u) , ((x∗, u∗) , (y∗, v∗) , α)) , taking u = 0, x∗ = y∗ = 0 and α > 0, we obtain

Φ(x, 0) + Φc ((0, u∗) , (0, v∗) , α) ≥ 0.

∗ ∗ ∗ ∗ Hence, the following inequality holds for all x ∈ X and for all (u , v , α) ∈ X × X × R++, Φ(x, 0) ≥ −Φc ((0, u∗) , (0, v∗) , α) ,

As a consequence, we have

inf Φ(x, 0) ≥ sup −Φc ((0, u∗) , (0, v∗) , α) . (21) ∈ ∗ ∗ ∗ x X u ,v ∈X α>0 Now, since the left-hand side in (21) coincides with v(P ), we associate to (P ) the dual problem given by the right-hand side in (21), that is,

(D) Sup −Φc ((0, u∗) , (0, v∗) , α) s.t. u∗, v∗ ∈ X∗, α > 0, and then, we have v(D) ≤ v(P ). On the other hand, let us consider the infimum value function p : X → R defined in (18), c ∗ ∗ ∗ ∗ ∗ ∗ and its c-conjugate p : X × X × R → R. Then, for all (u , v , α) ∈ X × X × R++, −Φc ((0, u∗) , (0, v∗) , α) = − sup {c¯((x, u) , ((0, u∗) , (0, v∗) , α)) − Φ(x, u)} (x,u)∈Z = − sup {c (u, (u∗, v∗, α)) − Φ(x, u)} x,u∈X = − sup {c (u, (u∗, v∗, α)) − p (u)} u∈X = −pc (u∗, v∗, α) .

Therefore, we get (D) Sup −pc (u∗, v∗, α) s.t. u∗, v∗ ∈ X∗, α > 0.

15 According to [20], a new dual problem can be associated with (P ), which is

(D′) Sup c (0, (u∗, v∗, α)) − pc (u∗, v∗, α) s.t. (u∗, v∗, α) ∈ W.

In fact, both dual problems (D) and (D′) have the same optimal value, since for every (u∗, v∗) ∈ X∗ × X∗ we have

c (0, (u∗, v∗, α)) − pc (u∗, v∗, α) = +∞ − (+∞) = −∞

if α ≤ 0, and c (0, (u∗, v∗, α)) = 0 if α > 0. Hence

v(D′) = sup {c (0, (u∗, v∗, α)) − pc (u∗, v∗, α)} (u∗,v∗,α)∈W = sup {c (0, (u∗, v∗, α)) − pc (u∗, v∗, α)} u∗,v∗∈X∗ α>0 = sup {−pc (u∗, v∗, α)} = v(D). u∗,v∗∈X∗ α>0 Moreover, if the above optimal value is finite, then (D) is solvable if and only if so is (D′). In this case, both optimal solution sets coincide. Hence, conditions which guarantee strong duality for (P )–(D) will also provide strong duality for (P )–(D′) and conversely.

Theorem 16. Let A ⊂ X be a nonempty e-convex set and{ let f : X →}R be a proper e-convex ∩ ̸ ∅ | ∈ Ee c c function. Assume that A dom f = . If f + δA = sup a a f,δA and epi f + epi δA is e′-convex, then strong duality holds for (P )–(D), i.e.,

inf f (x) = max −pc (u∗, v∗, α) . x∈A u∗,v∗∈X∗ α>0

Proof. Since A is a nonempty e-convex set, δA is a proper e-convex function. Then, according to Corollary 15, we can write

{ } {− c ∗ ∗ − c − ∗ − ∗ } inf f (x) + δA (x) = max f (u , v , α1) δA ( u , v , α2) . x∈X u∗,v∗∈X∗ α1+α2>0

Firstly, observe that infx∈X {f (x) + δA (x)} = infx∈A f (x). On the other hand, let us take ∗ ∗ ∗ ∗ any (u , v , α) ∈ X × X × R++. We have

pc (u∗, v∗, α) = sup {c (u, (u∗, v∗, α)) − p (u)} u∈X ∗ ∗ = sup {c (u, (u , v , α)) − f (x + u) − δA (x)} x,u∈X ∗ ∗ = sup {c (r − x, (u , v , α)) − f (r) − δA (x)} , x,r∈X

where r := x + u ∈ X. It is easy to check that

∗ ∗ ∗ ∗ ∗ ∗ c (r − x, (u , v , α)) ≤ c (r, (u , v , α1)) + c (x, (−u , −v , α2)) ,

16 for all α1, α2 ∈ R such that α1 + α2 = α. Then,

c ∗ ∗ ∗ ∗ ∗ ∗ p (u , v , α) ≤ sup {c (r, (u , v , α1)) + c (x, (−u , −v , α2)) − f (r) − δA (x)} x,r∈X ∗ ∗ ∗ ∗ = sup {c (r, (u , v , α1)) − f (r)} + sup {c (x, (−u , −v , α2)) − δA (x)} r∈X x∈X c ∗ ∗ c − ∗ − ∗ = f (u , v , α1) + δA ( u , v , α2) . As a consequence, we obtain

− c ∗ ∗ ≥ − c ∗ ∗ − c − ∗ − ∗ p (u , v , α) f (u , v , α1) δA ( u , v , α2) , (22)

for all α1, α2 ∈ R such that α1 + α2 = α. Therefore, − c ∗ ∗ ≥ {− c ∗ ∗ − c − ∗ − ∗ } sup p (u , v , α) sup f (u , v , α1) δA ( u , v , α2) . u∗,v∗∈X∗ u∗,v∗∈X∗ α>0 α1+α2>0 We have proved the following inequalities and equalities,

{− c ∗ ∗ − c − ∗ − ∗ } v(P ) = inf f (x) = max f (u , v , α1) δA ( u , v , α2) x∈A u∗,v∗∈X∗ α1+α2>0 ≤ sup −pc (u∗, v∗, α) = v(D) ≤ v(P ). u∗,v∗∈X∗ α>0 Hence, v(P ) = v(D). Moreover, since

− c ∗ ∗ − c − ∗ − ∗ v(P ) = f (¯u , v¯ , α¯1) δA ( u¯ , v¯ , α¯2)

∗ ∗ ∗ for someu ¯ , v¯ ∈ X ,α ¯1, α¯2 ∈ R such thatα ¯ :=α ¯1 +α ¯2 > 0, we have − c ∗ ∗ − c ∗ ∗ − c − ∗ − ∗ sup p (u , v , α) = f (¯u , v¯ , α¯1) δA ( u¯ , v¯ , α¯2) , u∗,v∗∈X∗ α>0 and − c ∗ ∗ ≤ − c ∗ ∗ − c − ∗ − ∗ p (¯u , v¯ , α¯) f (¯u , v¯ , α¯1) δA ( u¯ , v¯ , α¯2) . Applying inequality (22) we obtain

− c ∗ ∗ − c ∗ ∗ − c − ∗ − ∗ p (¯u , v¯ , α¯) = f (¯u , v¯ , α¯1) δA ( u¯ , v¯ , α¯2) . Therefore, sup −pc (u∗, v∗, α) = −pc (¯u∗, v¯∗, α¯) , u∗,v∗∈X∗ α>0 and strong duality holds for (P )–(D).

Corollary 17. Let A ⊂ X be a nonempty e-convex set and{ let f : X →}R be a proper e-convex ∩ ̸ ∅ | ∈ Ee c c function. Assume that A dom f = . If f + δA = sup a a f,δA and epi f + epi δA is e′-convex, then there exists strong duality for (P )–(D′).

17 Proof. We already know that v (D′) = v (D) ≤ v(P ). Applying the previous theorem, we will obtain v (D′) = v(P ), that is,

inf f (x) = sup c (0, (u∗, v∗, α)) − pc (u∗, v∗, α) . ∈ x A (u∗,v∗,α)∈W

On the other hand, according to Theorem 16 we know that

v(P ) = v(D) = max −pc (u∗, v∗, α) = −pc (¯u∗, v¯∗, α¯) u∗,v∗∈X∗ α>0 for someu ¯∗, v¯∗ ∈ X∗,α ¯ > 0. This implies

sup c (0, (u∗, v∗, α)) − pc (u∗, v∗, α) = (u∗,v∗,α)∈W

= −pc (¯u∗, v¯∗, α¯) = c (0, (¯u∗, v¯∗, α¯)) − pc (¯u∗, v¯∗, α¯) . Hence, strong duality holds for (P )–(D′).

Corollary 18. Let A ⊂ X be a nonempty e-convex set and{ let f : X →}R be a proper e-convex ∩ ̸ ∅ | ∈ Ee c c function. Assume that A dom f = . If f + δA = sup a a f,δA and epi f + epi δA is e′-convex, then the infimum value function p is e-convex at 0. Proof. For proving this result it suffices to combine Corollary 17 and Theorem 6.7 in [20], which asserts that strong duality for (P )–(D′) is equivalent to the e-convexity of p at 0.

Remark 4. If there exists strong duality for (P )–(D) and the optimal value is finite, then, ′ applying again [20, Theorem 6.7], the optimal solution set to (D ) is ∂c p(0).

Remark 5. In the literature, there exist several type of conditions in order to guarantee strong duality for the primal-dual pair

(P ) Inf f (x) + g (x) x∈X

(D) Sup −f ∗ (u∗) − g∗ (−u∗) u∗∈X∗ where f and g are proper convex functions with dom f ∩ dom g ≠ ∅. We can find conditions involving the continuity of one of the two functions at a point of dom f ∩ dom g, and some classical generalized interior point ones [1, 14, 28]. Recently, it has been introduced regularity conditions involving lower semicontinuity of the two functions, named closedness-type regu- larity conditions [5, 6], which are not comparable with the most recent point-interior ones in the infinite dimensional setting (see [2]). In the e-convex context, the conditions obtained in this paper for the fulfilment of strong Fenchel duality can be regarded as the closedness-type ones in the classical convex setting, hence the problem of obtaining interiority-type conditions and the comparison of both type is a very interesting problem to work on in a future paper. Acknowledgments. The authors are very grateful to the reviewers for their careful reading and valuable comments and suggestions for improving the quality of the manuscript.

18 References

[1] Attouch, H., Brezis, H. (1986) Duality of the sum of convex functions in general Ba- nach spaces. In: Barroso, J.A. (ed) Aspects of mathematics and its applications. North- Holland Publishing Company, Amsterdam, pp. 125–133

[2] Bot¸, R.I. (2010) Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, 637. Springer-Verlag, Berlin

[3] Bot¸, R.I., Grad, S.M., Wanka, G. (2009) Duality in vector optimization. Springer Verlag, Berlin

[4] Bot¸, R.I., Grad, S.M., Wanka, G. (2009) Generalized Moreau–Rockafellar results for composed convex functions. Optimization 58(7):917–933

[5] Bot¸, R.I., Wanka, G. (2006) A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Analysis 64:2787–2804

[6] Burachik, R.S., Jeyakumar, V. (2005) A dual condition for the convex subdifferential sum formula with applications. Journal of Convex Analysis 12:279–290

[7] Burachik, R.S., Jeyakumar, V. (2005) A new geometric condition for Fenchel’s duality in infinite dimensional spaces. Mathematical Programming, Ser. B 104:229–233

[8] Daniilidis, A., Mart´ınez-Legaz,J.E. (2002) Characterizations of evenly convex sets and evenly quasiconvex functions. J. Mathematical Analysis and Applications 273:58–66

[9] Ekeland, I., Temam, R. (1976) Convex Analysis and Variational Problems. North- Holland Publishing Company, Amsterdam

[10] Fenchel, W. (1952) A remark on convex sets and polarity. Comm. S`em.Math. Univ. Lund (Medd. Lunds Algebra Univ. Math. Sem.), Tome Suppl´ementaire, 82–89

[11] Goberna, M.A., Jeyakumar, V., L´opez, M.A. (2008) Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities. Nonlinear Analysis 68:1184–1194

[12] Goberna, M.A., Jornet, V., Rodr´ıguez, M.M.L. (2003) On linear systems containing strict inequalities. Linear Algebra and its Applications 360:151–171

[13] Goberna, M.A., Rodr´ıguez,M.M.L. (2006) Analyzing linear systems containing strict inequalities via evenly convex hulls. European Journal of Operational Research 169:1079– 1095

[14] Gowda, M.S., Teboulle, M. (1990) A comparison of constraint qualifications in infinite- dimensional convex programming. SIAM J. Control Optim. 28(4):925–935

[15] Jeyakumar, V., Goberna, M.A., Dinh, N. (2006) Dual characterizations of sets contain- ments with strict convex inequalities. Journal of Global Optimization 34:33–54

19 [16] Klee, V., Maluta, E., Zanco, C. (2007) Basic properties of evenly convex sets. Journal of Convex Analysis 14(1):137–148

[17] Li, C., Fang, D., L´opez, G., L´opez, M.A. (2009) Stable and total Fenchel duality for convex optimization problems in locally convex spaces. Siam J. Optim. 20(2):1032–1051

[18] Mart´ınez-Legaz,J.E. (1983) A generalized concept of conjugation. In: Hiriart-Urruty, J.-B., Oettli, W., Stoer, J. (eds) Optimization: Theory and Algorithms, Lecture Notes in Pure and Applied Mathematics 86, Marcel Dekker, New York, pp. 45–59

[19] Mart´ınez-Legaz, J.E. (1988) Quasiconvex duality theory by generalized conjugation methods. Optimization 19:603–652

[20] Mart´ınez-Legaz,J.E. (2005) Generalized convex duality and its economic applications. In: Hadjisavvas, N., Koml´osi,S., Schaible, S. (eds) Handbook of Generalized Convexity and Generalized Monotonicity, Springer, New York, pp. 237–292

[21] Mart´ınez-Legaz,J.E., Vicente-P´erez,J. (2011) The e-support function of an e-convex set and conjugacy for e-convex functions. J. Math. Anal. Appl. 376:602–612

[22] Moreau, J.J. (1970) Inf-convolution, sous-additivit´e,convexit´edes fonctions num´eriques. J. Math. Pures et Appl. 49:109–154

[23] Ng, K.F., Song, W. (2003) Fenchel duality in infinite-dimensional setting and its appli- cations. Nonlinear Analysis 55:845–858

[24] Passy, U., Prisman, E.Z. (1984) Conjugacy in quasiconvex programming. Mathematical Programming 30:121–146

[25] Phelps, R.R. (1993) Convex Functions, Monotone Operators and Differentiability. Springer Verlag, New York

[26] Rockafellar, R.T. (1966) Extension of Fenchel’s duality theorem for convex functions. Duke Math. J. 33:81–89

[27] Rockafellar, R.T. (1970) Convex Analysis. Princeton University Press, Princeton, New Jersey

[28] Rockafellar, R.T. (1974) Conjugate Duality and Optimization. CBMS-NSF Regional Conference Series in Applied Mathematics, No 16, SIAM Publications

[29] Rodr´ıguez, M.M.L., Vicente-P´erez,J. (2011) On evenly convex functions. Journal of Convex Analysis 18(3), to appear

[30] Z˘alinescu,C. (2002) Convex analysis in general vector spaces. World Scientific, New Jersey

20