Infimal convolution,
c-subdifferentiability, and Fenchel duality in evenly convex optimization
M.D. Fajardo1,3, J. Vicente-P´erez2 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-Legaz in [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-Legaz presented 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.
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)
s.t. x ∈ A, (1)
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} .
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∈dom f{⟨x, x∗⟩ − f(x)} , 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∗ = ∪
ε≥0
{(v∗,⟨a, v∗⟩ + ε − f(a)) | v∗ ∈ ∂εf (a)} . (2)
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
x1+x2=x{f (x1) + g (x2)} ,
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+ε2=ε ε1,ε2≥0
∂ε1f (x) + ∂ε2g (x) .
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
c(x, (y∗, z∗, α)) :=
{ ⟨x, y∗⟩ if ⟨x, z∗⟩ < α, +∞ otherwise.
For any f : X → R, its c-conjugate fc: W → R is defined by fc((y∗, z∗, α)) := sup
x∈X{c(x, (y∗, z∗, α))− f(x)} . (4) Similarly, the c′-conjugate of g : W → R is the function gc′ : X → R defined by
gc′(x) := sup
(y∗,z∗,α)∈W{c′((y∗, z∗, α), x)− g(w)} , (5) 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 ) fc is e′-convex ; gc′ is e-convex.
(ii ) If f has a proper e-convex minorant, eco f = fcc′; e′co g = gc′c.
(iii ) If f does not take on the value−∞, then f is e-convex if and only if f = fcc′; g is e′-convex if and only if g = gc′c.
(iv ) fcc′ ≤ f ; gc′c ≤ g.
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
a (x) =
{ ⟨x, y∗⟩ − β if ⟨x, z∗⟩ < α,
+∞ 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 fc+ 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 fc+ 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 fc+ 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 fc and epi gc, we have that (0, 1) belongs also to the recession cone of eco (epi fc+ 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 fc+ epi gc)}
is an e-convex function, and epi k = eco (epi fc+ 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 fc+ epi gc), and it may not coincide with eco (epi fc+ 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.
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,
e′co D = epi (e′co fD) = epi fDc′c. (6)
Definition 3. Consider two functions f, g : X → R. A function a : X → R belongs to the set eEf,g if there exist a1 ∈ Ef, a2 ∈ Eg such that, if
a1(·) =
{ ⟨·, y1∗⟩ − β1 if ⟨·, z1∗⟩ < α1,
+∞ otherwise, and a2(·) =
{ ⟨·, y∗2⟩ − β2 if ⟨·, z2∗⟩ < α2,
+∞ otherwise,
then
a (·) =
{ ⟨·, y1∗+ y2∗⟩ − (β1+ β2) +∞
if ⟨·, z1∗+ z2∗⟩ < α1+ α2, otherwise.
Obviously, every a∈ eEf,g is an e-affine function and also a≤ a1+ a2 ≤ f + g. Consequently, Eef,g is a new set of e-affine minorants of f + g, and the inclusion eEf,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
h := sup {
a| a ∈ eEf,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 fc+ epi gc) = epi hc.
Proof. First of all, we shall show that epi fc+ epi gc⊂ epi hc. Consider (y∗1, z∗1, α1, β1)∈ epi fc and (y∗2, z2∗, α2, β2)∈ epi gc. Then,
c (x, (y∗1, z1∗, α1))− β1 ≤ f(x) and c (x, (y2∗, z2∗, α2))− β2 ≤ g(x)
4Although the results in [29] are established on functions defined onRn, [21, Remark 12] allows to extend most of them to the framework of locally convex spaces.
for all x∈ X. The e-affine functions c (·, (y∗1, z∗1, α1))− β1 and c (·, (y∗2, z2∗, α2))− β2 belong to Ef and Eg, respectively, so c (·, (y1∗+ y2∗, z1∗+ z∗2, α1+ α2))− (β1+ β2) ∈ eEf,g. Therefore, the following inequality holds for all x∈ X,
c (x, (y1∗+ y2∗, z1∗+ z∗2, α1 + α2))− h(x) ≤ (β1 + β2) . In consequence, hc(y∗1+ y∗2, z1∗+ z2∗, α1+ α2)≤ β1+ β2 and
(y1∗+ y2∗, z1∗+ z∗2, α1+ α2, β1+ β2)∈ epi hc.
Since hc is an e′-convex function, its epigraph will be an e′-convex set, hence
e′co (epi fc+ 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 fc+ 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 fc+ epi gc⊂ e′co (epi fc+ epi gc) = epi kc′c. (10) Now, we shall prove that eEf,g ⊂ Ekc′. Pick any a∈ eEf,g, then there exist a1 ∈ Ef, a2 ∈ Eg
such that a1(x) =
{ ⟨x, y1∗⟩ − β1 if ⟨x, z∗1⟩ < α1,
+∞ otherwise, a2(x) =
{ ⟨x, y∗2⟩ − β2 if ⟨x, z2∗⟩ < α2,
+∞ otherwise,
and
a (x) =
{ ⟨x, y1∗+ y∗2⟩ − (β1+ β2) if ⟨x, z1∗+ z2∗⟩ < α1+ α2,
+∞ otherwise,
for all x ∈ X. Hence, c (x, (y1∗, z1∗, α1))− f (x) ≤ β1 for all x ∈ X and fc(y∗1, z1∗, α1) ≤ β1. Analogously, gc(y∗2, z2∗, α2)≤ β2. We have (y1∗, z1∗, α1, β1)∈ epi fc and (y2∗, z2∗, α2, β2)∈ epi gc, therefore, (y1∗+ y∗2, z∗1+ z2∗, α1+ α2, β1+ β2)∈ epi kc′c according to (10). This means that
c (x, (y∗1+ y∗2, z1∗+ z2∗, α1+ α2))− (β1+ β2)≤ kc′(x) , for all x∈ X, which actually means a ≤ kc′. Consequently, a∈ Ekc′.
The set containment eEf,g ⊂ Ekc′ implies h ≤ kc′. Hence, kc′c ≤ hc and epi hc ⊂ epi kc′c, which entails epi hc⊂ e′co (epi fc+ 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 fc+ 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.
4 The infimal convolution of c-conjugates
In this section, the e′-convex hull of the infimal convolution fc⊕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 fc 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 ,
fc(y∗, z∗, α) =
{ f∗(y∗) if ⟨x, z∗⟩ < α for all x ∈ dom f, +∞ otherwise.
Moreover, dom fc= 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= (fc⊕ gc)c′c. (ii ) h = (fc⊕ gc)c′.
Proof. (i ) According to (6), we can write e′co (epi fc+ 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 fc+ epi gc} . From (3), it follows that k = fc⊕ gc, and applying Theorem 4, we have
epi hc = e′co (epi fc+ epi gc) = epi (fc⊕ gc)c′c. Consequently, hc = (fc⊕ gc)c′c.
(ii ) Since h is an e-convex function, then h = hc′c. According to Proposition 2, we also know that
(
(fc⊕ gc)c′c )c′
= (fc⊕ 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 fc⊕ gc if and only if f + g = h.
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, epi fc= ∪
ε≥0
{(u∗, v∗, ω,⟨x0, u∗⟩ + ε − f (x0)) | (u∗, v∗, ω)∈ ∂c,εf (x0)} .
Proof. Pick any (u∗, v∗, ω, β)∈ epi fc. 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.
Hence, fc(u∗, v∗, ω)≤ ⟨x0, u∗⟩ + ε − f (x0) and (u∗, v∗, ω,⟨x0, u∗⟩ + ε − f (x0))∈ epi fc. 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= fc⊕ gc and the infimal convolution is exact.
(ii ) epi fc+ epi gc is e′-convex.
Proof. (i ) ⇒ (ii). We shall prove that epi fc+ 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 fc+ epi gc⊂ epi hc. For proving the reverse inclusion, consider (y∗, z∗, α, β)∈ epi hc. Applying (i ) we have
β ≥ hc(y∗, z∗, α) = (fc⊕ gc) (y∗, z∗, α) = fc(y1∗, z1∗, α1) + gc(y2∗, z2∗, α2) ,
for certain (y∗1, z1∗, α1), (y2∗, z2∗, α2)∈ W such that (y1∗, z1∗, α1) + (y2∗, z2∗, α2) = (y∗, z∗, α). Con- sider now the scalars β1 := fc(y1∗, z1∗, α1) and β2 := β− β1. Then,
(y1∗, z1∗, α1, β1)∈ epi fc and (y∗2, z2∗, α2, β2)∈ epi gc. Hence, (y∗, z∗, α, β) = (y1∗, z∗1, α1, β1) + (y2∗, z2∗, α2, β2)∈ epi fc+ epi gc.
(ii ) ⇒ (i). As the set epi fc+ epi gc is e′-convex, by means of Theorem 4 we conclude that epi fc+ 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 fc+ epi gc}
= (fc⊕ gc) (y∗, z∗, α) ,
and consequently, hc= fc⊕ gc. Let us check that the infimal convolution is exact. Since h is a proper convex function, then fc⊕ gc is a proper function according to Lemma 6. Suppose that (fc⊕ gc) (y∗, z∗, α) = +∞ for some (y∗, z∗, α) ∈ W . Since gc(0, 0, 0) = +∞ and fc is proper, then we can write
(fc⊕ gc) (y∗, z∗, α) = fc(y∗, z∗, α) + gc(0, 0, 0) , and therefore, the infimal convolution is exact at (y∗, z∗, α).
Now, assume that β := (fc⊕ gc) (y∗, z∗, α)∈ R. Then, (y∗, z∗, α, β)∈ epi hc= epi fc+ epi gc, and there exist (¯y1∗, ¯z∗1, ¯α1, β1) ∈ epi fc and (¯y2∗, ¯z2∗, ¯α2, β2) ∈ epi gc such that (y∗, z∗, α, β) = (¯y1∗, ¯z1∗, ¯α1, β1) + (¯y2∗, ¯z∗2, ¯α2, β2) and
fc⊕ gc(y∗, z∗, α) = β1+ β2
≥ fc(¯y∗1, ¯z∗1, ¯α1) + gc(¯y∗2, ¯z2∗, ¯α2)
≥ inf {fc(y1∗, z1∗, α1) + gc(y2∗, z∗2, α2) | (y∗1, z∗1, α1) + (y2∗, z2∗, α2) = (y∗, z∗, α)}
= fc⊕ 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 fc+ epi gc, is linked to the infimal convolution of fc 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= fc⊕ gc and the infimal convolution is exact.
(ii ) epi fc+ epi gc is e′-convex.
(iii ) For each ε≥ 0 and for each x ∈ dom f ∩ dom g,
∂c,ε(f + g) (x) = ∪
ε1+ε2=ε ε1,ε2≥0
∂c,ε1f (x) + ∂c,ε2g (x) . (12)
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
(u∗, v∗, ω,⟨x0, u∗⟩ + ε − (f + g) (x0))∈ epi (f + g)c = epi fc+ epi gc. Hence, there exist (u∗1, v∗1, ω1, β1)∈ epi fc and (u∗2, v∗2, ω2, β2)∈ epi gc such that
(u∗1, v∗1, ω1, β1) + (u∗2, v∗2, ω2, β2) = (u∗, v∗, ω,⟨x0, u∗⟩ + ε − (f + g) (x0)) . Applying again Lemma 9, there exist ε1, ε2 ≥ 0 such that
(u∗1, v∗1, ω1)∈ ∂c,ε1f (x0) , β1 =⟨x0, u∗1⟩ + ε1− f (x0) , (u∗2, v2∗, ω2)∈ ∂c,ε2g (x0) , β2 =⟨x0, u∗2⟩ + ε2− g (x0) ,
which implies ε1 + ε2 = ε. As a consequence, (u∗, v∗, ω) ∈ ∂c,ε1f (x0) + ∂c,ε2g (x0), and therefore,
∂c,ε(f + g) (x0)⊂ ∪
ε1+ε2=ε ε1,ε2≥0
∂c,ε1f (x0) + ∂c,ε2g (x0) .
Conversely, consider (u∗1, v∗1, ω1) ∈ ∂c,ε1f (x0) and (u∗2, v2∗, ω2) ∈ ∂c,ε2g (x0), where ε1, ε2 ≥ 0 are such that ε1+ ε2 = ε. Lemma 9 is used again to obtain
(u∗1+ u∗2, v1∗+ v2∗, ω1 + ω2,⟨x0, u∗1+ u∗2⟩ + ε − (f + g) (x0))∈ epi fc+ epi gc.
Since epi fc+ epi gc = epi (f + g)c, then (u∗1, v∗1, ω1) + (u∗2, 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 fc+ epi gc) = epi (f + g)c. So, for prov- ing (ii ) it suffices to show that epi (f + g)c ⊂ epi fc+ 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 (u∗1, v∗1, ω1) ∈ ∂c,ε1f (x0), (u∗2, v∗2, ω2) ∈
∂c,ε2g (x0) verifying (y∗, z∗, α) = (u∗1, v∗1, ω1) + (u∗2, v2∗, ω2), name β1 := ⟨x0, u∗1⟩ + ε1 − f (x0) and β2 :=⟨x0, u∗2⟩ + ε2 − g (x0). Observe that β1+ β2 = β. Again by Lemma 9, we obtain
(u∗1, v∗1, ω1, β1)∈ epi fc, (u∗2, v2∗, ω2, β2)∈ epi gc. Therefore, (y∗, z∗, α, β)∈ epi fc+ 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 fc+ epi gc is e′-convex, then
∂c(f + g) (x) = ∂cf (x) + ∂cg (x) , ∀x ∈ dom f ∩ dom g.
