Formulae for the Conjugate and the Subdifferential of the Supremum Function
Pedro Pérez-Aros
math.OC 90C25, 90C34, 46N10







Pedro Pérez-Aros Instituto de Ciencias de la Ingeniería - Universidad de O'higgins

              Rancagua, Chile

              pedro.perez@uoh.cl 

              This work was  supported in part by  CONICYT-PCHA/doctorado Nacional/2014-21140621.       






Formulae for the  Conjugate and the Subdifferential  of the  Supremum Function
    Pedro Pérez-Aros
    Received: date / Accepted: date
==============================================================================




The aim of this work is to provide formulae for the subdifferential and the conjungate function of the supremun function over an arbitrary family of functions. The work is principally motivated by  the case when  data functions are  lower semicontinuous proper and convex. Nevertheless, we explore the case when the family of functions is arbitrary, but satisfying that the biconjugate of the supremum functions is equal to the supremum of the biconjugate  of the data functions. The study focuses its attention on functions defined in finite-dimensional spaces, in this case the formulae can be simplified under certain qualification conditions. However, we show how to extend these results to arbitrary locally convex spaces without any qualification condition.


90C25 90C34 46N10





§ INTRODUCTION


	Convex analysis has been one of the most studied topics in nonsmooth analysis; currently, chapters  referring to convex analysis can be found in many monographic books related to optimization and variational  analysis. The development of convex analysis has brought many tools to establish necessary and sufficient conditions for optimality. A remarkable tool in this scenario is the notion of the subdifferential of a convex function. Everybody interested in nonsmooth analysis is aware of classical calculus rules and qualification conditions, together with practical algorithms, which involve this mathematical object.
	
		Another important tool in (convex) subdifferential  calculus is the so called  approximate subdifferential of convex functions (see, e.g., <cit.>). This operator has played a central role in optimization; its impact can be seen in applications such as  linear programming, convex and non-convex programming, stochastic programming and semi-infinite programming among many other topics, because this mathematical object has been used to provide calculus rules without qualification conditions (see, e.g., <cit.>),  characterize points of  sub-optimality (see, e.g., <cit.>). It has been used in many minimization algorithms (see, e.g., <cit.>) and also can be found in connection with integration of subdifferentials (see e.g <cit.>). Also, the nonemptiness of this operator is guaranteed in arbitrary locally convex spaces under weaker conditions (<cit.>) and it represents, in some sense, a continuous multifunction (see, e.g., <cit.>). All of these properties have motivated some authors to give generalizations of this  object to vector optimization  (see, e.g., <cit.> and the references therein).
		
		A functional, which appears commonly in applications, is the supremum functional of a family of functions, for this reason many authors have shown several formulae for the calculus of the  subdifferential of the supremun function (see, e.g.,  <cit.>  and the reference therein). Moreover in <cit.> the   authors have shown that the supremum function appears to be fundamental in the theory of convex subdifferential calculus,  in the sense that the general formula presented in <cit.> allows the authors to recover classical and important formulae available in the literature (see <cit.>, where the authors recover as a simple corollary of their general formula the calculus rules for the sum and composition). However, none of them have studied the ε-subdifferential of the supremum function. This observation motivates this work to provide calculus for the subdifferential  together with formulae for the conjugate function of the supremum function. 
		
		This work is organized as follows. In Section <ref>, we present some classical  notation and definitions of convex analysis which agree with many monograph books (see, e.g., <cit.>).  In Section <ref>, we give preliminary results concerning the calculus of the conjugate and the subdifferential of the  supremum function.  Section <ref> is devoted to studying formulae for the conjugate function of the supremun function of an arbitrary  family of (possibly non-convex) functions using the additional assumption that the index set is an ordered set and the functions  are epi-pointed; this allows us to give formulae for the conjugate of the lim sup and lim inf function. Finally, we divide Section <ref>, where the main results are  established, in two subsections. Subsection <ref> is focused on when the  functions are defined in a  finite-dimensional space; in this scenario we investigate the subdifferential  and the conjugate function of the pointwise supremum under the standard assumption that the normal cone of the domain of the supremum function at the point of interest  does not contain lines.  The main results about the calculus of the subdifferential of the supremum function is Theorem  <ref>; later under assumptions of compactness of the index set and some continuity property, which are classical in semi-infinite programming, we derive Theorem <ref>. Posteriorly  we get a result that can be understood as the conjugate counterpart of the last two theorems (see Theorem <ref>).  Due to the fact that this research is principally motivated by <cit.>, in this section  we bypass the convexity of the data function  by the weaker assumption that the biconjugate of the  supremum function is equal to the supremum of the biconjugate of the data functions. This kind of hypothesis has been recently used in several works (see, e.g., <cit.>). Subsection <ref> is motivated by the completeness of this work . In this subsection we show how to generalize the  mentioned results (given in a  finite-dimensional space) to a general locally convex space and without any qualification condition using the family of all  finite-dimensional subspaces containing a given point. 



§ NOTATION

Throughout the paper X and X^∗ will denote  two (separated) locally convex spaces (lcs) in duality by the bilinear form ⟨·,·⟩ : X^∗× X  →ℝ, given by  ⟨ x^∗, x ⟩ =x^∗(x). In  X^∗ the weak topology is denoted by  w(X^∗,X)  (w^∗ for short) and  the Mackey topology  on X^∗ is  denoted by  τ(X^∗,X), the space X will be endowed with a compatible initial topology τ (i.e., (X,τ)^∗=X^∗). The family of all closed convex and balanced neighborhoods of zero for the topology τ will be denoted by 𝒩_0(τ) and we omit the symbol τ when there is no confusion, similar terminology is adapted for the space X^∗.   We will write ℝ:= [-∞,∞ ] and we adapt the convention α/0=+∞ for all α>0. The symbol X means the dimension of the space X.


 The closure of A will be denoted by A. We denote by (A),  conv(A) and  (A), the interior, the convex hull and the closed convex  hull of A, respectively. The polar of A is the set  A^o := { x^∗∈ X^∗ : ⟨ x^∗,x⟩≤ 1, ∀ x∈ A}.
When the space X is  finite-dimensional, the norm on X and X^∗ will be denoted by · and ·_∗, respectively. Given x ∈ X (or x^∗∈ X) and r≥ 0 we denote 𝔹(x,r):={y ∈ X: x-z≤ r  } (𝔹(x^∗,r):={y^∗∈ X^∗: x^∗-z^∗_*≤ r  }) .


For a given function f:X→ℝ, the (effective) domain and the  epigraph  of f are 
    f := { x∈ X: f(x) < +∞} and  f := { (x,λ)∈ X× : f(x)≤λ},
 respectively. We say that f is proper if f ≠∅ and f>-∞, and  inf-compact if for every λ∈ℝ the set [f≤λ]:={ x ∈ X: f(x)≤λ} is compact. We denote Γ_0(X)  the class of proper lower semicontinuous (lsc) convex functions on X. The Fenchel conjugate of f is the function f^∗:X^∗→ℝ defined by

    f^∗(x^∗) := sup_x∈ X{⟨ x^∗,x⟩ - f(x)},

and the biconjugate  of f is f^∗∗:= (f^∗)^∗:X →.  The closed hull, the convex hull and   the closed convex hull of  f are denoted  by  f, f and  f respectively, and they are defined  as the functions such that f= f, f= f,  and  f =  f respectively. We recall that f^∗∗≤ f and whenever f^∗∗ is proper one has f^∗∗=  f. 

For ε≥0, the ε-subdifferential (or the approximate subdifferential of convex functions)   of f at a point x∈ X, where it is finite, is  the set

    ∂_ε f(x) := { x^∗∈ X^∗ : ⟨ x^∗,y-x⟩≤ f(y) - f(x) +ε, ∀ y∈ X  };

if f(x) is not finite, we set ∂_ε f(x):=∅. The special case ε=0 is the classical convex subdifferential, also called  the  Moreau-Rockafellar Subdifferential  and  is denoted by ∂ f(x). 

The indicator and the support functions of a set A (⊂ X,X^∗) are, respectively,

    δ_A(x) := 
    0,        x∈ A,
    
    +∞,    x∉ A,
            σ_A :=δ_A^∗.

The asymptotic cone of A is defined by A_∞:=⋂_ε >0 (   ] 0,ε] A ) (see, e.g., <cit.>). When the space X is a Banach space the   asymptotic cone is commonly  defined as A_∞= { d ∈ X : ∃ s_k ↘ 0  and  x_k ∈ A  such that limx_k/s_k = d}. When the set  A is convex A_∞ is also known as the recession cone; in this case it admits the following representation 
    A_∞= { d ∈ X:   x+ λ d∈ A  for some x  and all λ≥ 0  }.
 We set 

    A^-   :={ x^∗∈  X^∗: ⟨ x^∗ , x ⟩≤ 0, ∀ x∈ A   },
    
    A^⊥   :={ x^∗∈  X^∗: ⟨ x^∗ , x ⟩ = 0, ∀ x∈ A   }
 the negative dual cone, and the orthogonal subspace (or
annihilator) of A, respectively.

The ε-normal set of A at a point x is defined and denoted by 
    N^ε_A(x):=∂_εδ_A(x).


Now let us introduce the class of epi-pointed functions. This class of functions has shown that it shares  important properties with the class of convex function, we refer to <cit.>  and the references therein. 


	
	A function f:X→ is said to be epi-pointed if f^∗ is proper and  τ(X^∗,X)-continuous at some point of its domain. 


 As far as we know, this family of functions was introduced in finite dimensions in <cit.>. However, the the above extension was introduced in <cit.> with the name of Mackey epi-pointed functions.

In a finite-dimensional Banach space X the reader can understand the above property in terms of the pointedness of the epigraph of  its asymptotic function, which is  given by 

    f^∞ (u):=lim inf_ s ↘ 0
     w → u  s f( s^-1 w	) 	,

or in terms of a  coercive property, which is the  existence of   x^∗∈ X, α >0 and r ∈ such that f(x) ≥⟨ x^∗ , x ⟩ +α x +r for all x∈ X  (see, e.g., <cit.>). 

For a set T, 𝒫_f(T)  denotes the set of all F⊆ T such that F is finite. We  define 
    f(x):=sup_t ∈ T f_t,  for x∈ X
 as the pointwise supremum of the family {f_t : t ∈ T }. 	When T is a directed set  ordered  by ≼ (i.e.,  (T,≼) is an ordered set and for every t_1,t_2∈ T there exists t_3 ∈ T with t_1≼ t_3 and t_2 ≼ t_3) we say that the family of functions is increasing provided that  for all t_1,t_2 ∈ T

    t_1 ≼ t_2  f_t_1≤ f_t_2 (i.e.,   f_t_1(x) ≤ f_t_2(x) ,  ∀ x∈ X).
   	
  For  convenience we also use the notation t_2 ≽ t_2 iff t_1 ≼ t_2.

Following the standard notation (see, e.g., <cit.>), for a subset Z of   let Z^(T) be the set of all (λ_t) ∈ Z^T such that λ_t≠0 for finitely many t∈  T. The support of λ is defined as λ :={t ∈ T : λ_t ≠ 0  }. The generalized simplex on T is the set Δ(T):={λ∈  [0,1]^(T)  : ∑_t∈ Tλ_t =1 }, also it will be convenient to denote Δ^ε(T): ={λ∈  [0,ϵ]^(T)  :  ∑_t∈ Tλ_t =ε}.  For a  family of functions { f_t }_t∈ T⊆^T,  a point x̅ and ε≥ 0,  the set of ε-active indexes at x̅ is defined by T_ε(x̅):={ t∈ T : f_t(x̅) ≥ f(x̅) -ϵ}.






§ PRELIMINARY RESULT




 §.§ Basic Properties of the Conjugate and the ε-Subdifferential of the Pointwise Supremum


The next lemma shows some basic  properties of the the pointwise supremum  function.

	Let {f_t : t ∈ T } be an arbitrary family of functions and define h:=inf_t ∈ T f^∗_t. Then
	
		
		
  *    h^∗=sup f^∗∗_t, consequently if h^∗∗≠ -∞, then h=h^∗∗. Therefore,  if the functions f_t ∈Γ_0(X) and f is proper one has 
    f^∗= ⋃_t∈ T f^∗_t .

	
		
  *  h (x^∗)={  f_t^∗ : t∈ T  }(x^∗), where 
    {  f_t^∗ : t∈ T  }(x^∗):= inf{∑_t∈ Tλ_t f_t(x^∗_t) : [          λ∈Δ(T)  and; ∑_t∈ tλ_t x^∗_t =x^∗ ]}.
 Moreover,  the infimum can be taken only in λ∈Δ(T) with 
    #λ≤min{ X + 1, # T     }.

			
  *    If  f^∗ is proper and   f^∗∗=sup_t∈ T f^∗∗_t, then  f^∗={ f_t^∗: t∈ T}.
		
  *   If {f_t : t ∈ T }  is an increasing family of functions, then inf_t∈ T f^∗_t is convex.
		
		
  *  If f_t ≤ f_s and f_t is epi-pointed and f^∗_s is proper, then f_s is epi-pointed.
		
  *   For every t∈ T we have that  f_t^∗⊆ f^∗ and f_t^∗⊆ f^∗. In addition, if f^* is proper,  f^∗∗=sup_t∈ T f^∗∗_t and {f_t : t ∈ T }  is an increasing family of  functions one has 
		
    ⋃_ t∈ T f_t^∗ =  f^∗ and ⋃_ t∈ T  f_t^∗ =  f^∗ .

		
	


Proof
	
		
  * It is not difficult to see that  h^∗=sup f^∗∗_t, then h=h^∗∗ whenever h^∗∗ is proper (see, e.g., <cit.>).   Moreover, when {f_t }_t∈ T⊆Γ_0(x) we have f_t^∗∗ =f_t for each t∈ T, in particular,  h^∗=f. Then, using the fact that  f^∗ is proper we have h^∗∗ is also proper and consequently  
    f^∗ = ⋃_t∈ T f^∗_t  and  f^∗=inf_t∈ T f_t^∗.
 
		
		
  *  First, h ≤{  	 f^∗_t : t∈ T} due to the convexity of h. Now, we notice that 
    _s h = ⋃_ t∈ T_s f^∗_t,

			where _s denotes the strict epigraph. It follows that for every x^∗∈ h and every and   ϵ >0, the element  (x^∗,α_ϵ) ∈ ( h), where 
			
    α_ϵ:={[        h(x^∗),   if  h(x^∗)∈,;          ϵ^-1, if  h(x^∗)=-∞. ]
 
			Then,   there exists λ_i ≥ 0 and (x_i^*, β_i ) ∈ h with i=1,...,p and ∑_ i=1  ^p λ_i =1 such that (x^∗,α_ϵ) = ∑_ i=1  ^p λ_i (x_i^*, β_i ). Moreover, by (<ref>) there exists t_i ∈ T  such that (x_i^*, β_i + ϵ ) ∈_s f^*_t_i. It implies that
		
    α_ϵ + ϵ≥∑_ i=1  ^p λ_i f^*_t_i (x^*_i) ≥{ f_t^∗ : t∈ T }.

		Since α_ϵ + ϵ converges to h(x^∗) as ϵ↘ 0 we conclude the equality.
	 	
 Moreover, when the space is  finite-dimensional we can use Carathéodory's Theorem to consider only λ∈Δ(T) with #suppλ≤ X + 1.
		
		
  * Since f^∗ is proper and   h^∗∗≥ f^∗ we have h^∗∗≠ -∞, then by <ref> we get  h^∗∗ =  h. Finally, let us prove that h ≤  f^∗, indeed
		
    h^∗∗   = sup_x∈ X{⟨· ,x ⟩ - h^*(x) }=sup_x∈ X{⟨· ,x ⟩ - f^∗∗(x) } = f^∗.

		Therefore f^∗= h^∗∗=  h= { f_t^∗ : t ∈ T}.
		
  * Because the family of functions is increasing, we have  f_t^∗⊆ f_s^∗ for t≤ s, therefore ⋃_ t∈ T f^∗_t is convex and then 
		inf_t∈ T f_t^∗ is convex.
		
  * If f_t ≤ f_s one has f^∗_t ≥ f_s^∗ and since f^∗_t is bounded in ( f_t^∗), so is f_s^∗, which implies the continuity of f_s^∗.
		
  * First, one  has ⋃_ t∈ T f_t^∗⊆ f^∗ and ⋃_t∈ T f^∗_t⊆ f^∗, consequently ⋃_ t∈ T f_t^∗⊆ f^∗ and ⋃_t∈ T f^∗_t ⊆ f^∗. Now, by <ref>  and <ref>    one has 
		
    f^∗ = {⋃_t∈ T f^∗_t}=  {⋃_t∈ T f^∗_t}

	 and then f^∗⊆{⋃_ t∈ T f_t^∗}. Fix   (u^∗_0,β_0) ∈⋃_t∈ T f^∗_t and take  (x_0^∗, α_0)  ∈ f^∗; then there exists   γ >0 such that (x,α):=(x_0^∗,α_0) + γ ( (x_0^∗,α_0) - (u^∗_0,β_0) ) ∈ f^∗, because  ⋃_t∈ T f^∗_t is a convex open  dense subset in f^∗ (see (<ref>)) one has 
		
    1/1+γ  (x,α) + γ/1+γ (u^∗_0,β_0)= (x_0^∗,α_0) ∈⋃_t∈ T f^∗_t.

		Using the same argument one also can conclude  ⋃_ t∈ T f_t^∗=  f^∗.
	
  


	  Lemma <ref> <ref> appears to be very simple. Nevertheless, it represents a key point in  the entire work and it allows us to use a weaker condition than the lsc and convexity of the data function (i.e. f_t^∗∈Γ_0(X) for all t∈ T). In this work we use the hypothesis that f^∗∗= sup_t∈ T  f^∗∗_t,  which has been used recently by some authors (see  <cit.>) and it is weaker than the hypothesis used in <cit.>. More precisely in <cit.> the authors use the hypothesis that {f_t}_t∈ T are proper convex functions satisfying f =sup_t∈ T f_t. Posteriorly in <cit.> the authors use the  hypotheses that {f_t}_t∈ T are proper convex functions and f(x) =sup_t∈ T f_t(x) for all x∈ ( f). Recently, in <cit.> the authors presented an improvement of the last results, in this work the functions  {f_t}_t∈ T are assumed to be proper convex functions and f(x) =sup f_t(x) for all x∈ f (see <cit.>). It is worth mentioning that to prove Lemma <ref> <ref>  one can  assume the weaker hypothesis that f^∗ is proper and  f^∗∗(x)= sup_t∈ T  f^∗∗_t(x) for all x ∈ D, where D is a graphically dense subset of f^∗∗ (which is satisfied under the assumption of <cit.>). For the sake of simplicity, we keep in mind  these sophisticated assumptions for a  future work. 


The next lemma is a slight extension of <cit.> without the assumption of convexity.


	Let g:X → be an epi-pointed function.  Consider η >0 and x ∈ X such that  η +g^∗∗(x)>g(x). Then
	
    ∂_η g(x)= ^w^∗{∂_η g(x)∩ g^∗}.

Consequently  for every η≥ 0

    ∂_η g(x)= ⋂_γ >   0^w^∗{∂_η+γ g(x)∩ g^∗}.



Proof 	First, if |g^∗∗(x)|=+∞ one has the empty set in both sides of (<ref>). Now consider g^∗∗(x)∈ and take γ >0  such that η-γ +g^∗∗(x)>g(x). Because g^∗∗ is a proper convex and lsc function and γ>0, one has ∂_γ  g^∗∗(x) is non-empty, and by  <cit.>  we get  ∂_γ  g^∗∗(x) ∩ g^∗≠∅.

 Then taking u^∗∈∂_γ  g^∗∗(x) ∩ g^∗ one gets u^∗∈∂_η g(x), which implies, ∂_η g(x) ∩ g^∗≠∅.   Since g^∗ is open, convex and dense in the convex set g^∗, we have that
	
    {∂_η g(x)∩ g^∗}= ∂_η g(x) ∩ g^∗ = ∂_η g(x).
 Finally,  (<ref>) follows noticing that if ∂_ηg(x) ≠∅ we have γ + η + g^∗∗(x) > g(x) and  ∂_ηg(x) = ⋂_γ >   0∂_η + γ  g(x).


<cit.>
	Let h be an extended-real-valued convex function defined over X^∗. Then, for all r ∈,
	
    {  x^∗∈ X^∗ :  h(x^∗)≤ r      }=⋂_γ >0 { 	x^∗∈ X^∗ : h(x^∗) < r +γ}.

	Moreover, if r > inf_X^∗ h, then 
	
    {  x^∗∈ X^∗ :  h(x^∗)≤ r      }={ 	x^∗∈ X^∗ : h(x^∗) < r 	}.


 The following results give us a first representation of the ϵ-subdifferential of f, this result corresponds to a slight extension  of  <cit.>.

	Let {f_t : t ∈ T } be an arbitrary family of functions such that f^∗∗=sup_T  f^∗∗_t. 	If  ε > inf_x^∗∈ X^∗{ f^∗(x^∗) +f(x) -⟨ x^∗ ,x ⟩}, then
	
    ∂_ε f(x) =     {∑_t∈ Tλ_t ∂_η_t   f_t(x) :  [                                 λ∈Δ(T),  η_t ≥ 0,;                       ∑_t∈ Tλ_t ·η_t ∈ [0,ε)  and; ∑_t∈ Tλ_t ·  f_t  (x) > f(x) + ∑_t∈ Tλ_t ·η_t - ε ]}.

	Consequently, for every  ε≥ 0
	
    ∂_ε f(x) = ⋂_γ >   ε{∑_ t∈ Tλ_t ∂_η_t   f_t(x) : [                               λ∈Δ(T),  η_t ≥ 0,;                   ∑_t∈ Tλ_t ·η_t ∈ [0,γ),
    	 and; ∑_t∈ Tλ_t  f_t  (x) ≥ f(x) + ∑_t∈ Tλ_t ·η_t - γ ]}.

	
 Moreover, when the space X is  finite-dimensional  the infimum can be taken only λ∈Δ(T) with  #λ≤min{ X + 1,#T }.

Proof
	Since  the right side of (<ref>) and (<ref>) are included in _ε f(x) we must focus on the opposite inclusion. Assume that 
	∂_ε f(x)≠∅, then f^∗ is proper and by Lemma <ref><ref>, f^∗={ f^∗_t : t∈ T }. Hence, using  Lemma <ref>
	
    ∂_ε f(x) = {   x^∗∈ X^∗ : { f_t^∗ : 	 t∈ T}(x^∗) +f(x) < ⟨  x^∗ , x ⟩ + ε}.

	Now let x^∗∈ X^∗ satisfying { f_t^∗ : 	 t∈  T}(x^∗) +f(x)  -⟨  x^∗ , x ⟩ <   γ, then by Lemma <ref> <ref> there exist λ∈Δ(T) (with #suppλ≤ X + 1) and x^∗_t ∈ X^∗ for t∈λ  such that ∑_t∈ Tλ_t x^∗_t  = x^∗ and 
	
    { f_t^∗ : 	 t∈ T}(x^∗) +f(x) - ⟨  x^∗ , x ⟩≤   ∑_t∈ Tλ_t (  f_t^∗(x_t^∗)  +f_t(x) - ⟨ x^∗_t, x⟩)  +
       ∑λ_t ( f(x)- f_t(x) )		<  ε,

	then taking  η_t := f_t^∗(x_t^∗)  +f_t(x) - ⟨ x^∗_t, x⟩ if t ∈λ and η_t:=0 if t∉λ one gets that x^∗_t ∈∂_η_t  f_t(x) for every t∈λ,  ∑_t∈ Tλ_t ·η_t ∈ [0,ε) and ∑_t∈ Tλ_t ·  f_t  (x) > f(x) + ∑_t∈ Tλ_t ·η_t - ε, which concludes the proof of  (<ref>).
	
	Finally, if  ε≥ 0 is any real number such that _ε f(x)≠∅ one has that inf_x^∗∈ X^∗{ f^∗(x^∗) +f(x) -⟨ x^∗ ,x ⟩}≤ε, consequently using (<ref>) for  γ >ε and  the fact that ∂_ε f(x) =   ⋂_γ >ε∂_γ f(x) we obtain

    ∂_ε f(x)    = ⋂_γ >ε{∑_t∈ Tλ_t ∂_η_t   f_t(x) :  [                                 λ∈Δ(T),  η_t ≥ 0,;                          ∑_t∈ Tλ·η_t ∈ [0,γ)  and; ∑_t∈ Tλ_t ·  f_t  (x) > f(x) + ∑_t∈ Tλ_t ·η_t - γ ]}.
	


	

 §.§ Characterization of Global ε-Minimum of Pointwise Supremum

The intention of this subsection is to introduce the definition of ε-Robust infimum, which appears to be promising  to characterize ε-minimums of the supremum function; we also refer a sufficient condition to guarantee the mentioned property. The introduction of this concept  allows us to cover many ideas reflected in  max-min theorems that have been broadly studied, this notation is motivated by the concept of Robust infimum or Decoupled Infimum used in subdifferential theory to get fuzzy calculus rules (see, e.g., <cit.>). To be  more precise the reader can observe that when ε=0 the  definition below corresponds to a max-min equation, that is to say, 
    sup_Tinf_X f_t(x)= inf_Xsup_Tf_t(x).
 

	We will say that the family of functions {f_t : t ∈ T} has an  ε-Robust infimum on B⊆ X at x̅∈ B provided that 
	
    f(x̅) ≤sup_t∈ Tinf_x ∈ Bf_t(x) +ε.

	The special case ε=0 is simply  called a Robust minimum. 



The min-max problem has been studied in many papers with different states of generality, for this reason it is impossible to recall all the sufficient conditions that establish the interchange  between supremum and infimum; we refer to  <cit.>) for some results and further discussions.  However, we establish the next lemma which guarantees this class of max-min results and fits perfectly with the framework of our study.  

First we need the following result.

 <cit.> 
	Let  X be a topological space and let (f_α)_α∈ D be a net  of lsc proper functions defined on X such that
	
    α,β∈ D, α≤β⇒ f_α≤ f_β.

	For (ε_α)_α∈ D↘ 0, let (x_α)_α∈ D be a relatively compact net such that 
    x_α∈ε_α-argminf_α for each α.
  Then
	
    inf_x ∈ Xsup_α∈ D f_α(x)= sup_α∈ Dinf_x∈ Xf_α(x),

	and every accumulation point of   (x_α) is a minimizer of the function  sup_α∈ D f_α.


[Sufficient condition for robust local minimum]
	Let (X,τ) be a topological space and  B⊆ X. Suppose (T,≼) is a directed set and the family   of  τ-lsc functions {f_t : t ∈ T} is  increasing, 	B is τ-closed and there exists some t_0 such that f_t_0  is τ-infcompact on B. Then the family {f_t : t ∈ T} has a Robust minimum on B.

Proof
	It is easy to see that inf_x∈ Bsup_t∈ T f_t(x) ≥sup_t∈ Tinf_x ∈ Bf_t(x)=:η. For the opposite inequality let us assume that η <+∞.   Consider  ε_t ∈ (0,1) and  points x_t ∈ B such that ε_t→ 0  and x_t belongs to ε_t-argmin_B{ f_t},  then  
    inf_x ∈ Bf_t(x) ≥inf_x ∈ Bf_t_0(x)>-∞ and x_t  ∈{ x :  f_t_0(x) ≤η + 1  } for all t ≽ t_0,
 so the set (x_t) is relatively compact,  and by Lemma <ref>  we get the result.
We finish this section with a very simple result of  the ε-robust infimum in terms of the ε-subdifferential of the initial data.

	Let  {f_t : t ∈ T} be a family of functions which has an  ε-Robust infimum on B⊆ X at x, then
	
    0 ∈⋂_γ >   0⋃_t ∈ T∂_ε + γ (f_t+δ_B)(x).

	The condition is also sufficient if f(x)=f_t(x) for all t∈ T.

Proof
	Let x such that f(x)≤sup_t∈ Tinf_y∈ Bf_t (y) + ε, then for a given γ >0 one can take t∈ T such that f_t(x)≤ f(x)≤inf_y∈ B f_t(y)+ ε + γ, that is, 0 ∈∂_ε + γ (f_t+δ_B)(x). Conversely, if f(x)=f_t(x) for all t∈ T and 0 belongs to ⋂_γ >   0⋃_t ∈ T∂_ε + γ (f_t+δ_B)(x), one has that for every γ >0 there exists some t∈ T such that  f(x)=f_t(x) ≤ f_t(y) + ε + γ for all y∈ B. Then, f(x) ≤sup_t ∈ Tinf_y∈ B f_t(y) + ε + γ, so the arbitrariness  of γ gives the result.


	

§ POINTWISE SUPREMUM FUNCTION OF  AN ORDERED FAMILY OF EPI-POINTED FUNCTIONS

	In this section, we investigate the ε-subdifferential of the pointwise supremum under the assumption that the  data is epi-pointed and the set T is a directed set. This extra hypothesis allows us to give better estimations for the conjugate function of the supremum function as well as formulae for the lim sup and lim inf functions. It is worth mentioning that the property of being a directed set can be  satisfied using the family 𝒫_f(T) and the  family of functions  defined by g_A(·) := max_s ∈ A f_s, then f(x)= sup_A ∈𝒫_f(T) g_A(·). Moreover the epi-pointed property can be also obtained using an appropriate perturbation of the functions f_t as will be used in some of the proofs.
	
		Let  {f_t : t ∈ T }  be an increasing family  of epi-pointed functions such that f^∗ is proper and  f^∗∗ = sup_t∈ T f^∗∗.
		Then for every x^∗∈ f^∗ one has
		
    f^∗(x^∗)=inf{ f_t^∗(x^∗) : t∈ T }.

	
	Proof
	 Let x_0^∗∈ f^∗,	by Lemma <ref> <ref> f^∗(x_0^∗)≤inf{ f_t^∗(x_0^∗) : t∈ T }. Now, using Lemma <ref> <ref> and <ref>  we have that f^∗ is the closed hull of the function inf{ f_t^∗(·) : t∈ T }, then one can take   x^∗_α→ x_0^∗  such that 
    f^∗(x_0^∗)= lim_αinf{ f_t^∗(x_α^∗)  :t ∈ T	},
 besides by Lemma <ref> <ref>  x^*_0 ∈ f_t_0 for some t_0 ∈ T. Because the family of epi-pointed functions { f^∗_s}_s≽ t_0 is decreasing, they are (uniformly) bounded on x_0^∗+ Ũ_t_0 (recall that x^∗_0 ∈ f_t_0^∗ and f_t_0^∗ is continuous at x_0^∗), then  by <cit.> we can  find a neighbourhood U_t_0∈𝒩_0, K_t_0 >0 and a continuous seminorm ρ  such that 
		
    |f^∗_s(x^∗)-f_s^∗(y^∗)| ≤ K_t_0ρ(x^∗ - y^∗)  for all  x,y ∈ x^∗_0  +  U_t_0,  for all  s ≽ t_0.

		Now let γ_α→ 0 and t_α∈ T with t_α≽ t_0  such that γ_α + f^∗(x^∗_0)  ≥ f^∗_t_α (x_α^∗) (recall (<ref>)), then using (<ref>) one yields
		
    γ_α + f^∗(x^∗_0)  ≥ -Kρ(x_α - x_0^∗) + f_t_α(x_0^∗) ≥  -Kρ(x_α - x_0^∗)  + inf{ f_t^∗(x^∗) : t∈ T }.

		Then taking the limits we conclude the result.
	Now based on  Theorem <ref> together with classical calculus rules for the conjugate of the sum of convex functions we give one answer to the question proposed in <cit.>.
	
	For this purpose we introduce the set ℓ^1(,X^∗) as the set of all sequences (x_n^∗) ∈ X^∗ such that 
	∑_n∈ | ⟨  x^∗_n , x ⟩|  <+∞ for every x∈ X   and the linear functional   ⟨ x^∗ , ·⟩  := ∑_n∈⟨  x^∗_n , ·⟩ belongs to X^∗ (see <cit.> for more details). 
	
	
		Consider  (x^∗_n) ∈ℓ^1(,X^∗), (α_n) ∈ℓ^1(,) and  a sequence of  functions f_n ∈Γ_0(X) such that 
		
    f_n(x) ≥⟨ x^∗_n , x ⟩  + α_n  ∀ x∈ X, ∀ n ∈.

		Consider f(x):=∑_n∈ f_n(x). If f and f_n are epi-pointed functions, then, for every x^∗∈ f^∗ one has
		
    f^∗(x^∗)   = inf{∑_n=1 ^N f_n^∗(x_n^∗)  : N∈ and ∑_n=1 ^N x_n^∗ =x^∗}
       = inf{∑_n∈ f_n^∗(x_n^∗)  :  ∑_n∈ x_n^∗ =x^∗}.

	
	Proof
		We (may) assume f_n≥ 0 for all n∈ (otherwise we can consider f̃_n := f_n - x_n^∗ - α_n). Hence, f^∗_n(0) ≤ 0  for all n∈.	Thus, f=sup_n∈ g_n, where   g_n:= ∑_k≤ n f_k.  First we recall (see, e.g., <cit.>) that for every n ∈ one has 
		
    (g_n )^∗(x^∗)   =  ^w^∗□_k=1^n f^∗_k (x^∗),  ∀ x^∗∈ X^∗,
     (g_n )^∗(x^∗)   =  □_k=1^n f^∗_k (x^∗),  ∀ x^∗∈(□_k=1^n f^∗_k),

		where □_k=1^n f^∗_k (·)=  inf{ f_1^∗(x^∗_1)+...+f_n^∗(x^∗_n)  : ∑_k=1^n x_k^∗=·}.   Now (<ref>) implies that 
    ∑_k=1^n f^∗_k⊆g_n^∗⊆( ∑_k=1^n f^∗_k),

		whence 
		(□_k=1^n f^∗_k)=  g^∗_n.
		
		Now, consider x^∗∈ f^∗. It is not difficult to prove that  
    f^∗(x^∗) ≤inf{∑_n∈ f_n^∗(x_n^∗)  :  ∑_n∈ x_n^∗ =x^∗}.
  For the other inequality we notice that  { g_n} form an  increasing family of proper lsc convex epi-pointed   functions  and f=sup_n∈ g_n, then using Theorem <ref> one has that
		f^∗(x^∗)=inf{ g_n^∗(x^∗) : n ∈},  besides by  Lemma <ref>  <ref> x^∗∈⋃_n∈ g_n^∗. Whence   (<ref>) yields 
		f^∗(x^∗)= inf{∑_k=1 ^n f_k^∗(x_k^∗)  : N∈ and ∑_k=1 ^n x_k^∗ =x^∗}, which concludes the proof of (<ref>). Finally, in order to prove (<ref>)  we recall f_n^∗(0) ≤ 0, thus  (<ref>) implies 
		f^∗(x^∗) ≥inf{∑_n∈ f_n^∗(x_n^∗)  :  ∑_n∈ x_n^∗ =x^∗} and that concludes the proof.
	A straightforward application  of Lemma <ref> and Theorem <ref> gives us an estimation for the conjugate of  lim sup and lim inf functions.
				
				
					Consider a directed set T  and an arbitrary family of functions  {f_t : t ∈ T }⊆Γ_0(X).
					
						
  * If h(x):=lim sup_t∈ T f_t (x)  is proper, then
						
    h^∗ =⋂_t ∈ T{⋃_s≽ t  f_t^∗}.

						
  * If g(x):=lim inf_t∈ T f_t(x)  is proper  and there exists t_0∈ T such that (inf_s≽ t_0 f_s))(·) is epi-pointed and 	 g^∗∗= sup_t∈ T g_t^∗∗,
					where 	g_t:= inf_s≽ t f_s, then 
						
    g^∗(x^∗):=lim sup f_t^∗(x^∗)  for all  x^∗∈g^∗

						and  g^∗ = ⋃_t ∈ T⋂_s≽ t f_s^∗. 
						
				
				Proof
					First define h_t:= sup_ s ≽ t f_s. Then
					
						
  * h^∗ =sup_x∈ X{⟨· , x ⟩ - h(x) }= sup_t∈ Tsup_x∈ X{⟨· , x ⟩ - h_t(x)}=sup_t∈ Th_s^∗. Then, we notice that  h_t must  be  proper for some t ∈ T, so by  Lemma <ref> <ref>   h_s^∗={⋃_s≽ t  f_t^∗}, and so (<ref>) holds.
						
  * g=sup_t∈ T g_t and { g_t }_t≽ t_0 is an increasing family of epi-pointed functions  satisfying g^∗∗= sup_t∈ T g_t^∗∗. Then, Theorem  <ref> and Lemma <ref> <ref> yield (<ref>), also Lemma <ref> <ref> implies g_t^∗ =  		⋂_s≽ t f_s^∗, and consequently g^∗ =⋃_t ∈ T g_t^∗ =⋃_t ∈ T⋂_s≽ t f_s^∗.
					
				
					Let  {f_t : t ∈ T }  be an increasing family of  epi-pointed functions such that  f^∗∗=sup_t∈ Tf_t^∗∗,  then  for all x∈ X.
					
    _ε f(x)   =⋂_t∈ T 
    γ >0 {⋃_ s ≽   t_ε+γ f_s(x)    }
       =	 ⋂_t∈ T 
    γ >0 {⋃_ s_0 ≽   t⋂_ s ≽   s_0_ε+γ f_s(x)    }.

					Moreover, the above formulae hold if the functions f_t are not necessarily , but they belong to Γ_0(X). 
				
				Proof First we check that the right side of (<ref>) is included in the ε-subdifferential of f at x. Indeed, let γ >0 and t∈ T and pick x^∗∈⋃_ s_0 ≽   t⋂_ s ≽   s_0∂_ε+γ f_s(x). Then ⟨ x^∗ , y - x ⟩≤ f_s(y) -f_s(x) +ε + γ≤ f(y)-f_s(x) +ε +γ for all y ∈ X and  for all s≽ s_0, then ⟨ x^∗ , y - x ⟩≤  f(y)-f(x) +ε +γ,  which means x^∗∈_ε + γ f(x). Therefore 
					
    ⋂_γ>0⋂_t∈ T  {⋃_ s_0 ≽   t⋂_ s ≽   s_0_ε+γ f_s(x)    }⊆⋂_γ>0_ε + γ f(x) =∂_ε f(x).

			 Now, it is easy to see that (<ref>) is included in (<ref>). For the opposite inclusion we notice that for any co-final set T̃⊆ T we have
					
    ⋂_t∈ T 
    γ >0 {⋃_ s_0 ≽   t⋂_ s ≽   s_0_ε+γ f_s(x)    }    = ⋂_t∈T̃
    γ >0 {⋃_ s_0 ≽   t⋂_ s ≽   s_0_ε+γ f_s(x)    }.

				  Then consider γ >0 and t ∈ T_γ(x) arbitrary. Pick s_2 ≽ s_1 ≽  t (it implies that s_1,s_2 ∈ T_γ(x)) and  x^∗∈_ε+γ f_s_1(x), then 
				
    ⟨ x^∗ , y - x⟩   ≤ f_s_1(y) - f_s_1(x) + ε+γ
       ≤ f_s_2(y)  - f_s_1(x) +f_s_2(x)   - f_s_2(x)+  ε+γ
       ≤ f_s_2(y)  - f_s_1(x) +f(x)   - f_s_2(x) +ε+γ
       ≤ f_s_2(y)   - f_s_2(y)+ ε+2γ,

				which means x^∗∈_ε+2γ f_s_2(x), since it is for every  s_2 ≽ s_1 ≽  t we conclude that 
				
    ⋃_ s ≽   t_ε+γ f_s(x)   ⊆⋃_ s_0 ≽   t⋂_ s ≽   s_0_ε+2γ f_s(x),

				and consequently
					
    ⋂_t∈ T 
    γ >0 {⋃_ s ≽   t_ε+γ f_s(x)    }   ⊆⋂_t∈ T_γ(x) 
    γ >0 {⋃_ s ≽   t_ε+γ f_s(x)    }
       ⊆⋂_t∈  T_γ(x)  
    γ >0 {⋃_ s_0 ≽   t⋂_ s ≽   s_0_ε+γ f_s(x)    }
       ⊆⋂_t∈ T 
    γ >0 {⋃_ s_0 ≽   t⋂_ s ≽   s_0_ε+γ f_s(x)    }
 
					
					Now we focus on proving that  ∂_ε f(x) is included in the right side of (<ref>),  so w.l.o.g. we can assume that ∂_ε f(x)≠∅, in particular f^* is proper,  and by  Lemma <ref> f is epi-pointed. First we prove this in the case that the functions f_t are epi-pointed. Thanks to Lemma <ref> it is enough to prove that for every γ >0 and t∈ T
					
					
    _γ + ε f(x) ∩ f^∗⊆⋃_ s ≽   t_ε+2γ f_s(x)  .

					
					
					Then, take x^∗ in the left side of (<ref>), then  f^*(x^∗) + f(x) ≤⟨ x^∗ ,x ⟩  + ε+γ, then using Theorem <ref> (recall x^∗∈ f^∗) and due to  the fact that  the family is increasing,  we can take  s ≽ t such that 
					 f_s^∗(x) + f_s(x) ≤⟨ x^∗ ,x ⟩  + ε+2γ, which implies x^∗∈_ε+2γ f_s(x).
					
					Now when the functions are not necessarily epi-pointed one  can use the following argumentation;  in order to simplify the notation we assume that x=0. For the sake of  understanding the proof first consider that X is a  space. Then, for every α >0 we define the epi-pointed  function 
					g^α_t(·) := f_t(·) + α·  (see, e.g., <cit.>) and  consider g^α=sup_t∈ T g_t^α = f + α· .  Then, _ε f(0) = ⋂_α > 0∂_ε g^α (0), therefore 
					
    ∂_ε  f(0)     = ⋂_α > 0∂_ε g^α (0)=⋂_α > 0⋂_t∈ T 
    γ >0 {⋃_ s ≽ t_ε+γ g_s^α (0)    }
       = ⋂_α > 0⋂_t∈ T 
    γ >0 {⋃_ s ≽ t( _ε+γ f_s(0) + α𝔹(0,1)) }
       = ⋂_t∈ T 
    γ >0 ⋂_α > 0{(⋃_ s ≽ t_ε+γ f_s(0)) + α𝔹(0,1) }
       = ⋂_t∈ T 
    γ >0 {⋃_ s ≽ t_ε+γ f_s(0) }

						Finally, if the space  X is infinite-dimensional we must identify the elements of 
					∂ (f +δ_L)(0) with ∂ f_|_L (0); here L is a finite-dimensional subspace of X and f_|_L is the restriction of f to L.
					Indeed, consider  t∈ T and γ>0, then take  e_k ∈ X for k=1,...,p and  define V={ x^∗∈ X^∗ : |⟨ x^∗, e_k |   ≤ 1 } and let L be the linear subspace generated by { e_k }_k=1^p. Now, we are going to prove that there exists s ≽ t such that ∂ (f +δ_L)(0)⊆∂_ε+γ f_s(x) + V. Let x^∗∈∂ (f +δ_L)(0) and consider P:X→ L a continuous projection and P^∗ its adjoint operator, 
					then x^∗_|_L ∈∂ f_|_L(0), then by the last part there exists  s ≽ t such that x^∗_|_L ∈∂ (f_s)_|_L(0) +W^∗,
					where W^∗:={ x^∗∈ L^∗ : |⟨ x^∗, e_k |   ≤ 1 }. Then
					x^∗ = P^∗(x^∗_|_L ) + x^∗ - P^∗(x^∗_|_L ) ∈∂_ε+γ( f_s+δ_L)(0) +V+  L^⊥, besides Hiriart-Urruty and Phelps's formula (see, e.g., <cit.>) implies 
    ∂_ε+γ( f_s+δ_L)(0) ⊆∂_ε+γ( f_s+δ_L)(0) + V^∗ + L^⊥.

					Therefore x^∗∈∂_ε+γ f_s(x) + V.
					
				The following example shows that the closure operator in the above result is necessary even for functions defined in the real line.
				
					Consider the functions f_n(x)=(1-1/n)| x|, then f(x)=|x| and _ε f_n(0)=[-1+1/n,1-1/n] and _ε  f(0)=[-1,1] for all ϵ≥ 0.  Whence ⋃_n∈_ε f_n(0) = (-1,1)≠ [-1,1]=  f(0).
				
				The next result corresponds to a general formula for the ϵ-normal set of an intersection of arbitrary closed and convex sets.
			
				
					Consider a family of closed convex subsets { C_t : t∈ T } and C:=⋂_t∈ T C_t. Then for every ε≥ 0 and   x∈ C
					
    N^ε_C(x) =⋂_γ >   0{∑_ t ∈ A  N^η_t_C_t(x) : [             A ⊆ T,  #A < +∞,; η_t ≥ 0  and ∑_t∈ Aη_t = ε+γ ]}.

				
			Proof
			First the right-hand side of (<ref>) is included in N^ε_C(x). We focus on the opposite inclusion.	Consider the sets G_A:=∩_t∈ A C_t for A ∈𝒫_f(T). Then 
				
				δ_C = sup_A ∈𝒫_f(T)δ_G_A, then by Theorem <ref> we have that
				
				
				
    N^ε_C(x)= ⋂_γ >   0{N^ε+γ_G_A(x)  : A ⊆ T,  #A <+∞},

				moreover
				N^ε+γ_G_A(x)= _ε+γ( ∑_t∈ Aδ_C_t)(x)
				and by Hiriart-Urruty and Phelps's  formula
				
    _ε+γ( ∑_t∈ Aδ_C_t)(x)=^w^∗{∑_t∈ AN^η_t_C_t(x)   : η_t ≥ 0  and ∑_t∈ Tη_t = ε + γ}.

				Consequently, mixing (<ref>) and (<ref>) we get the result.
				

§ CALCULUS  FOR   Ε-SUBDIFFERENTIAL  AND THE FENCHEL CONJUGATE  OF POINTWISE SUPREMUM FUNCTION

			This section is devoted to the study of the ε-subdifferential  of the pointwise supremum function. In the first part  we  assume that the space X is  ; this condition allows us to use Carathéodory's Theorem to bound the cardinal of the support of the elements λ∈Δ(T) in Proposition <ref> and with this we obtain more precise calculus rules for the ε-subdifferential and the Fenchel conjugate of the pointwise supremum function. In the second part of this section,  the  finite-dimensional condition over X  is removed using some reduction to  finite-dimensional subspaces; for this reason we prefer to use the notation X in Subsection <ref> for a finite-dimensional space also instead of using  the Euclidean space ^n, putting  emphasis on the fact that X could be an abstract  finite-dimensional space.
			
			

 §.§ Finite-Dimensional Banach Spaces

		In this section we give simplifications of Proposition <ref> under some classical qualification conditions in a  finite-dimensional Banach space X. In this subsection we assume that the functions f_t satisfy the relation  f^∗∗=sup_t ∈ T  f^∗∗_t.
			
		Let us introduce the following  qualification condition at a point x∈ X.
			
			
    N_ f(x)  does not contain lines.

	
		It is important to recall that for any arbitrary function f:X→ with proper conjugate the following statements are equivalent (i) N_ f(x) does not contains lines, (ii) f is continuous at some point and (iii) f^∗ is epi-pointed. Moreover, [ _ε f(x) ]_∞ =N_ f(x) for all ε≥ 0 and all x∈ X such that _ε f(x) ≠∅.
		
	
	The key tool to establish this exact formulation is the pointedness of the normal cones involved and  Carathedory's Theorem, which allow us  to give a limiting representation of (<ref>) and (<ref>). The next lemma represents the major ideas in the proofs of the main results for this section.

	
		Consider ε≥ 0, n:=min{#T,  X + 1} and   x^∗∈∂_ε f(x), then  there are sequences t_i,k∈ T, (λ_i,k) ∈Δ({1,...,n}),  x^∗_i,k∈∂_η_i,k f_t_i,k (x) and η_i,k≥ 0  with  i∈{1,...,n }, k∈ such that ∑_i=1^n lim_k  (λ_i,k·η_i,k)  ≤ε,  
		
    ∑_i=1^nlim_kλ_i,k· ( f(x)-f_t_i,k(x))  ≤ε  - ∑_i=1^nlim_kλ_i,kη_i,k

		
		and ∑_i=1^n λ_i,k· x_i,k^∗→ x^∗. Moreover, one of the following conditions holds.
		
			
  *  There exists n_1 ∈ with n_1 ≤  n such that λ_i,kk →∞⟶λ_i > 0, x_i,kk →∞⟶ x_i^∗, η_i,k→η_i for i ≤ n_1   and λ_i,kk →∞⟶  0, λ_i,k· x_i,kk →∞⟶ x_i^∗ for n_1 < i ≤ n, 
			
    ∑_i=1^n_1λ_i (lim_k →∞( f_t_i,k(x) -f(x))  + ∑_i>n_1lim_k →∞(λ_i,k  f_t_i,k(x) -f(x)) 
    ≤ε  - ∑_i=1^n_1λ_i lim_k →∞η_i,k -∑_i>n_1lim_k →∞λ_i,kη_i,k,

	∑_i=1^n_1λ_i =1	and  x^∗=∑_i=1^n_1λ_i x^∗_i + ∑_i>n_1^n x_i^∗, or
			
  *  There are ν_k ↘ 0 such that ν_k ·λ_i,k· x^∗_i,kk →∞⟶ x_i^∗ and ∑_i=1^n_1x^∗_i =0 with not all x_i^∗ equal to zero.
			
			Moreover, if one assumes  (<ref>), then  <ref> always holds.
		
	
	Proof Consider x^∗∈∂_ε f(x) and γ_k → 0. Then by  Proposition <ref> there are sequences t_i,k∈ T, (λ_i,k) ∈Δ({1,...,n}), η_i,k≥ 0 and x_i,k∈∂_η_i,k f_t_i,k (x)   such that ∑_t∈ Tλ_t_i,k·η_t_i,k∈ [0,ε +γ_k]
					and 	∑_t∈ Tλ_t ·  f_t  (x) ≥ f(x) + ∑_t∈ Tλ_t ·η_t - γ and u_k^∗:=∑_i=1^nλ_i,k· x_i,k^∗→ x^∗. We may assume (up to a subsequence)  that  for every i=1,...,n  (i) λ_i,k→λ_i, (ii) lim_k  (λ_i,k·η_i,k)  exists and (iii) lim_kλ_i,k· ( f(x)-f_t_i,k(x)) exists. 
				Then,  ∑_i=1^n lim_k  (λ_i,k·η_i,k)  ≤ε and 
				
    ∑_i=1^nlim_kλ_i,k· ( f(x)-f_t_i,k(x))  ≤ε  - ∑_i=1^nlim_kλ_i,kη_i,k.

					  Moreover (reordering if it is necessary),
				we  assume that λ_i >0 for  i=1,...,n_1 and λ_i >0 for i > n_1.
				
					Now, on the one hand, if we assume that sup_k, iλ_i,k· x^∗_i,k_*  <+∞, then up to a subsequence one can assume  that λ_i,kk →∞⟶λ_i > 0, x_i,kk →∞⟶ x_i^∗, for i ≤ n_1   and λ_i,kk →∞⟶  0, λ_i,k· x_i,kk →∞⟶ x_i^∗ for n_1 < i ≤ n, 
					
    ∑_i=1^n_1λ_i (lim_k →∞( f_t_i,k(x) -f(x))  + ∑_i>n_1lim_k →∞(λ_i,k  f_t_i,k(x) -f(x)) 
    ≤ε  - ∑_i=1^n_1λ_i lim_k →∞η_i,k + ∑_i>n_1lim_k →∞λ_i,kη_i,k

					and  x^∗=∑_i=1^n_1λ_i x^∗_i + ∑_i>n_1^n x_i^∗. 
					
					On the other hand, if sup_k, iλ_i,k· x^∗_i,k_* =sup_k {max_iλ_i,k· x^∗_i,k}=+∞ up to a  subsequence one can assume that ν_k:=(max_iλ_i,k· x^∗_i,k_* )^-1↘ 0 and ν_k λ_i,k· x^∗_i,kk →∞⟶ x_i^∗  with not all x_i^∗ equal to zero, thus ν_k u_k^∗→∑_i=1^n x_i^∗= 0.
					
					Finally, it is not difficult to see that <ref> contradicts (<ref>).
					
					
				Now we present the main result of this section.
			
				Assume that (<ref>) holds at x∈ f. Then for every  ε≥ 0 one has
				
    ∂_εf(x)    =⋃{ S(x,ε_1)+N^ε_2_ f(x)  | 	(ε_1,ε_2) ∈Δ^ε({1,2})	},
       = ⋃{S(x,ε_1)+N^ε_2_ f(x)  |  (ε_1,ε_2) ∈Δ^ε({1,2})	},

				where
				
    S(x,ε_1)     :=⋂_γ >0 {∑_t∈λλ_t  ∂_ε_t/λ_t + γ f_t(x)  : [ λ∈Δ(T),   
    				(ε_t)∈Δ^ε_1 (T)  and;           f_t(x) +ε_t/λ_t +γ≥ 	f(x) ]}
    S(x,ε_1)     :={∑_t∈λλ_t x_t^∗  : [ λ∈Δ(T), 
    				    (ε_t) ∈Δ^ε_1 (T)  and;                   x^∗_t ∈ A(x,ε_t,λ_t) ]}
 
				and A(x,ε,λ):={ x^∗∈ X^∗ :	[ ∃ t_k ∈ T,   x_k^∗∈∂_ε/λ + γ_k  f_t_k(x),  γ_k ↘ 	0,;   such that  x_k^∗→ x^∗ and lim f_t_k(x) + ε/λ≥ f(x) ]}
				
				
			
Proof
	We focus on the nontrivial  inclusions.  Let x^∗∈∂_ε f(x), then consider sequences   t_i,k∈ T, (λ_i,k) ∈Δ({1,...,n}), η_i,k≥ 0 and x_i,k∈∂_η_i,k f_t_i,k (x)    as in  Lemma <ref>, and by (<ref>)  condition  <ref>  must hold. Now, using the notation of Lemma <ref> <ref>, define 
	y^∗:= ∑_i=1^n_1λ_i x^∗_i, z^∗:= ∑_i=n_1+1^n x^∗_i, ε_1:= 	∑_i=1^n_1 p_i, with p_i:= λ_i ( f(x)-lim_k →∞f_t_i,k(x)   +  η_i) for  i=1,..., n_1 and 
	
    ε_2:=	  ∑_i>n_1lim_k →∞λ_i,k ( f(x) -f_t_i,k(x))  + ∑_i>n_1lim_k →∞λ_i,kη_i,k,

	so x^∗=y^∗+z^∗ and ε_1+ε_2≤ε. It follows that y^∗∈S(x,ε_1). Indeed,  define  γ_i,k= |p_i/λ_i - (f(x)-f_t_i,k(x)   +  η_i,k)|, then one gets x^∗_i,k∈∂_p_i/λ_i + γ_i,k f_t_i,k(x) and lim_k f_t_i,k(x) +p_i/λ_i = f(x) + η_i ≥ f(x), which means  x^∗_i ∈ A(x,p_i,λ_i).
	
	  Now we show that z^∗∈N^ε_2_ f(x) and  y^∗∈S(x,ε_1). Let γ any number in  (0, min{λ_i : i=1,...,n_1 }) and define  M= sup{  x^∗_i,k_* : k∈,   i=1,...,n_1 }, let k ∈ be such that y^∗ - ∑_i=1^n_1λ_i,k x^∗_i,k_*≤γ, (1- ∑_i=1^n_1λ_i,k) M < γ, 
	
    λ_i,k (f(x)- f_t_i,k(x)  +  η_i,k )< p_i +γ^2


	and ε_1 ( 1 /λ_i,k- 1/ (1- ∑_i=1^n_1λ_i,k)  + λ_i,k )< γ for all i =1,...,n_1. Then, we set λ∈Δ(T), (ε_t) ∈Δ^ε_1(T) and η∈_+^(T) by 
	
	
    λ_t =   {[ λ_i,k + (1- ∑_i=1^n_1λ_i,k),                                              if  t=t_1,k,;                       λ_i,k,                              if  t=t_i,k for i=2,...,n_1,;                           0,                                                otherwise, ].
    ε_t =   {[         p_i,              if  t=t_i,k,;           0,                otherwise, ].
    η_t =   {[                       η_i,k,                              if  t=t_i,k for i=1,...,n_1,;                           0,                                                otherwise, ].

	respectively. Then y^∗∈∑_t∈λλ_t ∂_η_t f_t(x) + 𝔹(0,2γ), moreover  (recall (<ref>) and γ < λ_i,k) for t≠ t_1,k,  f_t(x) +ε_t/λ_t +γ≥ 	f(x)    +η_t, and for t= t_1,k, f_t(x) +ε_t/λ_1,k +γ≥ 	f(x)    +η_t, besides
	  
    ε_t/λ_1,k = ε_t/λ_t   + ε_t (    1/λ_1,k - 1/λ_1,k + (1- ∑_i=1^n_1λ_i,k)) ≤ε_t/λ_t   + γ.

	 Therefore y^∗∈∑_t∈λλ_t ∂_ε_t/λ_t + 2γ f_t(x) + 𝔹(0,2γ)
	
	Finally, for every y∈ f, one has
	
    ⟨ z^∗ , y- x⟩   = lim_k →∞∑_i=n_1+1^n λ_i,k⟨  x^∗_i,k ,  y-x ⟩
       ≤lim_k →∞∑_i=n_1+1^n λ_i,k( 			f_t_i,k (y) - f_t_i,k(x)	 + η_i,k) 
       ≤lim_k →∞∑_i=n_1+1^n λ_i,k ( f(y) - f(x) )  +	lim_k →∞∑_i=n_1+1^n λ_i,k ( f(x)   - f_t_i,k (x))
       ≤ε_2   (recall  (<ref>)).

Although the assumption that the normal does not contain  lines, or some of its equivalent statements (see Remark <ref>), is standard in convex analysis, for some applications it is better to understand this condition in terms of the functions f_t, for this reason we introduce the next  qualification condition in terms of the normal cones of the data function { f_t}_t∈ T. 

    [                                                   For every  A ⊆𝒫_f(T)   one has; x^∗_t ∈N_ f_t  (x)  for  t∈ A  and ∑_t ∈ A x^∗_t = 0 ⟹  x_t^∗ =0  for all  t∈ A. ]

Under the additional assumptions of the compactness of T and some continuity property of the function t → f_t(w) we can prove that (<ref>) is equivalent to (<ref>). More precisely we get the following result.

	 Assume that T is a compact space, t → f_t(w) is upper semicontinous for every w ∈ X and (<ref>) holds at x. 
	Then, for every ε≥ 0
	
    N^ε_ f(x,f(x))   =  ⋃{∑_t∈ (ε_t)N^ε_t_ f_t(x,f(x)): 
    	[ (ε_t) ∈Δ^ε(T) ]}
    ∂_εf(x)    =⋃{𝒮(x,ε_1, T) + 𝒩(x,ε_2,T) : (ε_1,ε_2) ∈Δ^ε({1,2}) }

	where 
	
    𝒮(x,ε_1)    := {∑_t∈λλ_t ∂_ε_t/λ_t f_t(x) : [      λ∈Δ(T),  (ε_t) ∈Δ^ε_1(T); and  f_t(x) +ε_t/λ_t  ≥ 	f(x) ]},
    𝒩(x,ε_2)   :=  {∑_t∈ηN^η_t_ f_t (x) : 
    	(η_t) ∈Δ^ε_2(T) }.


Proof   First we prove (<ref>), the result is trivial if f^∗∗=-∞, or f^∗∗=+∞. Therefore, we assume that f^∗∗ is proper. Consider the set T̃:={ t : f^∗∗_t ≠-∞}
and the functions g_t := δ_ f_t and g :=sup_t∈T̃ g_t.  The relation f^∗∗=sup f^∗∗ is equivalent to g^∗∗=sup_t∈T̃ g^∗_t. Then, we pick  (x^∗,α) in N^ε_ f(x,f(x))=∂_ε g (x,f(x)), so  applying Lemma <ref> (and following its notation) there are sequences t_i,k∈ T, (λ_i,k) ∈Δ({1,...,n}), η_i,k≥ 0 and elements (x^∗_i,k,α_i,k ) in ∂_η_i,k g_t_i,k (x,f(x))=N^η_i,k_  f_t_i,k (x,f(x))   with  i∈{1,...,n }, k∈ such that ∑_i=1^n lim_k  (λ_i,k·η_i,k)  ≤ε, 
and ∑_i=1^n λ_i,k· (x_i,k^∗,α_i,k) → (x^∗,α). We recall that necessarily α≤ 0 and  α_i,k≤ 0 for all i,k.  Since T is compact we may assume (up to a subnet) that t_i,k→ t_i, let us define ε_t=∑_i:   t_i =tlim_k  (λ_i,k·η_i,k) if there exists some t_i=t, and ε_t=0 if t≠ t_i for all i.

  Now suppose  that condition <ref> of Lemma <ref>  holds, then the elements ν_k λ_i,k· (x_i,k^∗,α_i,k) converges to  (x^∗_i,β_i) and ∑_i=1^n (x^∗_i,β_i) =(0,0) for some ν_k ↘ 0 and not all  (x^∗_i,β_i) are equal to zero, then necessarily β_i =0, because β_i ≤ 0 for all i=1,...,n. Now we check that  x_i^∗∈N_ f_t_i(x), indeed let y ∈ f_t_i, because t → f_t(y) is upper semicontinuous (usc) we have that there exists r ∈ such that (y,r) ∈f_t_i,k for large enough k, consequently
  
    ⟨ x^∗_i, y - x  ⟩   = lim_k(  ⟨ν_k    λ_i,k· x_i,k^∗, y -x ⟩ +  ν_k   λ_i,kα_i,k (r - f(x)) )
       = lim_kν_k    λ_i,k(  ⟨   x_i,k^∗, y -x ⟩ +  α_i,k (r - f(x)) )
       ≤lim_kν_k λ_i,kη_i,k≤lim_kν_k ε =0,

the last means x_i^∗∈N_ f_t_i(x) and ∑_i=1^n x_i^∗=0 with not all x^∗_i equal to zero,  this contradicts (<ref>). Therefore, condition <ref> of Lemma <ref> must hold,  then λ_i,k· (x_i,k^∗,α_i,k)  → (x^∗_i,β_i) and  using that  t → f_t(w) is usc we get 
  
    (u^∗_t,α_t ):=∑_j: t_j=t (x^∗_i,β_i)    ∈N^ε_t_f_t (x,f(x)),   if there exists some   t_i=t,
    
      (u^∗_t,α_t ):=(0,0)   ∈N^ε_t_f_t (x,f(x)),   otherwise,

  and  ∑_t ∈ T (u^∗_t,α_t )= (x^∗,α). Consequently using (<ref>) (with ε=0) we get that (<ref>) implies  (<ref>). Then, we can  apply Theorem <ref> and following its notation we use the compactness of T and the upper semicontinuity of t → f_t(w) (together with similar arguments as in the first part)  to prove  that S(x,ε_1) is contained in 𝒮(x,ε_1). Moreover, using the fact that (x^∗,0 )∈N^ε_2_ f(x,f(x))  iff x^∗∈N^ε_2_ f(x)   together with  (<ref>) we derive that N^ε_2_ f (x) is equal to 𝒩(x,ε_2). This concludes the proof of (<ref>). 

	It has not escaped our notice that using (<ref>) one can prove that 
	
    ∂_ε f(x)    =	⋃{𝒮(x,ε_1, T_1) + 𝒩(x,ε_2,T_2) : [ (ε_1,ε_2) ∈Δ^ε({1,2}),;      T_1 ∩ T_2 = ∅ and;  #T_1+ # T_2 ≤dim(X)+1 ]},

where 

    𝒮(x,ε_1, T_1)    := {∑_t∈λλ_t ∂_ε_t/λ_t f_t(x) : [    λ∈Δ(T),  (ε_t) ∈Δ^ε_1(T_1); and  f_t(x) +ε_t/λ_t  ≥ 	f(x) ]},
    𝒩(x,ε_2,T_2)    :=  {∑_t∈ηN^η_t_ f_t (x) : 
    	(η_t) ∈Δ^ε_2(T_2) }.

	Indeed, to prove  (<ref>) we notice that by the finite dimension of X every element  x^∗∈_ε f(x) must be expressed as x^∗= ∑_t∈ T_1 λ_ t x^∗_t  + ∑_s ∈ T_2 w_s^∗ with some  x^∗_t ∈∂_ε_t/λ_t f_t(x), t∈ T_1, w^∗_s ∈N^ν_s_ f_s, with ∑_t∈ T_1ε_t + ∑_s∈ T_2ν_s ≤ε,  T_1,T_2 ⊆ T,  #(T_1 ∪ T_2) ≤ X+1 and f_t(x) +ε_t/λ_t  ≥ 	f(x) for all t ∈ T_1. Then λ_t ∂_ε_t/λ_t f_t(x) +  N^ν_t_ f_t(x) ⊆λ_t ∂_(ε_t+ ν_t)/λ_t  f_t(x) for t ∈ T_1∩ T_2, consequently x^∗∈𝒩(x,ε_1 ,T_1) + 	𝒩(x, ε_2,T_2\ T_1), where ε_1:= ∑_t∈ T_1ε_t  + ∑_t∈ T_1∩ T_2ν_t and ε_2:=   ∑_s∈ T_2\ T_1ν_s.
	

The final goal of this section is to give the following characterization of the epigraph of  f^∗ and consequently an expression for itself. This result can be understood as the conjugate counterpart of Theorems <ref> and <ref>.

	
	
			
  * If f^∗ is epi-pointed, one has 
		
    f^∗   = ⋃{ f_t: t ∈ T } + ( f^∗)_∞.

	 Consequently f^∗ = ℓ  +(f^∗)^∞, where ℓ is the lsc function such that its epigraph is  ⋃{ f_t: t ∈ T }.
		
  *  If T is compact and t → f_t(x) is upper semicontinous for every x∈ X and for every B ∈𝒫_f(T)  and every (x^∗_t,α_t) ∈ ( f_t^∗)_∞ with t∈ B one has 
		
    ∑_t ∈ B (x^∗_t,α_t) = (0,0) ⟹  (x_t^∗,α_t) =(0,0)  for all  t∈ B.
	
	Then
		
    f^∗   ={ f^∗_t  : t∈ T} + {  ( f_t^∗)_∞ : t∈ T }.

	
 
Proof
	Assume that f^∗ is epi-pointed. Consider (x^∗,α) ∈ f^∗, then by Lemma <ref> <ref>  we have (x^∗,α)=lim (x^∗_n,α_n) for some (x^∗_n,α_n) ∈{⋃_t∈ T f^∗_t }. Moreover,  we can write 
	
    (x^∗_n,α_n) = ∑_k=1^N λ_t_k,n (y^∗_k,n,β_k,n)
 where N:= X+2, (λ_t_k,n ) ∈Δ(T) and (y^∗_k,n,β_k,n) ∈ f^∗_t_k,n for some t_k,n∈ T. Now by a similar argumentation as the one given in Lemma <ref>, up to a subsequence  λ_t_k,n (y^∗_k,n,β_k,n) must converge to a point (w^∗_k,β_k) and λ_k,n→λ_k, otherwise there exists (u^∗_k,μ_k) ∈ ( f^∗)_∞= (f^∗)^∞ not all equal to zero such that ∑_k=1^N (u_k^∗,μ_k)=(0,0) which contradicts the fact that f^* is epi-pointed.  Now, on the one hand for every k =1,...,N with λ_k≠ 0 one has that 
    (y^∗_k,n,β_k,n) → (y^∗_k , β_k) ∈⋃{ f_t: t ∈ T } .
  On the other hand for every k =1,...,N with λ_k= 0  one has that 
    λ_k,n (y^∗_k,n,β_k,n)  → (w^∗_k, γ_k) ∈ ( f^∗)_∞.
 Therefore, 
    x^∗ = ∑_k: λ_k>0λ_k (y^∗_k , β_k) + (w^∗,β),
 where (y^∗_k , β_k)  ∈⋃{ f_t: t ∈ T } and  (w^∗,β)=∑_k: λ_k=0 (w^∗_k, γ_k)  ∈( f^∗)_∞. 
	The proof of   <ref> follows
	 similar arguments as the previous one, together with the argumentation about the compactness of T and the upper semicontinuity as was given in the proof of Theorem <ref>, so we omit the proof.

	A relevant point to consider is that all the convex combinations given in the results of this subsection can be  written as  convex combinations of no more than X +1 points (or X+2  for formulas related to epigraphs). In the same form,  condition (<ref>) and (<ref>) are  equivalent to  consider A ⊆ T with #A ≤ X and B⊆ T with  #B ≤ X +1 in  (<ref>) and (<ref>) respectively.
	


 §.§ Infinite-Dimensional Locally Convex Spaces


For the sake of simplicity  we assume that the functions f_t∈Γ_0(X) for all t∈ T. 

Consider x∈ X,  by ℱ_x we denote the family of all finite-dimensional linear subspaces of X  containing  x.


	Consider a  family of  functions { f_t}_t∈ T⊆Γ_0(X). Then, for every x∈ X and ε≥ 0 one has
	
    ∂_εf(x)   =⋂_L∈ℱ_x
    γ >0 ^w^∗⋃{𝔖(x,ε_1,γ)+N^ε_2_ f∩ L(x)  | 	(ε_1, ε_2)∈Δ^ε({1,2})	},

	where
	
    𝔖(x,ε_1,γ)   := {∑_t∈ Tλ_t  ∂_ε_t/λ_t+γ f_t(x)  : [      λ∈Δ(T),  	(ε_t) ∈Δ^ε_1(T),; and 
    	f_t(x) +ε_t/λ_t +γ≥ 	f(x) ]}
 

Proof 
	Since the right side of the equation is contained in the ε-subdifferential of f at x,  we must focus on the opposite inclusion. Without loss of generality we (may) assume that x=0. 
	
	Now take  γ >0, 	V^∗∈𝒩_0  and L∈ℱ_0 such that L^⊥⊆ V^∗. Take x^∗ in ∂_ε f(0), consider F:= spn{ L∩ f}	and   P:X → F a continuous linear projection and let us denote by P^∗ its adjoint operator. Then x^∗ = y^∗ + z^∗, where y^∗:=P^∗(x_|_F^∗) and z^∗: =x^∗ - y^∗∈ F^⊥,  then x_|_F^∗ belongs to ∂ f_|_F(0). Since the relative interior of f_|_F with respect to F is nonempty, the hypotheses of  Theorem <ref> hold (in F), then there exists ε_1,ε_2 ≥ 0 with ε_1+ε_2=ε together with elements u^∗,  λ^∗ in S(0,ε_1) and  N^ε_2_ f_|_F (0) respectively such that x_|_F^∗=u^∗ + λ^∗, which implies the existence of λ∈Δ(T), (ε_t) ∈Δ^ε_1(T) such that 
    f_t(x) +ε_t/λ_t +γ≥ 	f(x)  and  u^∗∈∑_t ∈λ∂_ε_t/λ_t+γ (f_t)_|_F(x) +(P^∗)^-1(V^∗).
 Whence P^∗(u^∗) ∈∑_t ∈λ∂_ε_t/λ_t+γ (f_t+ δ_F)(x) + V^∗ and by <cit.> we have 
	
    ∑_t ∈λ∂_ε_t/λ_t+γ (f_t+ δ_F)(x)   =^w^∗( 	∑_t ∈λ∂_ε_t/λ_t+γ f_t(0) + L^⊥)
       ⊆∂_ε_t/λ_t+γ f_t(0) +V^∗.


	Moreover,   P^∗(λ^∗) + z^∗∈ N_ f ∩ L(x). Therefore, 
	
    x^∗∈𝔖(x,ε_1,γ) + N^ε_2_ f∩ L(x) +V^∗+V^∗.

	From the arbitrariness  of  L, V^∗ and γ>0 we get the result.

	In <cit.> the author has proposed changing the family ℱ_x for some family  of convex sets C ⊆ f which covers f. It is not difficult to see that N^ε_ f  ∩ L(x) ⊆N^ε_ f ∩ C ∩ L(x) for every  set C containing x. Then, in order to get  	(<ref>) with some special class of sets C one needs a formula in the following form
	
    N^η_ f ∩ C ∩ L(x) = ^w^∗(	N^η_ f ∩ C (x) +(L-x)^⊥).

	or some ν-enlargement of the above expression 
		
    N^η_ f ∩ C ∩ L(x) = ⋂_ν >   0^w^∗(	N^η+ν_ f ∩ C (x) +(L-x)^⊥).

	Then, in order to obtain a new formula for the ε-subdifferential of  f one can consider a family of sets { C_α : α∈𝒜} satisfying  (<ref>) for large enough L  and all η≤ε which cover f, or if one prefers  the use of the  ν-enlargement, one can consider a family   of sets { C_α : α∈𝒜} satisfying  (<ref>) for large enough L and all η≤ε which cover f. In particular it happens in <cit.>, where the author has used a family of closed convex sets . 
	 
	 A more concrete idea is to understand the ε-normal set of the intersection as the ε-subdifferential of the sum of the indicators of the respective sets, that is,
	
    N^η_ f ∩ C ∩ L (x) ={[      ∂_η (δ_ f ∩ C ∩ L )(x) = ∂_η (δ_ f∩ C +δ_L)(x),   if η >0,; N__+( f)  ∩ L (x)= ∂ (δ__+ ( f∩ C -x) +x +δ_L )(0),    if η =0, ].

then one can  apply  Hiriart-Urruty and Phelps's formulae. Nevertheless, these formulae need  that the sets be closed, or at least satisfied

    ( f∩ C ∩ L)   =  ( f∩ C) ∩ L
    (  (_+ ( f∩ C -x) +x) ∩ L)   = (_+ ( f∩ C -x) +x) ∩ L

  (see, e.g., <cit.>,<cit.>, where   Hiriart-Urruty and Phelps's formulae have been imposed with weaker assumptions, in particular (<ref>)  and (<ref>) under  (<ref>)  and (<ref>)).  Then, in order to obtain formulae for the ε-subdifferential (with ε>0) of f one defines  𝒜_x^ε as some family of sets C satisfying (<ref>) and (<ref>)  for large enough L. Similarly, for the Moreau-Rockafellar Subdifferential one can consider 𝒜_x as some family of sets C satisfying  (<ref>)  for large enough L. 
  
  Typical examples of sets that satisfy (<ref>) and (<ref>) are sets which satisfy the accessibility lemma, that is there exist some x_0 ∈ f∩ C such that  
    [x_0, x[ :={ (1-λ) x_0 + λ x: λ∈ [0,1]	}⊆ f∩ C  for every  x ∈( f∩ C).
 This assumption is satisfied under the nonemptiness of some notations of relative interior (see <cit.>).  In this scenario, we can underline <cit.> where the authors assume that the relative interior of the domain is not empty, in this case it is enough to use 𝒜_x ={ f  }. Another assumption is that   _+ ( -x) is closed (see, e.g., <cit.>), again it is enough to consider 𝒜_x ={ f  }.
  
  

As a  corollary to our formula we recover  the result given in  <cit.> (see also <cit.>).

	In the setting of  Theorem <ref> one has for all x∈ X
	
    f(x)    =⋂_L∈ℱ_x
    γ >0 ^w^∗{{⋃_t ∈ T_γ(x) _γ f_t(x)}   +N_ f∩ L(x) },




	In the setting of Theorem <ref> assume that f is continuous at some point of its domain, then for ε≥ 0 and  all x ∈ X one has
	
    _εf(x)    =⋃{⋂_γ >0 𝔖(x,ε_1+γ,γ) +N^ε_2_ f(x)  | ε_1 +ε_2=ε and ε_1,ε_2≥ 0	},


Proof
Let x^∗∈_ε f(x) and γ >0, then by  Theorem <ref> there exist  nets y^∗_ν,L∈𝔖(x,ε_1(ν,L),γ) and λ^∗_ν,L∈N^ε_2(ν,L)_∩ L(x) such that x^∗= lim y^∗_ν,L  + λ^∗_ν,L; because f is continuous at some point the net (y^∗_ν,L ) must be bounded, so (up to a subnet) we may assume y^∗_ν,L→ y^∗ and λ^∗_ν,L→λ^∗ and ε_1(ν,L) →ε_1 and ε_2(ν,L) →ε_2. It follows that  λ^∗∈N^ε_2_ f(x). Now it only remains to show that y^∗∈𝔖(x,ε_1+γ,γ). Indeed, consider (ν_0,L_0) such that  for every (ν,L) ≽ (ν_0,L), |ε_1(ν,L) - ε_1| ≤γ. Whence y^∗_ν,L∈∑_t∈λλ_t _ε_t/λ_t + γ f_t (x) and  f_t(x) + ε_t/λ_t + γ≥ f(x) for some (ε_t) ∈Δ^ε(ν,L)(T), then considering the real numbers ε̃_t:= ε_t + λ_t (γ + ε_1-  ε_1(ν,L) we get  y^∗_ν,L∈𝔖(x,ε_1+γ,γ) for every (ν,L) ≽ (ν_0,L), that concludes the proof.


Due to the fact that our formulae involve the ε-normal set of f ∩ L at a point x, we present the following result, where the set is expressed  in terms of the data function f_t. The proof of the following lemma  follows  <cit.>.

	In the setting of Theorem <ref> assume that f is proper. Then for every x∈ I_f and every ε≥ 0, we have that:
	
    N^ε_ f(x)
    	   ={ x^∗∈ X^∗ :   (x^∗,⟨ x^∗,x⟩+ε) ∈ (σ_ f)    }
       ={ x^∗∈ X^∗ :   (x^∗,⟨ x^∗,x⟩+ε) ∈  ( f^∗)_∞}
       ={ x^∗∈ X^∗ :   (x^∗,⟨ x^∗,x⟩+ε) ∈[ ( ⋃_ t∈ T f^∗_t:)]_∞}
       ={ x^∗∈ X^∗ :  (x^∗,⟨ x^∗,x⟩+ε) ∈[ [ ( ⋃_ t∈ T f^∗_t  ) ]_∞;             + {0 }×[0,ε] ]}.
 
	Consequently for every L∈ℱ_x,  N^ε_ f ∩ L(x) can be expressed as
		
    { x^∗∈ X^∗ :   (x^∗,⟨ x^∗,x⟩+ε) ∈[ (  L^⊥×_+∪⋃_t∈ T f^∗_t )]_∞},  or
    { x^∗∈ X^∗ :  (x^∗,⟨ x^∗,x⟩+ε) ∈[ [ (  L^⊥×{0}∪⋃_t ∈ T f^∗_t ) ]_∞;                     + {0 }×[0,ε] ]}.
 
	 

Proof
	 The  equalities (<ref>) and  (<ref>)  follow from the definition of N^ε_ I_f(x) and from the fact that ( f^∗)_∞ =(σ_ f) (see <cit.>). The third equation is given by the fact that  f^∗= { f_t^∗ : t∈ T} (see Lemma   <ref>). Now we  have to prove (<ref>), from the fact that 
    ( ⋃_t ∈ T f_t^∗) ⊇( ⋃_t∈ T f^∗_t) + {0}× [0,ε]
  the inclusion ⊇  holds in (<ref>), then we   only have to prove the opposite inclusion. We have that 
	
    ^w^∗[  ( ⋃_t∈ T f^∗_t )  +{0 }×_+]= ( ⋃_t∈ T f_t^∗) .

	Since f^∗ is proper, we have 

    [       [ ( ⋃_t∈ T f^∗_t )    ]_∞∩ - [{0 }×_+]_∞;                                               ; ⊆[ ( ⋃_t∈ T f_t^∗) ]_∞∩ - [{0 }×_+]_∞={(0,0)}, ]

	so  Dieudonné's Theorem (see <cit.>) implies that 
    ( ⋃{ f^∗_t: t∈ T })  +{0 }×_+
 is closed, besides (  ⋃_ t∈ T f_t^∗)= (  ⋃_ t∈ T f^∗_t )  +{0 }×_+ and 
	
    [ (  ⋃_ t∈ T f_t^∗) ]_∞ = [(  ⋃_ t∈ T f^∗_t ) ]_∞ + {0 }×_+.

	Then take x^∗∈ X^∗ such that   (x^∗,⟨ x^∗,x⟩+ε) ∈[ (  ⋃_ t∈ T f_t^∗) ]_∞, by (<ref>) there exist (y^∗,γ)∈[(  ⋃_ t∈ T f^∗_t ) ]_∞ and η≥ 0 such that (x^∗,⟨ x^∗,x⟩+ε)=(y^∗,γ+η), so x^∗=y^∗. Moreover, since 
	
    f×{-1}⊆[ [  f^∗]_∞]^-   =[[ (  ⋃_ t∈ T f_t^∗) ]_∞]^-
       ⊆[[(  ⋃_ t∈ T f^∗_t ) ]_∞]^-,

 we get  ⟨ (x^∗, γ), (x,-1)⟩≤ 0, so η≤ε, and then 
    (x^∗,⟨ x^∗,x⟩+ε) ∈[{ f^∗_t : t∈ T} ]_∞ +  {0 }×[0,ε].

 
 Finally, (<ref>) and (<ref>) follow applying (<ref>) and (<ref>)  to the family of functions {f_t : t ∈ T}∪{δ_L}.
 


§ CONCLUSIONS

We have derived formulae for the ε-subdifferential and the conjugate functions of the supremum function. We studied two different methods to get these expressions. 

The first one corresponds to the case when the index set is a directed set and the family of functions  is an increasing family of epi-pointed functions. Nevertheless, these  hypotheses can be obtained, as we have shown in Theorem   <ref> and Corollary <ref>, considering the maximum function over all the finite parts of the index set and doing some appropriate perturbation of the function in order to get the epi-pointedness property, in this case the formula obtained does not involve the use of normal cones of the supremum function, which for example, allows us to give a formula for the normal cone of an arbitrary intersection only considering the normal cones of the data, which (at least not directly) cannot be derived from the formulae expressed in Section <ref>. 

The second one corresponds to obtaining general formulae for the conjugate  and the ϵ-subdifferential  of f in  spaces. Next, we have shown how to derive these results in general locally convex spaces by a reduction to  finite-dimensional subspaces. It is important to say that this reduction to  finite-dimensional subspaces allows us to use standard arguments in subdifferential theory to get   the results, which in particular extend some of the most general formulae to the Moreau-Rockafellar subdifferential of the supremum function, to the calculus of the ε-subdifferential of the supremum function. 





spmpsci_unsrt


