1 Introduction

The characterization of the subdifferential of the pointwise supremum of a family of functions has attracted the attention of many researchers. Their interest comes from the fact that a huge number of important functions in convex analysis and optimization (like the Fenchel conjugate, the sum, the composition with affine mappings, etc.) can be expressed as suprema of this type. Accordingly, many publications in the last decades dealt with supremum functions and their subdifferentials and, among the most remarkable, we quote here the following ones: Brøndsted [1], Ioffe [9], Ioffe and Levin [10], Ioffe and Tikhomirov [11], Levin [12], Pschenichnyi [16], Rockafellar [17], Valadier [19], etc. See [18] to trace out the historical evolution of the topic.

More precisely, given the pointwise supremum \(f:=\sup _{t\in T}f_{t}\) of a family of convex functions \(f_{t}:X\rightarrow {\mathbb {R}}\cup \{+\infty \}\), \(t\in T\), T being a non-empty and arbitrary set, defined on a separated locally convex space X,  many authors addressed the problem of characterizing the subdifferential of the supremum, \(\partial f(x)\), at any point \(x\in {\text {*}}{dom}f,\) the effective domain of f. These characterizations are usually given in terms of (approximate-) subdifferentials of the data functions, \(\partial _{\varepsilon }f_{t}(x),\ t\in T,\) \(\varepsilon \ge 0,\) and, in the most general cases, in terms also of the normal cone to (finite-dimensional sections of) the effective domain of f,  \(\mathrm {N}_{L\cap {\text {*}}{dom}f}(x).\) For instance, if \(f_{t}\in \Gamma _{0}(X)\), \(t\in T\), where \(\Gamma _{0}(X)\) is the family of proper convex and lower semicontinuous (lsc, in brief) functions, then the following key formula is proved in [7, Theorem 4] (see [14, Theorem 4] and [13] for related formulas):

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{L\in {\mathcal {F}}(x), \varepsilon >0}} \overline{{\text {*}}{co}}\left( {\textstyle \bigcup \nolimits _{t\in T_{\varepsilon }(x)}} \partial _{\varepsilon }f_{t}(x)+\mathrm {N}_{L\cap {\text {*}}{dom} f}(x)\right) , \end{aligned}$$
(1)

where \(\overline{{\text {*}}{co}}\) stands for the \(w^{*}\)-closed convex hull,

$$\begin{aligned} T_{\varepsilon }(x):=\{t\in T:\ f_{t}(x)\ge f(x)-\varepsilon \}, \end{aligned}$$
(2)

and

$$\begin{aligned} {\mathcal {F}}(x):=\{L\subset X:\ L\text { is a finite-dimensional linear subspace such that }x\in L\}. \end{aligned}$$

In the so-called compact setting, which stands for assuming that T is compact and the mappings \(t\mapsto f_{t}(z),\) \(z\in X,\) are upper semicontinuous (usc, in brief), the following result, involving only the active functions at the reference point x, is established in [4, Theorem 3.8]:

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{L\in {\mathcal {F}}(x),\varepsilon >0}} \overline{{\text {*}}{co}}\left( {\textstyle \bigcup \nolimits _{t\in T(x)}} \partial _{\varepsilon }f_{t}(x)+\mathrm {N}_{L\cap {\text {*}}{dom} f}(x)\right) , \end{aligned}$$
(3)

where \(T(x):=T_{0}(x)\) (see (2)).

In order to get simpler formulas, without these normal cones, one possibility is to impose additional assumptions as the continuity of f at x, in which case (1) gives rise to ([7, Corollary 10]; see, also, [20], for normed spaces):

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} \overline{{\text {*}}{co}}( {\textstyle \bigcup \nolimits _{t\in T_{\varepsilon }(x)}} \partial _{\varepsilon }f_{t}(x)). \end{aligned}$$

The operation of taking the pointwise supremum is exclusive to convex analysis and has no equivalence in differential calculus. Since the sum operation is fundamental in classical calculus, many authors have been naturally led to establish a relationship between these two operations. In other words, they aimed to transform the supremum into a sum, in order to use the classical tools dealing with differentiable functions like Fermat’s rule and many others.

In the case of finitely many functions \(f_{1},\) \(\cdots ,f_{n},\ \)with \(f=\max _{1\le k\le n}f_{k},\) it is well-known that for every \(x\in X\) and \(\varepsilon \ge 0\) ([21, Corollary 2.8.11], see also Lemma 11 in Appendix for an alternative proof based on the minimax theorem)

$$\begin{aligned} \partial _{\varepsilon }f(x)=\bigcup \nolimits _{\eta \in \left[ 0,\varepsilon \right] ,\uplambda \in S(x,\varepsilon -\eta )}\partial _{\eta }\left( \sum \nolimits _{1\le k\le n}\uplambda _{k}f_{k}\right) (x), \end{aligned}$$
(4)

with

$$\begin{aligned} S(x,\varepsilon -\eta ):=\{\uplambda \in \Delta _{n}:\sum \nolimits _{1\le k\le n}\uplambda _{k}f_{k}(x)\ge f(x)+\eta -\varepsilon \}, \end{aligned}$$

and \(\Delta _{n}\) being the canonical simplex in \({\mathbb {R}}^{n}.\)

The purpose of this paper is to establish new characterizations of \(\partial f(x),\) in which only the data functions \(f_{t}\)’s appear and without involving the extra term \(\mathrm {N}_{{\text {*}}{dom}f}(x);\) namely, we provide the following more general formulas

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} \overline{{\text {*}}{co}}\left( \left( \bigcup \nolimits _{t\in T_{\varepsilon }(x)}\partial _{\varepsilon }f_{t}(x)\right) +\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \{0,\varepsilon \}\partial _{\varepsilon }f_{J}(x)\right) \right) , \end{aligned}$$
(5)

where

$$\begin{aligned} {\mathcal {T}}_{\varepsilon }(x):=\{J\subset T:J\text { finite and\ }\max _{t\in J}f_{t}(x)\ge f(x)-\varepsilon \}, \end{aligned}$$

or equivalently, using (4),

$$\begin{aligned} \partial f(x)= {\displaystyle \bigcap _{\varepsilon >0}} \overline{{\text {*}}{co}}\left( \left( \bigcup _{t\in T_{\varepsilon } (x)}\partial _{\varepsilon }f_{t}(x)\right) +\left( {\displaystyle \bigcup \limits _{\begin{array}{c} J\in \mathcal {T}_{\varepsilon }(x)\\ \uplambda \in S_{J}(x,\varepsilon ) \end{array}}}\{0,\varepsilon \}\partial _{\varepsilon }\left( \sum _{t\in J}\uplambda _{t} f_{t}\right) (x)\right) \right) ,\nonumber \\ \end{aligned}$$
(6)

where

$$\begin{aligned} S_{J}(x,\varepsilon ):=\{\uplambda \in \Delta _{\left| J\right| } :\sum \nolimits _{t\in J}\uplambda _{t}f_{t}(x)\ge f(x)-\varepsilon \}. \end{aligned}$$

Preliminary results in this direction have been obtained in [5] for the compact setting.

Both formulas (5) and (6) highlight the role played by the almost active functions at the reference point, whereas the normal cone which is present in (1) is now replaced by weighted finite maxima and sums. Formula (6) naturally covers some other formulas from the literature, as those established in [8, Theorem 2] for the case of exact subdifferentials (see, also, [15, Theorem 1]). At the same time, we prove that the choice of the involved convex combinations in (5) and (6) can be made more precise in the so-called compact setting; in fact, we establish that for any fixed \(t_{0}\in T(x)\) we have that

$$\begin{aligned} \partial f(x)\!=\! {\textstyle \bigcap \limits _{\varepsilon >0}} \overline{{\text {*}}{co}}\left( \left( {\textstyle \bigcup \limits _{t\in T(x)}} \partial _{\varepsilon }f_{t}(x)\right) \cup \left( {\textstyle \bigcup \limits _{t\in T\setminus T(x)}} \partial _{\varepsilon }(\rho _{t,\varepsilon }f_{t}\!+\!(1-\rho _{t,\varepsilon })f_{t_{0}})(x)\right) \right) , \qquad \end{aligned}$$
(7)

with

$$\begin{aligned} \rho _{t,\varepsilon }:=\frac{\varepsilon }{2f(x)-2f_{t}(x)+\varepsilon }, t\in T\setminus T(x),\varepsilon >0. \end{aligned}$$

The paper is structured as follows. After the section devoted to present the notation and preliminary results used in the paper, Sect. 3 provides, in Theorem 2, a representation of \(\partial f(x),\) by means of specific convex combinations of the \(f_{t}\)’s which involve at most two functions. Proposition 3, first result in Sect. 4, dealing with the non-compact setting, provides the reduction of the index set T to countable subsets. In this section, Theorems 4 and 5 give non-compact counterparts of the characterizations of \(\mathrm {N}_{{\text {*}}{dom}f}(x)\) and \(\partial f(x)\) established in [5]. Some technical results and/or proofs are transferred to appendix, with the purpose of simplifying the presentation of the more relevant results in the paper.

2 Notation and Preliminary Results

Let X be a (real) separated locally convex space (lcs, for short), whose topological dual space, \(X^{*},\) is endowed with the \(w^{*}\)-topology; hence, \(X^{**}:=(X^{*})^{*}\equiv X.\ \)The spaces X and \(X^{*}\) are paired in duality by the bilinear form \((x^{*},x)\in X^{*}\times X\mapsto \langle x^{*},x\rangle :=x^{*}(x).\) The zero vectors in X and \(X^{*}\) are denoted by \(\theta .\) We adopt the conventions \(\left( +\infty \right) +(-\infty )=\left( -\infty \right) +(+\infty )=+\infty \) and \(0(+\infty )=+\infty .\)

Given \(n\ge 1,\) the n-canonical simplex in \({\mathbb {R}}^{n}\) is

$$\begin{aligned} \Delta _{n}:=\left\{ (\uplambda _{1},\cdots ,\uplambda _{n})\ge 0:\uplambda _{1} +\cdots +\uplambda _{n}=1\right\} . \end{aligned}$$

Given two sets A and B in X (or in \(X^{*}\)), and \(\Lambda \subset {\mathbb {R}}\), we define

$$\begin{aligned} A+B:=\{a+b:\ a\in A, b\in B\}\text { and }\Lambda A:=\left\{ \uplambda a:\uplambda \in \Lambda , a\in A\right\} . \end{aligned}$$
(8)

By \({\text {*}}{co}(A)\) and \({\text {*}}{cone}(A)\), we denote the convex and the conical convex hulls of the non-empty set A, respectively. In the topological side, \({\text {*}}{cl}(A)\) and \({\overline{A}}\) are indistinctly used for denoting the closure of A. When \(A\subset X^{*}\), the closure is taken with respect to the \(w^{*}\)-topology, unless something else is explicitly stated.

Associated with a non-empty set \(A\subset X,\) we define the negative dual cone and the orthogonal subspace of A as follows

$$\begin{aligned} A^{-}&:=\left\{ x^{*}\in X^{*}:\ \langle x^{*},x\rangle \le 0\text { for all }x\in A\right\} ,\\ A^{\perp }&:=(-A^{-})\cap A^{-}=\left\{ x^{*}\in X^{*}:\ \langle x^{*},x\rangle =0\text { for all }x\in A\right\} , \end{aligned}$$

respectively. Observe that \(A^{-}=(\overline{{\text {*}}{cone}}(A))^{-}.\) These concepts are defined similarly for sets in \(X^{*}.\) The so-called bipolar theorem establishes that

$$\begin{aligned} A^{--}:=(A^{-})^{-}=\overline{{\text {*}}{cone}}(A). \end{aligned}$$
(9)

If \(A\subset X,\) we define the normal cone to A at x by

$$\begin{aligned} \mathrm {N}_{A}(x):=\left\{ \begin{array} [c]{ll} (A-x)^{-}, &{} \text {if \ }x\in A,\\ \emptyset , &{} \text {if }x\in X\setminus A. \end{array} \right. \end{aligned}$$

If \(A\ne \emptyset \) is convex and closed, \(A_{\infty }\) represents its recession cone defined by

$$\begin{aligned} A_{\infty }:=\left\{ y\in X:\ x+\uplambda y\in A\text { for some }x\in A\text { and all }\uplambda \ge 0\right\} . \end{aligned}$$

Given a function \(f:X\longrightarrow {\mathbb {R}}\cup \{\pm \infty \}\), its (effective) domain is \({\text {*}}{dom}f:=\{x\in X:\ f(x)<+\infty \},\) and f is proper when \({\text {*}}{dom} f\ne \emptyset \) and \(f(x)>-\infty \) for all \(x\in X\). The closed convex hull of f,  denoted by \(\overline{{\text {*}}{co}}f,\) is the largest lsc convex function dominated by f. If f is convex, then \(\overline{{\text {*}}{co}}f={\text {*}}{cl}f,\) the closed hull of f. For \(x\in X\) and \(\varepsilon \ge 0,\) the \(\varepsilon \)-subdifferential (or the approximate subdifferential) of f at x is

$$\begin{aligned} \partial _{\varepsilon }f(x)=\{x^{*}\in X^{*}:\ f(y)\ge f(x)+\langle x^{*},y-x\rangle -\varepsilon \text { \ for all }y\in X\}, \end{aligned}$$
(10)

when \(f(x)\in {\mathbb {R}},\) and \(\partial _{\varepsilon }f(x):=\emptyset \) when \(f(x)\notin {\mathbb {R}}.\) The subdifferential of f at x is \(\partial f(x):=\partial _{0}f(x)\). The \(\varepsilon \)-directional derivative of f at \(x\in f^{-1}(\mathbb {R)}\) in the direction \(u\in X\) is defined by

$$\begin{aligned} f_{\varepsilon }^{\prime }(x;u):=\inf \limits _{s>0}\frac{f(x+su)-f(x)+\varepsilon }{s}, \end{aligned}$$

so that

$$\begin{aligned} {\text {*}}{dom}f_{\varepsilon }^{\prime }(x,\cdot )={\mathbb {R}}_{+}\left( {\text {*}}{dom}f-x\right) . \end{aligned}$$
(11)

If \(f\in \Gamma _{0}(X),\) \(x\in {\text {*}}{dom}f,\) and \(\varepsilon >0,\) then \(\partial _{\varepsilon }f(x)\ne \emptyset \) and we have

$$\begin{aligned} \mathrm {N}_{{\text {*}}{dom}f}(x)=\left( \partial _{\varepsilon }f(x)\right) _{\infty }, \end{aligned}$$
(12)

and

$$\begin{aligned} {\sigma }_{\partial _{\varepsilon }f(x)}(\cdot )=f_{\varepsilon }^{\prime }(x,\cdot ). \end{aligned}$$
(13)

Formula (12) is also valid for \(\varepsilon =0\) provided that \(\partial f(x)\ne \emptyset .\)

The Fenchel conjugate of f is the function \(f^{*}:X^{*}\longrightarrow {\mathbb {R}}\cup \{\pm \infty \}\mathbb {\ }\)given by

$$\begin{aligned} f^{*}(x^{*}):=\sup \{\left\langle x^{*},x\right\rangle -f(x):\ x\in X\}, \end{aligned}$$

and it is well-known that, for all \(x\in f^{-1}({\mathbb {R}})\) and \(\varepsilon \ge 0,\)

$$\partial _{\varepsilon }f(x)=\{x^{*}\in X^{*}:\ f(x)+f^{*}(x^{*})\le \left\langle x^{*},x\right\rangle +\varepsilon \},$$

and \(\partial f(x)=\cap _{\varepsilon >0}\partial _{\varepsilon }f(x).\)

The support and the indicator functions of \(A\subset X\) are, respectively,

$$\begin{aligned} \sigma _{A}(x^{*}):=\sup \{\langle x^{*},x\rangle :\ x\in A\},x^{*}\in X^{*}, \end{aligned}$$

with \(\sigma _{\emptyset }\equiv -\infty \), and

$$\begin{aligned} \mathrm {I}_{A}(x):=\left\{ \begin{array} [c]{ll} 0 &{} \text {if }x\in A,\\ +\infty &{} \text {if }x\in X\setminus A. \end{array} \right. \end{aligned}$$

It is known that, if A is a closed convex set,

$$\begin{aligned} A_{\infty }=\left( {\text {*}}{dom}\sigma _{A}\right) ^{-}, \end{aligned}$$
(14)

or equivalently, by using (9),

$$\begin{aligned} \left( A_{\infty }\right) ^{-}={\text {*}}{cl}({\text {*}}{dom} \sigma _{A}). \end{aligned}$$
(15)

Next, given a finite family \(\{f_{k},\) \(1\le k\le n\}\subset \Gamma _{0}(X),\) we consider the maximum function \(f=\max _{1\le k\le n}f_{k}.\) We suppose that f is proper and denote

$$\begin{aligned} \varphi (\uplambda ,x):=\sum \nolimits _{1\le k\le n}\uplambda _{k}f_{k} (x)-\mathrm {I}_{{\mathbb {R}}_{+}^{n}}(\uplambda ), \uplambda \in {\mathbb {R}} ^{n},x\in X. \end{aligned}$$
(16)

The adopted convention \(0(+\infty )=+\infty \ \)entails \(0f_{k}=\mathrm {I} _{{\text {*}}{dom}f_{k}}.\) Then \(\varphi (\Delta _{n},{\text {*}}{dom} f)\subset {\mathbb {R}}\), \(\varphi (\cdot ,x)\) is concave and usc for every \(x\in {\text {*}}{dom}f\), and \(\varphi (\uplambda ,\cdot )\) is convex and lsc for every \(\uplambda \in \Delta _{n}\). Thus, since \(\Delta _{n}\) is compact in \({\mathbb {R}}^{n}\) and \({\text {*}}{dom}f\) is non-empty and convex, the minimax theorem ensures that (see, e.g., [21, Theorem 2.10.2])

$$\begin{aligned} \max _{\uplambda \in \Delta _{n}}\inf _{x\in {\text {*}}{dom}f}\varphi (\uplambda ,x)=\inf _{x\in {\text {*}}{dom}f}\max _{\uplambda \in \Delta _{n}} \varphi (\uplambda ,x). \end{aligned}$$
(17)

Moreover, since

$$\begin{aligned} f(x)=\max _{\uplambda \in \Delta _{n}}\varphi (\uplambda ,x),\text { for all } x\in {\text {*}}{dom}f\text {,} \end{aligned}$$

relation (17) leads us to

$$\begin{aligned} \max _{\uplambda \in \Delta _{n}}\inf _{x\in X}\varphi (\uplambda ,x)=\inf _{x\in X}f(x). \end{aligned}$$
(18)

As a consequence of this, for every \(x\in {\text {*}}{dom}f\) and \(\varepsilon \ge 0\) we obtain that (see Lemma 11 in Appendix)

$$\begin{aligned} \partial _{\varepsilon }f(x)&=\bigcup \nolimits _{\uplambda \in \Delta _{n} }\partial _{\varepsilon +\varphi (\uplambda ,x)-f(x)}\varphi (\uplambda ,\cdot )(x) \end{aligned}$$
(19)
$$\begin{aligned}&=\bigcup \nolimits _{\eta \in \left[ 0,\varepsilon \right] ,\uplambda \in S(x,\varepsilon -\eta )}\partial _{\eta }\varphi (\uplambda ,\cdot )(x), \end{aligned}$$
(20)

where \(S(x,\varepsilon -\eta ):=\{\uplambda \in \Delta _{n},\eta \in [0,\varepsilon ],\varphi (\uplambda ,x)\ge f(x)+\eta -\varepsilon \}.\) Notice that formula (19) constitutes a slight improvement of [21, Corollary 2.8.11] as it involves only one precise value of the parameter \(\eta .\)

The arguments used in Lemma 11 to prove (19) and (20) are specific to finite families of functions, and so they cannot be extended to families with infinitely many functions, where the following simplices in \({\mathbb {R}}^{T},\)

$$\begin{aligned} \left\{ \uplambda \in {\mathbb {R}}_{+}^{T}:\uplambda (t)\equiv \uplambda _{t}=0\text { except for finitely many }t\text {'s and }\sum \nolimits _{\uplambda _{t}>0} \uplambda _{t}=1\right\} , \qquad \end{aligned}$$
(21)

may be not compact.

3 The Compact Setting

We give in this section some additional results to those established in [5]. We consider a non-empty family \(\{f_{t},\ t\in T\}\subset \Gamma _{0}(X)\) such that

$$\begin{aligned} \begin{array} [c]{l} T\text { is Hausdorff compact, }\\ \text {and, for each }z\in X,\text { the mapping }t\longmapsto f_{t}(z)\text { is upper semicontinuous.} \end{array} \end{aligned}$$
(22)

The associated supremum function is

$$\begin{aligned} f:=\sup _{t\in T}f_{t}, \end{aligned}$$

and assumptions (22) ensure that (see [5, Lemma 5])

$$\begin{aligned} {\text {*}}{dom}f=\cap _{t\in T}{\text {*}}{dom}f_{t} \end{aligned}$$
(23)

and, for every \(x\in {\text {*}}{dom}f,\)

$$\begin{aligned} {\mathbb {R}}_{+}({\text {*}}{dom}f-x)=\cap _{t\in T}{\mathbb {R}}_{+} ({\text {*}}{dom}f_{t}-x). \end{aligned}$$
(24)

Moreover, (22) also yields

$$\begin{aligned} T(x):=\{t\in T:f_{t}(x)=f(x)\}\ne \emptyset . \end{aligned}$$

Assuming \(\inf _{t\in T}f_{t}(x)>-\infty ,\) we have proved in [5, Theorem 12] that

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} \overline{{\text {*}}{co}}\left( \left( {\textstyle \bigcup \nolimits _{t\in T(x)}} \partial _{\varepsilon }f_{t}(x)\right) +\left( {\textstyle \bigcup \nolimits _{t\in T\setminus T(x)}} \left\{ 0,\varepsilon \right\} \partial _{\varepsilon }f_{t}(x)\right) \right) . \end{aligned}$$
(25)

This formula involves the active functions \(f_{t},\) \(t\in T(x),\) as the same time as the non-active ones \(f_{t},\) \(t\in T\setminus T(x),\) but with these last ones being affected by the weighting parameter \(\varepsilon >0.\) The main ingredient we used to establish (25) is the following relation ([5, Theorem 6])

$$\begin{aligned} \mathrm {N}_{{\text {*}}{dom}f}(x)=\left[ \overline{{\text {*}}{co} }\left( {\textstyle \bigcup \limits _{t\in T}} \partial _{\varepsilon }f_{t}(x)\right) \right] _{\infty }\text {, for every }\varepsilon >0. \end{aligned}$$

We give next an equivalent description of the elements in \(\mathrm {N} _{{\text {*}}{dom}f}(x),\) which highlights the role played by the active and non-active functions.

Lemma 1

Assume that (22) holds. Consider \(x\in {\text {*}}{dom}f\) and fix \(t_{0}\in T(x).\) Then we have that

$$\begin{aligned} \mathrm {N}_{{\text {*}}{dom}f}(x)\subset \left[ \overline{{\text {*}}{co}}\left( \left( {\textstyle \bigcup \limits _{t\in T(x)}} \partial _{\varepsilon }f_{t}(x)\right) \cup \left( {\textstyle \bigcup \limits _{t\in T\setminus T(x)}} \partial _{\varepsilon }(\mu _{t}f_{t}+(1-\mu _{t})f_{t_{0}})(x)\right) \right) \right] _{\infty },\nonumber \\ \end{aligned}$$
(26)

for every \(\varepsilon >0\) and \(0<\mu _{t}<1.\)

Proof

We fix \(\varepsilon >0\) and \(0<\mu _{t}<1,\) for all \(t\in T\setminus T(x),\) and denote

$$\begin{aligned} E_{\varepsilon }:= {\textstyle \bigcup \nolimits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t}(x), \end{aligned}$$

where

$$\begin{aligned} {\tilde{f}}_{t}:=\left\{ \begin{array} [c]{ll} f_{t}, &{} \text {if }t\in T(x),\\ \mu _{t}f_{t}+(1-\mu _{t})f_{t_{0}}, &{} \text {if }t\in T\setminus T(x). \end{array} \right. \end{aligned}$$
(27)

The sets T(x) and \(E_{\varepsilon }\) are non-empty thanks to (22) and the lower semicontinuity of the \(f_{t}\)’s. Since

$$\begin{aligned} {\text {*}}{cl}({\text {*}}{dom}{\sigma }_{E_{\varepsilon } })=(\left[ \overline{{\text {*}}{co}}\left( E_{\varepsilon }\right) \right] _{\infty })^{-}, \end{aligned}$$

by (15), and \((\mathrm {N}_{{\text {*}}{dom}f}(x))^{-} =({\text {*}}{dom}f-x)^{--}={\text {*}}{cl}({\mathbb {R}}_{+} ({\text {*}}{dom}f-x)),\) by (9), desired relation (26) is equivalent to

$$\begin{aligned} {\text {*}}{cl}({\text {*}}{dom}{\sigma }_{E_{\varepsilon } })\subset {\text {*}}{cl}({\mathbb {R}}_{+}({\text {*}}{dom}f-x)). \end{aligned}$$

To prove this inclusion we take, using (13),

$$\begin{aligned} z\in {\text {*}}{dom}{\sigma }_{E_{\varepsilon }}&={\text {*}}{dom}\left( {\sigma }_{\cup _{t\in T}\partial _{\varepsilon }{\tilde{f}}_{t}(x)}\right) ={\text {*}}{dom}\left( \sup _{t\in T}{\sigma }_{\partial _{\varepsilon }{\tilde{f}}_{t}(x)}\right) \nonumber \\&={\text {*}}{dom}\left( \sup _{t\in T}({\tilde{f}}_{t})_{\varepsilon }^{\prime }(x;\cdot )\right) \subset \cap _{t\in T}{\text {*}}{dom}(\tilde{f}_{t})_{\varepsilon }^{\prime }(x;\cdot ). \end{aligned}$$
(28)

Hence, since for every \(t\in T\setminus T(x)\), by (11),

$$\begin{aligned} {\text {*}}{dom}({\tilde{f}}_{t})_{\varepsilon }^{\prime }(x;\cdot )&={\mathbb {R}}_{+}({\text {*}}{dom}{\tilde{f}}_{t}-x)\\&={\mathbb {R}}_{+}(({\text {*}}{dom}(\mu _{t}f_{t})\cap {\text {*}}{dom} ((1-\mu _{t})f_{t_{0}}))-x)\\&={\mathbb {R}}_{+}(\left( {\text {*}}{dom}f_{t}\cap {\text {*}}{dom} f_{t_{0}}\right) -x)\\&=({\mathbb {R}}_{+}{\text {*}}{dom}(f_{t}-x))\cap ({\mathbb {R}}_{+} {\text {*}}{dom}(f_{t_{0}}-x)), \end{aligned}$$

relation (28) entails

$$\begin{aligned} z&\in \left( \cap _{_{t\in T(x)}}{\text {*}}{dom}(f_{t})_{\varepsilon }^{\prime }(x;\cdot )\right) \cap \left( \cap _{_{t\in T\setminus T(x)} }{\text {*}}{dom}({\tilde{f}}_{t})_{\varepsilon }^{\prime }(x;\cdot )\right) \\&=\left( \cap _{_{t\in T(x)}}{\mathbb {R}}_{+}({\text {*}}{dom} f_{t}-x)\right) \cap \left( \cap _{_{t\in T\setminus T(x)}}({\mathbb {R}} _{+}{\text {*}}{dom}(f_{t}-x))\cap ({\mathbb {R}}_{+}{\text {*}}{dom} (f_{t_{0}}-x))\right) \\&=\cap _{_{t\in T}}{\mathbb {R}}_{+}{\text {*}}{dom}(f_{t}-x), \end{aligned}$$

and, so (24) gives rise to

$$\begin{aligned} z\in \cap _{t\in T}{\mathbb {R}}_{+}{\text {*}}{dom}(f_{t}-x)={\mathbb {R}} _{+}({\text {*}}{dom}f-x)\subset {\text {*}}{cl}({\mathbb {R}} _{+}({\text {*}}{dom}f-x)). \end{aligned}$$

Hence, \({\text {*}}{dom}{\sigma }_{E_{\varepsilon }}\subset {\text {*}}{cl}({\mathbb {R}}_{+}({\text {*}}{dom}f-x))\) and the desired inclusion follows. \(\square \)

The main purpose of this section is to obtain another representation of \(\partial f(x),\) which involves appropriate convex combinations of the non-active \(f_{t}\)’s. In the non-compact setting, instead of considering two-elements convex combinations as in the compact framework, we shall appeal to all finite-elements convex combinations of the \(f_{t} \)’s (see Theorem 5 below).

Theorem 2

Assume that hypothesis (22) fulfills. Consider \(x\in {\text {*}}{dom}f\) and choose \(t_{0}\in T(x).\) Then we have that

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \limits _{\varepsilon >0}} \overline{{\text {*}}{co}}\left( \left( {\textstyle \bigcup \limits _{t\in T(x)}} \partial _{\varepsilon }f_{t}(x)\right) \cup \left( {\textstyle \bigcup \limits _{t\in T\setminus T(x)}} \partial _{\varepsilon }(\rho _{t,\varepsilon }f_{t}+(1-\rho _{t,\varepsilon })f_{t_{0}})(x)\right) \right) , \nonumber \\ \end{aligned}$$
(29)

where

$$\begin{aligned} \rho _{t,\varepsilon }:=\frac{\varepsilon }{2f(x)-2f_{t}(x)+\varepsilon }, t\in T\setminus T(x), \varepsilon >0. \end{aligned}$$

Proof

Let us suppose, for simplicity, that \(f(x)=0,\ \)and observe that for each given \(\varepsilon >0\) we have, for every \(t\in T\setminus T(x),\)

$$\begin{aligned} 0<\rho _{t,\varepsilon }<1\text { and }\rho _{t,\varepsilon }f_{t}(x)>-\frac{\varepsilon }{2}. \end{aligned}$$
(30)

Let us also denote

$$\begin{aligned} {\tilde{f}}_{t,\varepsilon }:=\left\{ \begin{array} [c]{ll} f_{t}, &{} \text {if }t\in T(x),\\ \rho _{t,\varepsilon }f_{t}+(1-\rho _{t,\varepsilon })f_{t_{0}}, &{} \text {if }t\in T\setminus T(x). \end{array} \right. \end{aligned}$$

Then, for all \(t\in T\setminus T(x),\)

$$\begin{aligned} {\tilde{f}}_{t,\varepsilon }(x)=\rho _{t,\varepsilon }f_{t}(x)>-\frac{\varepsilon }{2}, \end{aligned}$$

and so, observing that \({\tilde{f}}_{t,\varepsilon }\le \max \{f_{t},f_{t_{0} }\}\le f,\)

$$\begin{aligned} {\textstyle \bigcup \limits _{t\in T\setminus T(x)}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)\subset \partial _{\frac{3\varepsilon }{2}}f(x)\subset \partial _{2\varepsilon }f(x). \end{aligned}$$

Thus, since we also have

$$\begin{aligned} {\textstyle \bigcup \limits _{t\in T(x)}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)\subset \partial _{\varepsilon }f(x)\subset \partial _{2\varepsilon }f(x), \end{aligned}$$

we conclude that

$$\begin{aligned} {\textstyle \bigcup \limits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)\subset \partial _{2\varepsilon }f(x), \end{aligned}$$

and the inclusion “\(\supset \)” follows by taking the closed convex hull and intersecting over \(\varepsilon >0.\)

To establish the inclusion “\(\subset \)”, we fix \(\varepsilon >0\) and L \(\in {\mathcal {F}}(x).\) Next, by applying Lemma 1 to the family \(\{{\tilde{f}}_{t,\varepsilon },\) \(t\in T;\) \(\mathrm {I}_{L}\}\ \)we obtain that

$$\begin{aligned} \mathrm {N}_{L\cap {\text {*}}{dom}f}(x)\subset \left[ \overline{{\text {*}}{co}}\left( {\textstyle \bigcup \limits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)\cup L^{\perp }\right) \right] _{\infty }=\left[ \overline{{\text {*}}{co}}\left( {\textstyle \bigcup \limits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)+L^{\perp }\right) \right] _{\infty }, \end{aligned}$$

where the last equality comes from (47). Therefore, by (3),

$$\begin{aligned} \partial f(x)&\subset \overline{{\text {*}}{co}}\left( {\textstyle \bigcup \nolimits _{t\in T(x)}} \partial _{\varepsilon }f_{t}(x)+\mathrm {N}_{L\cap {\text {*}}{dom} f}(x)\right) \\&=\overline{{\text {*}}{co}}\left( {\textstyle \bigcup \limits _{t\in T(x)}} \partial _{\varepsilon }f_{t}(x)+\left[ \overline{{\text {*}}{co}}\left( {\textstyle \bigcup \nolimits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)+L^{\perp }\right) \right] _{\infty }\right) \\&\subset \overline{{\text {*}}{co}}\left( {\textstyle \bigcup \nolimits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)+L^{\perp }\right) ={\text {*}}{cl}\left( {\text {*}}{co}\left( {\textstyle \bigcup \nolimits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)\right) +L^{\perp }\right) . \end{aligned}$$

Intersecting over the L’s in \({\mathcal {F}}(x)\) we get

$$\begin{aligned} \partial f(x)\subset {\displaystyle \bigcap \nolimits _{L\in {\mathcal {F}}(x)}} {\text {*}}{cl}\left( {\text {*}}{co}\left( {\textstyle \bigcup \nolimits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)\right) +L^{\perp }\right) =\overline{{\text {*}}{co}}\left( {\textstyle \bigcup \nolimits _{t\in T}} \partial _{\varepsilon }{\tilde{f}}_{t,\varepsilon }(x)\right) , \end{aligned}$$

where the last equality is due to the fact that, for every \(A\subset X^{*}\) (see ([3, Lemma 3])),

$$\begin{aligned} {\displaystyle \bigcap \nolimits _{L\in {\mathcal {F}}(x)}} {\text {*}}{cl}\left( A+L^{\perp }\right) ={\text {*}}{cl}\left( A\right) . \end{aligned}$$
(31)

\(\square \)

In the particular case when all the \(f_{t}\)’s are active at x,  that is, \(T(x)=T,\) formula (29) reduces to

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \limits _{\varepsilon >0}} \overline{{\text {*}}{co}}\left( {\textstyle \bigcup \limits _{t\in T}} \partial _{\varepsilon }f_{t}(x)\right) , \end{aligned}$$

which extends the well-known Brøndsted formula [1] to infinite index sets. Another illustration of Theorem 2 is the alternative proof of formula (51) in Appendix.

4 Non-Compact Framework

This section is devoted to give new characterizations of \(\mathrm {N} _{{\text {*}}{dom}f}(x)\) and \(\partial f(x),\) without any additional assumptions on the family \(\{f_{t},t\in T\}\subset \Gamma _{0}(X).\)

The first result, whose proof is postponed to Appendix, provides the reduction of the index set T to countable subsets within the normal cone of \({\text {*}}{dom}f\).

Proposition 3

Consider a family \(\{f_{t},t\in T\}\subset \Gamma _{0}(X)\) and \(f=\sup _{t\in T}f_{t}.\) Given \(x\in {\text {*}}{dom}f\) and \(u^{*} \in \mathrm {N}_{{\text {*}}{dom}f}(x),\) for each \(L\in {\mathcal {F}}(x)\) there is a sequence \((t_{n})_{n}\subset T\) such that

$$\begin{aligned} u^{*}\in \mathrm {N}_{{\text {*}}{dom}(\sup _{n\ge 1}f_{t_{n}})\cap L}(x). \end{aligned}$$

The following result provides the non-compact counterpart of the characterizations of \(\mathrm {N}_{{\text {*}}{dom}f}(x)\) established in [5].

Theorem 4

Consider the family \(\{f_{t},t\in T\}\subset \Gamma _{0}(X)\) and \(f:=\sup _{t\in T}f_{t}.\) Given \(x\in {\text {*}}{dom}f,\) for every \(\varepsilon >0\ \)we have that

$$\begin{aligned} \mathrm {N}_{{\text {*}}{dom}f}(x)\subset \left( \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}}} \partial _{\varepsilon }f_{J}(x)\right) \right) _{\infty }, \end{aligned}$$
(32)

where \({\mathcal {T}}:=\{J\subset T,\) \(\left| J\right| <+\infty \}\) and

$$\begin{aligned} f_{J}:=\max \{f_{t},t\in J\}, J\in {\mathcal {T}}. \end{aligned}$$
(33)

In addition, (32) becomes an equality when

$$\begin{aligned} \inf _{J\in {\mathcal {T}}}f_{J}(x)>-\infty . \end{aligned}$$
(34)

Remark 1

(Before the proof ) Condition (34) is not very restrictive, indeed, it suffices to choose \(t_{0}\in T\) and consider the family \(\{\max \{f_{t},f_{t_{0}}\},\) \(t\in T\}.\) This new family obviously satisfies condition (34),

$$\begin{aligned} \inf _{J\in \widetilde{{\mathcal {T}}}}f_{J}(x)\ge f_{t_{0}}(x), \end{aligned}$$

where \(\widetilde{{\mathcal {T}}}:=\{J\in {\mathcal {T}}:t_{0}\in J\},\) and, consequently, Theorem 4yields

$$\begin{aligned} \mathrm {N}_{{\text {*}}{dom}f}(x)=\left( \overline{{\text {*}}{co} }\left( {\displaystyle \bigcup \nolimits _{J\in \widetilde{{\mathcal {T}}}}} \partial _{\varepsilon }f_{J}(x)\right) \right) _{\infty }. \end{aligned}$$

Proof

Take \(u^{*}\in \mathrm {N}_{{\text {*}}{dom}f}(x)\) and \(\varepsilon >0.\) Then, by Proposition 3, for every fixed \(L\in {\mathcal {F}} (x)\ \)there exists a sequence \((t_{n})_{n}\subset T\) such that

$$\begin{aligned} u^{*}\in \mathrm {N}_{{\text {*}}{dom}(\sup _{n\ge 1}f_{t_{n}})\cap L}(x). \end{aligned}$$

We denote \(J_{n}:=\{t_{1},\cdots ,t_{n}\},\) \(n\ge 1,\) and introduce the functions

$$\begin{aligned} {\hat{f}}_{n}:=f_{J_{n}}+\mathrm {I}_{L},n\ge 1, \end{aligned}$$

where \(f_{J_{n}}=\max \{f_{t},\) \(t\in J_{n}\}\) (see (33)). So, \(({\hat{f}}_{n})_{n}\) is non-decreasing and

$$\begin{aligned} \sup \nolimits _{n\ge 1}\left( f_{t_{n}}+\mathrm {I}_{L}\right) =\sup \nolimits _{n\ge 1}{\hat{f}}_{n}\text { and }{\text {*}}{dom}\left( \sup \nolimits _{n\ge 1}f_{t_{n}}\right) \cap L={\text {*}}{dom}\left( \sup \nolimits _{n\ge 1}{\hat{f}}_{n}\right) . \end{aligned}$$

In addition, according to Lemma 12 and (46), we have that

$$\begin{aligned} \partial _{\frac{\varepsilon }{2}}(\sup \nolimits _{n\ge 1}{\hat{f}}_{n})(x)&= {\displaystyle \bigcap _{\delta >0}} {\text {*}}{cl}\left( {\displaystyle \bigcup \nolimits _{k\ge 1}} {\displaystyle \bigcap \nolimits _{n\ge k}} \partial _{\frac{\varepsilon }{2}+\delta }{\hat{f}}_{n}(x)\right) \\&\subset {\text {*}}{cl}\left( {\displaystyle \bigcup \nolimits _{k\ge 1}} \partial _{\varepsilon }(f_{J_{k}}+\mathrm {I}_{L})(x)\right) \\&\subset {\text {*}}{cl}\left( {\displaystyle \bigcup \nolimits _{k\ge 1}} {\text {*}}{cl}(\partial _{\varepsilon }f_{J_{k}}(x)+L^{\perp })\right) \\&\subset \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{k\ge 1}} \partial _{\varepsilon }f_{J_{k}}(x)+L^{\perp }\right) . \end{aligned}$$

Therefore, using (12),

$$\begin{aligned} u^{*}&\in \mathrm {N}_{{\text {*}}{dom}(\sup _{n\ge 1}f_{t_{n}})\cap L}(x)=(\partial _{\frac{\varepsilon }{2}}(\sup \nolimits _{n\ge 1}{\hat{f}} _{n})(x))_{\infty }\\&\subset \left( \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{k\ge 1}} \partial _{\varepsilon }f_{J_{k}}(x)+L^{\perp }\right) \right) _{\infty }, \end{aligned}$$

that is, for all \(L\in {\mathcal {F}}(x),\)

$$\begin{aligned} u^{*}\in \left( \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{k\ge 1}} \partial _{\varepsilon }f_{J_{k}}(x)+L^{\perp }\right) \right) _{\infty } \subset \left( \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}}} \partial _{\varepsilon }f_{J}(x)+L^{\perp }\right) \right) _{\infty }, \end{aligned}$$

and so

$$\begin{aligned} u^{*}&\in {\displaystyle \bigcap \nolimits _{L\in {\mathcal {F}}(x)}} \left( \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}}} \partial _{\varepsilon }f_{J}(x)+L^{\perp }\right) \right) _{\infty }\\&=\left( {\displaystyle \bigcap \nolimits _{L\in {\mathcal {F}}(x)}} \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}}} \partial _{\varepsilon }f_{J}(x)+L^{\perp }\right) \right) _{\infty }\\&=\left( \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}}} \partial _{\varepsilon }f_{J}(x)\right) \right) _{\infty }, \end{aligned}$$

where the last equality is a consequence of (31).

For the converse inclusion, observe that (34) implies the existence of a constant M such that

$$\begin{aligned} \inf _{J\in {\mathcal {T}}}f_{J}(x)\ge M(>-\infty ). \end{aligned}$$

Then, for every \(J\in {\mathcal {T}}\) and \(x^{*}\in \partial _{\varepsilon } f_{J}(x),\)

$$\begin{aligned} \left\langle x^{*},y-x\right\rangle&\le f_{J}(y)-f_{J}(x)+\varepsilon \\&\le f(y)-f(x)+(f(x)-M+\varepsilon ),\text { for all }y\in X; \end{aligned}$$

in other words, \(\partial _{\varepsilon }f_{J}(x)\subset \partial _{\varepsilon +f(x)-M}f(x)\) and so

$$\begin{aligned} \left( \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}}} \partial _{\varepsilon }f_{J}(x)\right) \right) _{\infty }\subset \left( \partial _{\varepsilon +f(x)-M}f(x)\right) _{\infty }=\mathrm {N} _{{\text {*}}{dom}f}(x), \end{aligned}$$

where the last equality comes from (12). \(\square \)

Next, we give the main result in this section, which constitutes a non-compact counterpart of Theorem 2.

Theorem 5

Consider the family \(\{f_{t},\) \(t\in T\}\subset \Gamma _{0}(X)\) and \(f:=\sup _{t\in T}f_{t}.\) Then for every \(x\in {\text {*}}{dom}f\) we have that

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} \overline{{\text {*}}{co}}\left( \left( \bigcup \nolimits _{t\in T_{\varepsilon }(x)}\partial _{\varepsilon }f_{t}(x)\right) +\{0,\varepsilon \}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \right) , \end{aligned}$$
(35)

where

$$\begin{aligned} {\mathcal {T}}_{\varepsilon }(x):=\{J\in {\mathcal {T}}:f_{J}(x)\ge f(x)-\varepsilon \}. \end{aligned}$$
(36)

Proof

Fix \(x\in {\text {*}}{dom}f\) and \(\varepsilon >0\) so that, by formula (1), and whichever \(L\in {\mathcal {F}}(x)\) we take, one has

$$\begin{aligned} \partial f(x)\subset \overline{{\text {*}}{co}}\left( {\textstyle \bigcup _{t\in T_{\varepsilon }(x)}} \partial _{\varepsilon }f_{t}(x)+\mathrm {N}_{L\cap {\text {*}}{dom} f}(x)\right) . \end{aligned}$$
(37)

Now we pick \(t_{0}\in T_{\varepsilon }(x),\) and denote \(\widehat{{\mathcal {T}} }:=\{J\in {\mathcal {T}}:t_{0}\in J\}\),

$$\begin{aligned} f_{J}:=\max \{f_{t},t\in J\}+\mathrm {I}_{L},J\in \widehat{{\mathcal {T}}}, \end{aligned}$$

so that

$$\begin{aligned} f_{J}(x)\ge f_{t_{0}}(x)\ge f(x)-\varepsilon ,\text {for every } J\in \widehat{{\mathcal {T}}}, \end{aligned}$$
(38)

and

$$\begin{aligned} \widehat{{\mathcal {T}}}\subset {\mathcal {T}}_{\varepsilon }(x). \end{aligned}$$
(39)

Then, by Remark 1, and taking into account (47 ), (48), and (39),

$$\begin{aligned} \mathrm {N}_{L\cap {\text {*}}{dom}f}(x)&=\left( \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in \widehat{{\mathcal {T}}}}} \partial _{\varepsilon }f_{J}(x)\right) \right) _{\infty }\\&\subset \left( \overline{{\text {*}}{co}}\left( \left( {\textstyle \bigcup _{t\in T_{\varepsilon }(x)}} \partial _{\varepsilon }f_{t}(x)\right) \bigcup \left( {\displaystyle \bigcup \nolimits _{J\in \widehat{{\mathcal {T}}}}} \partial _{\varepsilon }f_{J}(x)\right) \right) \right) _{\infty } \subset \left( \overline{{\text {*}}{co}}\left( E_{\varepsilon }\right) \right) _{\infty }, \end{aligned}$$

where we have denoted

$$\begin{aligned} E_{\varepsilon }:=\left( {\textstyle \bigcup _{t\in T_{\varepsilon }(x)}} \partial _{\varepsilon }f_{t}(x)\right) +\{0,\varepsilon \}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) . \end{aligned}$$
(40)

So, (37) gives rise to

$$\begin{aligned} \partial f(x)\subset \overline{{\text {*}}{co}}\left( {\textstyle \bigcup _{t\in T_{\varepsilon }(x)}} \partial _{\varepsilon }f_{t}(x)+\left( \overline{{\text {*}}{co}}\left( E_{\varepsilon }\right) \right) _{\infty }\right) \subset \overline{{\text {*}}{co}}\left( E_{\varepsilon }\right) , \end{aligned}$$

that is, the desired inclusion “\(\subset \)” follows once we intersect over \(\varepsilon >0.\)

To verify the opposite inclusion, by (36) we easily observe that

$$\begin{aligned} \left( {\textstyle \bigcup _{t\in T_{\varepsilon }(x)}} \partial _{\varepsilon }f_{t}(x)\right) \bigcup \left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \subset \partial _{2\varepsilon }f(x), \end{aligned}$$

and so,

$$\begin{aligned} {\textstyle \bigcap \nolimits _{\varepsilon>0}} \overline{{\text {*}}{co}}\left( E_{\varepsilon }\right)&\subset {\textstyle \bigcap \nolimits _{\varepsilon>0}} \overline{{\text {*}}{co}}\left( \partial _{2\varepsilon } f(x)+\{0,\varepsilon \}\partial _{2\varepsilon }f(x)\right) \\&\subset {\textstyle \bigcap \nolimits _{\varepsilon >0}} \left[ 1,1+\varepsilon \right] \partial _{2\varepsilon }f(x)=\partial f(x). \end{aligned}$$

\(\square \)

For \(x\in {\text {*}}{dom}f\), \(\delta \ge 0\) and \(J\in {\mathcal {T}},\) we denote

$$\begin{aligned} S_{J}(x,\delta ):=\left\{ \uplambda \in \Delta _{^{\left| J\right| }} :\sum \nolimits _{t\in J}\uplambda _{t}f_{t}(x)\ge f(x)-\delta \right\} . \end{aligned}$$

Observe that

$$\begin{aligned} S_{J}(x,0)=\left\{ \uplambda \in \Delta _{^{\left| J\right| }} :\sum \nolimits _{t\in J}\uplambda _{t}f_{t}(x)=f(x)\right\} . \end{aligned}$$

Theorem 5 leads us to the characterization below, involving the finite suprema \(f_{J}\) or sums \(\sum \nolimits _{t\in J} \uplambda _{t}f_{t}\).

Corollary 6

Consider the family \(\{f_{t},\) \(t\in T\}\subset \Gamma _{0}(X)\) and \(f:=\sup _{t\in T}f_{t}.\) Then for every \(x\in {\text {*}}{dom}f\) we have that

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \end{aligned}$$
(41)

and, consequently,

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}}, \uplambda \in S_{J} (x,\varepsilon )}} \partial _{\varepsilon }\left( \sum \nolimits _{t\in J}\uplambda _{t}f_{t}\right) (x)\right) . \end{aligned}$$
(42)

Proof

Fix \(x\in {\text {*}}{dom}f\) and \(\varepsilon >0.\) Since \(\{\{t\}:t\in T_{\varepsilon }(x)\}\subset {\mathcal {T}}_{\varepsilon }(x),\) we have that (recall the definition of \(E_{\varepsilon }\) in (40))

$$\begin{aligned} E_{\varepsilon }&=\left( {\textstyle \bigcup \nolimits _{t\in T_{\varepsilon }(x)}} \partial _{\varepsilon }f_{t}(x)\right) +\{0,\varepsilon \}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \\&\subset \left[ 1,1+\varepsilon \right] \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) , \end{aligned}$$

and so, by Theorem 5 and Lemma 13 (for the second inclusion),

$$\begin{aligned} \partial f(x)&\subset {\textstyle \bigcap \nolimits _{\varepsilon>0}} \left[ 1,1+\varepsilon \right] \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \\&= {\textstyle \bigcap \nolimits _{\varepsilon>0}} \overline{{\text {*}}{co}}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \\&\subset {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{2\varepsilon }f_{J}(x)\right) \\&\subset {\textstyle \bigcap \nolimits _{\varepsilon >0}} {\text {*}}{cl}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{2\varepsilon }(x)}} \partial _{2\varepsilon }f_{J}(x)\right) . \end{aligned}$$

Hence, the inclusion “\(\subset \)” in (41) follows.

To verify the opposite inclusion, take \(x^{*}\in \partial _{\varepsilon } f_{J}(x),\) \(J\in {\mathcal {T}}_{\varepsilon }(x),\) and \(\varepsilon >0.\) Then, for every \(y\in X,\)

$$\begin{aligned} \left\langle x^{*},y-x\right\rangle \le f_{J}(y)-f_{J}(x)+\varepsilon \le f(y)-(f(x)-\varepsilon )+\varepsilon =f(y)-f(x)-2\varepsilon , \end{aligned}$$

and so \(\partial _{\varepsilon }f_{J}(x)\subset \partial _{2\varepsilon }f(x).\) Thus,

$$\begin{aligned} {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\displaystyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x) }} \partial _{\varepsilon }f_{J}(x)\right) \subset {\textstyle \bigcap \nolimits _{\varepsilon >0}} \partial _{2\varepsilon }f(x)=\partial f(x), \end{aligned}$$

and we are done with the first statement.

Finally, using (20), formula (41) implies that

$$\begin{aligned} \partial f(x)&= {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x),\eta \in \left[ 0,\varepsilon \right] ,\uplambda \in S_{J}(x,\varepsilon -\eta )}} \partial _{\eta }\left( \sum \nolimits _{t\in J}\uplambda _{t}f_{t}\right) (x)\right) \\&\subset {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}}_{\varepsilon }(x), \uplambda \in S_{J}(x,\varepsilon )}} \partial _{\varepsilon }\left( \sum \nolimits _{t\in J}\uplambda _{t}f_{t}\right) (x)\right) \\&= {\textstyle \bigcap \nolimits _{\varepsilon >0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}},\uplambda \in S_{J} (x,\varepsilon )}} \partial _{\varepsilon }\left( \sum \nolimits _{t\in J}\uplambda _{t}f_{t}\right) (x)\right) , \end{aligned}$$

where we use the inclusions \(S_{J}(x,\varepsilon -\eta )\subset S_{J}(x,\varepsilon )\) and \(\partial _{\eta }g(x)\subset \partial _{\varepsilon }g(x),\) for any convex function g. Moreover, the converse inclusion follows by observing that

$$\begin{aligned} \partial _{\varepsilon }\left( \sum \nolimits _{t\in J}\uplambda _{t}f_{t}\right) (x)\subset \partial _{2\varepsilon }f(x), \end{aligned}$$

for all \(J\in {\mathcal {T}}\) and \(\uplambda \in S_{J}(x,\varepsilon ).\) \(\square \)

Remark 2

Let us emphasize at this point that the main feature of our approach is to provide characterizations of \(\partial f(x)\), which are independent of the effective domains of the involved functions and the associated normal cones. For comparative purposes, we quote here the following formula, given in [15, Theorem 1],

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}},\uplambda \in S_{J} (x,\varepsilon )}} \partial _{\varepsilon }\left( \sum \nolimits _{t\in J}\uplambda _{t}f_{t} +\mathrm {I}_{D}\right) (x)\right) , \end{aligned}$$
(43)

with D being any subset of X satisfying

$$\begin{aligned} {\text {*}}{dom}f\subset D\subset {\textstyle \bigcap \nolimits _{t\in T}} {\text {*}}{dom}f_{t}. \end{aligned}$$

Observe that formula (43) requires the use of the augmented functions \(f_{t}+\mathrm {I}_{D}\) and not the exact ones \(f_{t}\)’s as in (42). The following example illustrates the difference between (42) and (43).

Example 1

Consider the support function of a non-empty set \(T\subset X^{*},\)

$$\begin{aligned} \sigma _{T}(x):=\sup _{t\in T}\left\langle t,x\right\rangle . \end{aligned}$$

Here, \(f=\sigma _{T}=\sup _{t\in T}f_{t}\) with \(f_{t}(x):=\left\langle t,x\right\rangle ,\) \(t\in T,\) in \(\Gamma _{0}(X).\) On the one hand, for every \(x\in X,\) formula (42) yields

$$\begin{aligned} \partial f(x)&= {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}}, \uplambda \in S_{J} (x,\varepsilon )}} \partial _{\varepsilon }\left( \sum \nolimits _{t\in J}\uplambda _{t}f_{t}\right) (x)\right) \\&= {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}}, \uplambda \in S_{J} (x,\varepsilon )}} \left( \sum \nolimits _{t\in J}\uplambda _{t}t\right) \right) \\&= {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( \left\{ x^{*}\in {\text {*}}{co}T:\left\langle x^{*},x\right\rangle \ge f(x)-\varepsilon \right\} \right) \\&= {\textstyle \bigcap \nolimits _{\varepsilon >0}} \left\{ x^{*}\in \overline{{\text {*}}{co}}T:\left\langle x^{*},x\right\rangle \ge f(x)-\varepsilon \right\} , \end{aligned}$$

which is well-known (see, for instance, [6, (5) in page 834]); actually, it is a consequence of the Fenchel equality as

$$\begin{aligned} \sigma _{T}(x)+\mathrm {I}_{\overline{{\text {*}}{co}}T}(x^{*} )\le \left\langle x^{*},x\right\rangle +\varepsilon \Longleftrightarrow x^{*}\in \partial _{\varepsilon }\sigma _{T}(x)\text { for all } \varepsilon \ge 0. \end{aligned}$$

On the other hand, if we apply formula (43) choosing \(D={\text {*}}{dom}\sigma _{T},\) then we obtain that

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}}, \uplambda \in S_{J} (x,\varepsilon )}} \left( \sum \nolimits _{t\in J}\uplambda _{t}t\right) +\mathrm {N}_{D} ^{\varepsilon }(x)\right) . \end{aligned}$$

Hence, using Lemma 14, we derive the following alternative representation of \(\partial f(x),\)

$$\begin{aligned} \partial f(x)= {\textstyle \bigcap \nolimits _{\varepsilon >0}} {\text {*}}{cl}\left\{ \begin{array} [c]{c} x^{*}+y^{*}:x^{*}\in {\text {*}}{co}T,\left\langle x^{*},x\right\rangle \ge f(x)-\varepsilon ,\\ \quad \quad \quad \quad \quad y^{*}\in (\overline{{\text {*}}{co}}T)_{\infty },-\varepsilon \le \left\langle y^{*},x\right\rangle \le 0 \end{array} \right\} , \end{aligned}$$

which appeals to the extra term \(\{y^{*}\in (\overline{{\text {*}}{co} }T)_{\infty }:-\varepsilon \le \left\langle y^{*},x\right\rangle \le 0\}.\)

We apply Corollary 6 to provide a new proof for the characterization of the normal cone to sublevel sets given in [8, Corollary 7] (see, also, [2] and references therein).

Corollary 7

Consider a function \(g\in \Gamma _{0}(X)\) and let \(x\in X\) such that \(g(x)=0.\) Then we have that

$$\begin{aligned} \mathrm {N}_{\left[ g\le 0\right] }(x)= {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\textstyle \bigcup _{\mu >0}} \partial _{\varepsilon }\left( \mu g\right) (x)\right) . \end{aligned}$$
(44)

Proof

We define the functions

$$\begin{aligned} f_{t}:=tg,t>0,\text { and }f:=\sup _{t>0}f_{t}. \end{aligned}$$

Obviously, \(\{f_{t},\) \(t\in T\}\subset \Gamma _{0}(X)\) and \(f_{t}(x)=f(x)=0\) for all \(t>0.\) Therefore, since that \(f=\mathrm {I}_{\left[ g\le 0\right] },\) by formula (42) we obtain that

$$\begin{aligned} \mathrm {N}_{\left[ g\le 0\right] }(x)&=\partial f(x)\\&= {\textstyle \bigcap \nolimits _{\varepsilon >0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\subset \left] 0,+\infty \right[ , \left| J\right| <\infty , \uplambda \in S_{J}(x,\varepsilon )}} \partial _{\varepsilon }\left( \sum \nolimits _{t\in J}\uplambda _{t}tg\right) (x)\right) , \end{aligned}$$

where

$$\begin{aligned} S_{J}(x,\varepsilon )=\left\{ \uplambda \in \Delta _{^{\left| J\right| } }:\sum \nolimits _{t\in J}\uplambda _{t}tg(x)\ge -\varepsilon \right\} =\Delta _{^{\left| J\right| }}. \end{aligned}$$

Hence,

$$\begin{aligned} \mathrm {N}_{\left[ g\le 0\right] }(x)&= {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\textstyle \bigcup \nolimits _{J\subset \left] 0,+\infty \right[ , \left| J\right|<\infty ,\uplambda \in \Delta _{\left| J\right| }}} \partial _{\varepsilon }\left( \sum \nolimits _{t\in J}\uplambda _{t}tg\right) (x)\right) \\&= {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( \left\{ {\textstyle \bigcup } \partial _{\varepsilon }\left( \mu g\right) (x):\mu =\sum \nolimits _{t\in J}\uplambda _{t}t,J\subset \left] 0,+\infty \right[ ,\left| J\right| <\infty ,\uplambda \in \Delta _{\left| J\right| }\right\} \right) \\&= {\textstyle \bigcap \nolimits _{\varepsilon>0}} {\text {*}}{cl}\left( {\textstyle \bigcup _{\mu >0}} \partial _{\varepsilon }\left( \mu g\right) (x)\right) , \end{aligned}$$

and we are done. \(\square \)

The following corollary gives more insight to the conclusion of Corollary 6 in reflexive Banach spaces.

Corollary 8

If X is a reflexive Banach space, then (41) and (42) also hold when the closure is taken with respect to the strong (norm) topology.

Proof

It suffices to prove formula (41). Given \(x\in {\text {*}}{dom}f,\) by Corollary 6 we have that

$$\begin{aligned} \partial f(x)&= {\textstyle \bigcap _{\varepsilon>0}} {\text {*}}{cl}\left( {\textstyle \bigcup _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \\&\subset {\textstyle \bigcap _{\varepsilon>0}} \overline{{\text {*}}{co}}\left( {\textstyle \bigcup _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \\&= {\textstyle \bigcap _{\varepsilon >0}} \text {cl}^{\left\| \cdot \right\| _{*}}\left( {\text {*}}{co}\left( {\textstyle \bigcup _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \right) , \end{aligned}$$

due to Mazur’s theorem. Next, taking into account (54), we obtain that

$$\begin{aligned} \partial f(x)&\subset {\textstyle \bigcap _{\varepsilon>0}} \text {cl}^{\left\| \cdot \right\| _{*}}\left( {\textstyle \bigcup _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{2\varepsilon }f_{J}(x)\right) \\&\subset {\textstyle \bigcap _{\varepsilon >0}} \text {cl}^{\left\| \cdot \right\| _{*}}\left( {\textstyle \bigcup _{J\in {\mathcal {T}}_{2\varepsilon }(x)}} \partial _{2\varepsilon }f_{J}(x)\right) . \end{aligned}$$

Hence, using again Corollary 6, and taking into account that \({\text {*}}{cl}^{\left\| \cdot \right\| _{*}}(A)\subset {\text {*}}{cl}(A)\), for any \(A\subset X^{*},\)

$$\begin{aligned} \partial f(x)&\subset {\textstyle \bigcap _{\varepsilon>0}} \text {cl}^{\left\| \cdot \right\| _{*}}\left( {\textstyle \bigcup _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) \\&\subset {\textstyle \bigcap _{\varepsilon >0}} {\text {*}}{cl}\left( {\textstyle \bigcup _{J\in {\mathcal {T}}_{\varepsilon }(x)}} \partial _{\varepsilon }f_{J}(x)\right) =\partial f(x). \end{aligned}$$

\(\square \)

The following result shows that the subdifferential of the supremum can be reduced to the supremum of a countable family.

Proposition 9

Assume that X is a reflexive Banach space. Given an arbitrary family \(\{f_{t},t\in T\}\subset \Gamma _{0}(X)\) and \(f=\sup _{t\in T}f_{t},\) for every \(x\in {\text {*}}{dom}f\) we have that

$$\begin{aligned} \partial f(x)= {\textstyle \bigcup \nolimits _{J\in {\mathcal {T}}_{c}(x)}} \partial f_{J}(x), \end{aligned}$$

where

$$\begin{aligned} {\mathcal {T}}_{c}(x):=\{J\subset T: J\text { countable, }f_{J}(x)=f(x)\}. \end{aligned}$$

Proof

Take \(x^{*}\in \partial f(x).\) Then, by Corollary 8, for each \(n\ge 1\) there exists \(J_{n}\in {\mathcal {T}}_{\frac{1}{n}}(x)\) such that

$$\begin{aligned} x^{*}\in \partial _{\frac{1}{n}}f_{J_{n}}(x)+\frac{1}{n}B_{X^{*}}, \end{aligned}$$
(45)

where \(B_{X^{*}}\) is the closed unit ball in \(X^{*}.\) Moreover, denoting \(J:=\cup _{n\ge 1}J_{n},\) for every \(z^{*}\in \partial _{\frac{1}{n}}f_{J_{n}}(x)\) we have that

$$\begin{aligned} \left\langle z^{*},y-x\right\rangle&\le f_{J_{n}}(y)-f_{J_{n} }(x)+\frac{1}{n}\\&\le f_{J}(y)-f(x)+\frac{1}{n}+\frac{1}{n}\\&\le f_{J}(y)-f_{J}(x)+\frac{2}{n}, \end{aligned}$$

showing that \(z^{*}\in \partial _{\frac{2}{n}}f_{J}(x),\) that is, \(\partial _{\frac{1}{n}}f_{J_{n}}(x)\subset \partial _{\frac{2}{n}}f_{J} (x).\) Hence, (45) gives rise to

$$\begin{aligned} x^{*}\in \partial _{\frac{2}{n}}f_{J}(x)+\frac{1}{n}B_{X^{*}}, \end{aligned}$$

that is, \(x^{*}=u_{n}^{*}+v_{n}^{*},\) for \(u_{n}^{*}\in \partial _{\frac{2}{n}}f_{J}(x)\) and \(v_{n}^{*}\in \frac{1}{n}B_{X^{*}},\) \(n\ge 1.\) Hence, \(v_{n}^{*}\rightarrow \theta \) and we obtain, for every \(y\in X,\)

$$\begin{aligned} \left\langle x^{*},y-x\right\rangle&=\left\langle u_{n}^{*} +v_{n}^{*},y-x\right\rangle \\&=\lim _{n\rightarrow \infty }\left\langle u_{n}^{*}+v_{n}^{*},y-x\right\rangle \\&=\lim _{n\rightarrow \infty }\left\langle u_{n}^{*},y-x\right\rangle \\&\le \limsup _{n\rightarrow \infty }(f_{J}(y)-f_{J}(x)+\frac{2}{n})\\&=f_{J}(y)-f_{J}(x), \end{aligned}$$

which shows that \(x^{*}\in \partial f_{J}(x).\) Moreover, since that

$$\begin{aligned} f_{J}(x)\ge f_{J_{n}}(x)\ge f(x)-\frac{1}{n},\text { for all }n\ge 1, \end{aligned}$$

we deduce that \(f_{J}(x)=f(x),\) that is, \(J\in {\mathcal {T}}_{c}(x).\) We are done since the opposite inclusion holds straightforwardly. \(\square \)

5 Concluding Remarks

This paper is intended to establish new characterizations of the subdifferential of the pointwise supremum of an arbitrary family of convex functions which are free of the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it). These characterizations involve both (almost) active and non-(almost) active functions, the last ones being affected by a weighting parameter. Main formulas (5) and (6) highlight the role played by the almost active functions at the reference point. Formula (6) covers some other formulas in the literature; e.g., [8, Theorem 2] in the case of exact subdifferentials (see, also, [15, Theorem 1]).

The first part of the paper deals with the so-called compact scenario in which we assumed that the index set is compact and that the functions are upper semicontinuous with respect to the index. In this part, we first provide an explicit representation of the subdifferential of the supremum in Theorem 2, in terms of the active functions in one side, plus specific two-elements convex combinations in the other side.

In the second part of the paper, these compactness/upper semicontinuity assumptions are removed, and main Theorem 5 constitutes a non-compact counterpart of Theorem 2.We also aimed in the paper to emphasize the relationship of the subdifferential of the supremum function with the subdifferential of finite weighted sums. This is the purpose of (42) in Corollary 6.

Some consequences of the main results in the setting of reflexive Banach spaces are also analyzed. In particular, it turns out that formulas (41) and (42) are valid when the closure is taken with respect to the strong (norm) topology. The last proposition in the paper shows that, in this setting, the subdifferential of the supremum of the whole family can be reduced to the supremum of a countable subfamilies.