1 Introduction

Given a family of convex functions \(f_{t}:X\rightarrow {\mathbb {R}}_{\infty },\) \(t\in T\ne \emptyset ,\) defined on a locally convex space X, a well-known and classical result in convex analysis (see, e.g., [10]) provides an integral representation of the (convex) subdifferential \(\partial f(x)\) of the associated supremum function \(f:=\sup _{t\in T}f_{t}\). More specifically, the subdifferential of f at every continuity point x of f is expressed as

$$\begin{aligned} \partial f(x)= {\textstyle \bigcup \limits _{\alpha }} \partial \left( \int _{T}f_{t}(\cdot )d\alpha \right) (x), \end{aligned}$$
(1)

where the union is taken over all Radon probability measures \(\alpha \) defined on the index set T and having their support within the active set at the reference point x,  \(T(x):=\{t\in T:f_{t}(x)=f(x)\}.\) This integral representation is only valid within the compact-continuous framework, where T is Hausdorff compact and the mappings \(t\longmapsto f_{t}(z),\) \(z\in X,\) are upper-semicontinuous, by assuming the continuity of f (this last assumption is not needed when T is finite [2]). However, in general, outside the compact-continuous framework the set T(x) may be empty and Radon measures could not be defined appropriately. Moreover, without the continuity of f,  the integral function \(\int _{T}f_{t}d\alpha \) alone would not provide the complete description of \(\partial f(x)\), so that supplementary terms would be required.

The advantage of this integral-type representation is evident, as it enables us to transfer the problem of describing the variations and/or the subdifferential of the supremum function f to the conventional realm of differential calculus, which deals with integral functions incorporating parameters (for instance, series and infinite sums of functions).

It is worth observing here that the previous approach could not be directly extended beyond the compact-continuous framework. However, for a general index set T,  it is possible to endow T with the discrete topology, which makes it locally compact (but not compact unless it is finite). In a such a case the mappings \(f_{(\cdot )}(z),\) \(z\in X,\) are automatically continuous on T. Then, by considering the one-point (Hausdorff) compact extension of T,  denoted \(T_{\infty }:=T\cup \{\infty \},\) and appropriately choosing an extension \(\{f_{t},\) \(t\in T;\) \(f_{\infty }\}\) of the original family, the aforementioned result can be applied. Nevertheless, the resulting representation would do not provide the expected objective of explicitly describing the set \(\partial f(x)\) in term of the \(f_{t}\)’s, because the function \(f_{\infty }\) (see (25)) itself is a supremum-like function.

In this work, we aim to extend the aforementioned result to encompass a broader framework of arbitrary families of convex functions. Unlike the previous result, we do not assume any specific structure on the index set T and the index mappings \(f_{(\cdot )}(z),\) nor do we require continuity assumptions on the supremum function f. Additionally, our focus will be on representations of the \(\varepsilon \)-subdifferential of f,  \(\partial _{\varepsilon }f,\) rather than the exact subdifferential. Our characterizations are given by means of appropriate discrete sums performed on the data functions \(f_{t}\)’s together with specific singular measures operating on them. Taking a step further, provided that the underlying space X is a reflexive Banach space or a separable normed space, we substitute these additional measures with related limits that involve the data functions. All the objects involved in our characterizations rely intrinsically on the data functions that are (almost) active at the reference point. The derived formulas of \(\partial _{\varepsilon }f\) are intended to contribute in future research towards the elaboration of suitable duals for infinite convex optimization problems.

Our analysis offers a different perspective on the recent characterizations of \(\partial f(x)\) that have been discussed in various works such as [6,7,8,9, 11,12,13,14, 16], and references therein. These characterizations involve the \(\varepsilon \)-subdifferential of the \(f_{t}\)’s, which are almost active at the reference point, as well as the normal cone to finite-dimensional sections of the effective domain of f,  \({\text {dom}}f.\) Alternatively, first characterizations of \(\partial f(x)\) involve continuity assumptions either on the \(f_{t}\)’s or the function f [17, 18], while [1] requires that all the \(f_{t}\)’s are active at the reference point. For further details on this subject, we refer to the recent book [3].

In Theorem 5 we prove that, for every \(x\in {\text {dom}}f\) and\(\ \varepsilon \ge 0,\)

$$\begin{aligned} \partial _{\varepsilon }f(x)= {\textstyle \bigcup \limits _{\alpha }} \partial _{\varepsilon +f_{\lambda }(x)+\beta (F(x))-f(x)}\left( f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f}\right) (x), \end{aligned}$$
(2)

where the union is taken over all \(\alpha :=\lambda +\beta \) in the generalized canonical simplex \({\hat{\Delta }}(T)\) (see (4)); that is, \(\lambda \) (\(\in \ell _{1}^{+}(T)\) with support denoted by \({\text {*}}{supp} \lambda \)) and \(\beta \) (\(\in \ell _{+}^{s}(T)\)) are the respective regular and singular parts of \(\alpha .\) The proof of this result is based on the classical minimax theorem. The formula above involves the discrete sum \(f_{\lambda }:=\sum _{t}\lambda _{t}f_{t}\) and the function \(F(\cdot ):=(f_{t}(\cdot ))_{t\in T},\) associated with these \(\lambda \) and \(\beta ,\ \)along with the indicator function of \({\text {dom}}f,\) used to address the possible lack of continuity of the function f. The last formula above constitutes a generalization of the classical characterizations (1) given in the compact continuous framework. Indeed, for example, if T is a compact metrizable space and the index mappings \(f_{(\cdot )}(x):\) \(t\mapsto f_{t}(x),\) \(x\in X,\) are all finite and continuous, then the supremum function f is finite and continuous everywhere, and the indicator function \(\textrm{I} _{{\text {dom}}f}\) is removed from (2) (for \(\varepsilon =0\)). Moreover, since \(\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)\) defines an element of the dual space of \(C(T):=\{\varphi :T\rightarrow {\mathbb {R}}\) continuous\(\}\), by Riesz’s theorem there exists a unique Radon probability measure denoted in the same way, \(\alpha \), such that

$$\begin{aligned} f_{\lambda }(x)+(\beta \circ F)(x)=\alpha (f_{(\cdot )}(x))=\int _{T} f_{t}(x)d\alpha ,\text { for all }x\in X. \end{aligned}$$

Moreover, since the set \(\partial f(x)\) is not empty (due to the continuity of the convex function f), we must have \(f_{\lambda }(x)+\beta (F(x))-f(x)\ge 0\) in (2). Thus,

$$\begin{aligned} \int _{T}f_{t}(x)d\alpha =f_{\lambda }(x)+(\beta \circ F)(x)=f(x), \end{aligned}$$

and the support of \(\alpha \) is included in T(x). In other words, formula (2) reduces to that of (1) when \(\varepsilon =0,\) T is a compact metrizable space and the index mappings \(f_{(\cdot )}(x)\), \(x\in X\), are finite and continuous. We refer to Sect. 2 and 7 for further details on these objects.

Theorems 8 and 11 introduce alternative representations of \(\partial _{\varepsilon }f(x)\ \)that do not rely on singular measures. In these results, we employ countable sums of the \(f_{t}\)’s, as previously done in Theorem 5, along with new functions obtained through limiting processes performed on these countable functions. This approach is applicable when the underlying space X is a reflexive Banach space or a separable normed space. Finally, to illustrate the applicability of our characterizations, we introduce in Theorem 12 a new dual that establishes a strong duality relationship in infinite convex optimization.

The paper begins in Sect. 2 by presenting the essential notation and preliminary results. Further details on the material related to spaces of sequences are addressed in Sect. 7. Section 3 presents some preliminary integral-type characterizations of \(\partial _{\varepsilon }f(x)\) that can be derived from the recent results of [4] in certain particular scenarios. Section 4 gives the general characterization of \(\partial _{\varepsilon }f(x),\) by means of both regular and singular measures in the general canonical simplex \({\hat{\Delta }}(T)\). Subsequently, in the context of reflexive Banach spaces or separable normed spaces, Section 5 furnishes alternative descriptions of \(\partial _{\varepsilon }f(x)\) that do not involve these singular measures. Finally, Section 6 gives an application to duality in convex optimization.

2 Preliminaries

Given a real locally convex space (lcs, for short) X,  its topological dual \(X^{*}\) will be endowed with a compatible topology for the pairing \((x^{*},x)\in X^{*}\times X\mapsto \langle x^{*},x\rangle :=\left\langle x,x^{*}\right\rangle :=x^{*}(x)\) (for example, the \(w^{*}\)-topology \(\sigma (X^{*},X)\)). All zero vectors are written \(\theta \), and the family of closed convex balanced neighborhoods of \(\theta , \) called \( \theta \)-neighborhoods, is denoted \(\mathcal {N}_{X}.\) For instance, if X is given a norm \(\left\| \cdot \right\| \) and \(B_{X}\) is the associated closed unit ball, then \((X^{*},\left\| \cdot \right\| _{*})\) is the associated dual Banach space. We use the symbols \(\overline{{\mathbb {R}}}:={\mathbb {R}}\cup \{-\infty ,+\infty \}\) and \({\mathbb {R}}_{\infty }:={\mathbb {R}}\cup \{+\infty \},\) and we adopt the convention\(\left( +\infty \right) +(-\infty )=\left( -\infty \right) +(+\infty )=+\infty \) and \(0(+\infty )=+\infty .\)

Given a nonempty set T,  \({\mathbb {R}}^{T}\) and \({\mathbb {R}}_{+}^{T}\) are the real linear space of functions from T to \({\mathbb {R}}\) and its nonnegative orthant, respectively. We denote \({\mathbb {R}}^{(T)}:=\{\lambda \in {\mathbb {R}}_{+}^{T}:\) \({\text {*}}{supp}\lambda \) is finite} and \({\mathbb {R}}^{[T]}:=\{\lambda \in {\mathbb {R}}^{T}:\) \({\text {*}}{supp}\lambda \) is countable}, where \({\text {*}}{supp}\lambda :=\{t\in T:\ \lambda _{t}:=\lambda (t)\ne 0\}\) is the support of \(\lambda ;\) sometimes functions \(\lambda \in {\mathbb {R}}^{T}\) are written \((\lambda _{t})_{t\in T}.\) The sets \({\mathbb {R}}_{+}^{(T)}\) and \({\mathbb {R}}_{+}^{[T]}\) are the nonnegative orthants of \({\mathbb {R}}^{(T)}\) and \({\mathbb {R}}^{[T]}\), respectively. The function\(\ \chi _{S}\) is the characteristic of \(S\subset T;\) hence, the zero and one functions in \({\mathbb {R}}^{T}\) are defined as \(0_{T}:=\chi _{\emptyset }\) and \(1_{T}:=\chi _{T},\) respectively. We define \(\ell _{1} (T)\ \)and \(\ell _{\infty }(T)\) as the usual respective Banach spaces of functions \(\lambda \in {\mathbb {R}}^{T}\) such that

$$\begin{aligned} \left\| \lambda \right\| _{1}:=\sup _{S\subset T,\left| S\right|<+\infty }~ {\textstyle \sum \limits _{t\in S}} \left| \lambda (t)\right|<+\infty , \left\| \lambda \right\| _{\infty }:=\sup _{t\in T}\left| \lambda (t)\right| <+\infty , \end{aligned}$$

where \(\left| S\right| \) denotes the cardinal of S. We write \(\ell _{1}^{+}(T):={\mathbb {R}}_{+}^{T}\cap \ell _{1}(T),\) \(\ell _{\infty } ^{+}(T):={\mathbb {R}}_{+}^{T}\cap \ell _{\infty }(T).\) Observe that \(\ell _{\infty }(T)\) is the topological dual of \(\ell _{1}(T)\) with respect to the following pairing,

$$\begin{aligned} \left\langle \lambda ,u\right\rangle :=\lambda (u):= {\textstyle \sum \limits _{t\in T}} \lambda _{t}u_{t},\lambda \in \ell _{\infty }(T)\text { and }u\in \ell _{1}(T). \end{aligned}$$

The canonical simplices in \({\mathbb {R}}_{+}^{(T)}\) and \({\mathbb {R}}_{+}^{[T]}\) are

$$\begin{aligned} \Delta (T):=\left\{ \lambda \in {\mathbb {R}}_{+}^{(T)}: {\textstyle \sum \limits _{t\in T}} \lambda _{t}=1\right\} , \Delta [T]:=\left\{ \lambda \in {\mathbb {R}}_{+}^{[T]}: {\textstyle \sum \limits _{t\in T}} \lambda _{t}=1\right\} , \end{aligned}$$
(3)

respectively, where \(\sum _{t\in T}\lambda _{t}:=\left\langle \lambda ,1_{T}\right\rangle ;\) in particular, \(\Delta _{m}:=\Delta (T)\) if \(\left| T\right| =m\ge 1\). We will also use the linear space

$$\begin{aligned} \ell ^{s}(T):=\left\{ \beta \in (\ell _{\infty }(T))^{*}:\beta (u)=0,\text { for all }u\in \ell _{1}(T)\right\} , \end{aligned}$$

so that \((\ell _{\infty }(T))^{*}\) is the direct sum of \(\ell _{1}(T)\) and \(\ell ^{s}(T)\) (see Appendix), together with the cone

$$\begin{aligned} \ell _{+}^{s}(T):=\{\beta \in \ell ^{s}(T):\beta (\lambda )\ge 0\text { for all }\lambda \in \ell _{\infty }^{+}(T)\}. \end{aligned}$$

For simplicity of notation, for an element \(\beta \in \ell _{+}^{s}(T)\) and a set \(S\subset T\) we sometimes write \(\beta (S):=\beta (\chi _{S});\) therefore, abusing language, the elements of \(\ell _{+}^{s}(T)\) are called (nonnegative) singular measures on T,  while the elements (indeed, functions) in \(\ell _{1}(T)\) are sometimes called regular measures. The generalized canonical simplex in \((\ell _{\infty }(T))^{*}\) is the \(w^{*}\)-compact convex set defined as

$$\begin{aligned} {\hat{\Delta }}(T):\!=\!\left\{ \alpha \in (\ell _{\infty }(T))^{*}:\alpha \!=\!\lambda \!+\!\beta ,\lambda \in \ell _{1}^{\!+\!}(T), \beta \in \ell _{\!+\!} ^{s}(T), {\textstyle \sum \limits _{t\in T}} \lambda _{t}\!+\!\beta (\chi _{_{T}})\!=\!1\right\} . \end{aligned}$$
(4)

Further properties of the above objects are presented in Appendix. Next, we present some facts about convex functions (for more details, see, e.g., [3] and [19]). The algebraic sum of two nonempty sets A and B in X (or in \(X^{*}\)) is

$$\begin{aligned} A+B:=\{a+b:\ a\in A, b\in B\},\quad A+\emptyset =\emptyset +A=\emptyset . \end{aligned}$$
(5)

The sets \({\text {*}}{co}A\) and \(\overline{{\text {*}}{co}}A\) refer to the convex and the closed convex hulls of A, respectively, while \({\text {*}}{cl}A\) (or \(\overline{A})\) denotes the closure of A (closure with respect to a compatible topology, if \(A\subset X^{*}\)). Given a function \(f:X\longrightarrow \overline{{\mathbb {R}}},\) the sets \({\text {dom}}f:=\{x\in X:\ f(x)<+\infty \}\) and \({\text {*}}{epi} f:=\{(x,r)\in X\times {\mathbb {R}}:\ f(x)\le r\}\) are, respectively, its (effective) domain and epigraph. We write \(f\in \Gamma _{0}(X)\) if f is proper (\({\text {dom}}f\ne \emptyset \) and \(f>-\infty \)), convex (\({\text {*}}{epi}f\) is convex) and lower semicontinuous; lsc, for short (\({\text {*}}{epi}f\) is closed). We denote by \({\bar{f}}\) (or \({\text {*}}{cl}f\)) and \(\overline{{\text {*}}{co}}f\) the closed and the closed convex hulls of f,  respectively.

The Fenchel conjugate of a function \(f:X\rightarrow \overline{{\mathbb {R}}}\) is the function \(f^{*}:X^{*}\rightarrow \overline{{\mathbb {R}}}\) defined by \(f^{*}(x^{*}):=\sup _{x\in X}\{\left\langle x^{*},x\right\rangle -f(x)\}.\) Given \(\varepsilon \in {\mathbb {R}},\) the \(\varepsilon \)- subdifferential of f at \(x\in X\) is the set

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

with \(\partial _{\varepsilon }f(x):=\emptyset \) if \(x\notin f^{-1}({\mathbb {R}})\) or \(\varepsilon <0\), and \(\partial f(x):=\partial _{0}f(x).\) Equivalently, \(\partial f(x)=\cap _{\varepsilon >0}\partial _{\varepsilon }f(x)\) and \(x^{*}\in \partial _{\varepsilon }f(x)\) if and only if \(f(x)+f^{*}(x^{*} )\le \left\langle x^{*},x\right\rangle +\varepsilon .\) For every \(x\in X\) and \(\varepsilon \in {\mathbb {R}},\) we have

$$\begin{aligned} \partial _{\varepsilon }f(x)\subset \partial _{\varepsilon +{\bar{f}}(x)-f(x)}\bar{f}(x)\subset \partial _{\varepsilon }{\bar{f}}(x). \end{aligned}$$
(6)

More generally, if \(g:X\rightarrow \overline{{\mathbb {R}}}\) is another function such that \(f\le g\), then for every \(x\in {\text {dom}}g\) we have

$$\begin{aligned} \partial _{\varepsilon +f(x)}f(x)\subset \partial _{\varepsilon +g(x)}g(x). \end{aligned}$$
(7)

The indicator function of A is the function \(\textrm{I}_{A}\ \)such that \(\textrm{I}_{A}(x)=0\) if \(x\in A,\) and \(\textrm{I}_{A}(x)=+\infty \) otherwise. The support function of A is \(\sigma _{A}:=\sup _{a\in A}\left\langle a,\cdot \right\rangle \).

We close this section by recalling the classical minimax theorem (see, e.g., [19, Theorem 2.10.2]). Given another lcs Y,  a nonempty convex set \(B\times A\subset Y\times X\) and a function \(f:B\times A\subset Y\times X\rightarrow {\mathbb {R}}_{\infty },\) we assume that B is compact, the functions \(f(\cdot ,x),\) \({ x\in A},\) are upper semicontinuous (usc, for short) and concave, and the functions \(\{f(y,\cdot ),\) \(y\in B\}\) are convex. Then we have that

$$\begin{aligned} \max _{y\in B}\inf _{x\in A}f(y,x)=\inf _{x\in A}\max _{y\in B}f(y,x). \end{aligned}$$
(8)

3 Finite Sum Representations of the Subdifferential

The present section consists of describing the state of the art of the representation of \(\partial _{\varepsilon }f(x),\) where \(f:=\sup _{t\in T}f_{t}\) with \(f_{t}:X\rightarrow \overline{{\mathbb {R}}},\) \(t\in T,\) being convex functions defined on the lcs X. Specifically, the recent results obtained in [4] provide characterizations of \(\partial _{\varepsilon }f(x)\) through limiting processes that involve the functions

$$\begin{aligned} f_{\lambda }:= {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}f_{t}, \lambda \in \Delta (T^{p}), \end{aligned}$$
(9)

where \(T^{p}:=\{t\in T:{\bar{f}}_{t}>-\infty \},\) and assuming the following lower semicontinuity-like property ([7])

$$\begin{aligned} {\bar{f}}=\sup _{t\in T}{\bar{f}}_{t}. \end{aligned}$$
(10)

The following general result produces a satisfactory representation of \(\partial _{\varepsilon }f(x)\) even though it is not precisely formulated in the desired way, due to the closure involved within (11). Nonetheless, it serves as an inspiration to investigate situations in which we can eliminate such closure and, consequently, achieve finite sum representations of \(\partial _{\varepsilon }f(x).\)

Lemma 1

([3, Theorem 5.1.7]) Given a family \(\{f_{t}:X\rightarrow \overline{{\mathbb {R}}},\) \(t\in T\}\) of convex functions and \(f:=\sup _{t\in T}f_{t},\) we assume that (10) holds. Then, for all \(x\in {\text {dom}}f\) and \(\varepsilon >0,\) we have

$$\begin{aligned} \partial _{\varepsilon }f(x)={\text {*}}{cl}\left\{ {\textstyle \bigcup \limits _{\lambda \in \Delta (T^{p})}} \partial _{\varepsilon +f_{\lambda }(x)-f(x)}\left( f_{\lambda }+\textrm{I} _{{\text {dom}}f}\right) (x)\right\} . \end{aligned}$$
(11)

The first situation that allows eliminating the closure from (11) is associated with the so-called concept of concave-like families (see, for instance, [4, Definition 1]).

Definition 1

A family of functions \(\{f_{t}:X\rightarrow \overline{{\mathbb {R}}},\) \(t\in T\}\) is said to be concave-like if, for each \(\lambda \in \Delta (T),\) there exists some \(t\in T\) such that \(f_{\lambda }\le f_{t}.\)

The first finite-sum representation of \(\partial _{\varepsilon }f(x)\) is given in the following theorem, which has already been established in [4]; For the sake of completeness, we give a sketch of its proof.

Theorem 2

([4, Theorem 13]) Given a concave-like family \(\{f_{t}:X\rightarrow \overline{{\mathbb {R}}},\) \(t\in T\}\) of convex functions and \(f:=\sup _{t\in T}f_{t}\) satisfying (10), we suppose that \(T\ \)is compact and the mappings \(t\longmapsto f_{t}(x),\) \(x\in {\text {dom}}f,\) are usc. Then, for all \(x\in {\text {dom}}f\) and \(\varepsilon \ge 0,\)

$$\begin{aligned} \partial _{\varepsilon }f(x)= {\textstyle \bigcup \limits _{t\in T}} \partial _{\varepsilon +f_{t}(x)-f(x)}\left( f_{t}+\textrm{I} _{{\text {dom}}f}\right) (x). \end{aligned}$$
(12)

Proof

Only the proof of the inclusion “\(\supset \)” needs to be shown. Given \(x^{*}\in \partial _{\varepsilon }f(x),\) by Lemma 1 there are nets \(0<\varepsilon _{i}\rightarrow \varepsilon \) (with \(\varepsilon _{i}=\varepsilon \) when \(\varepsilon >0\)), \(x_{i}^{*}\rightarrow x^{*}\) in \(X^{*}\) and \((\lambda _{i})_{i}\subset \Delta (T^{p} )\subset \Delta (T)\) such that

$$\begin{aligned} x_{i}^{*}\in \partial _{\varepsilon _{i}+f_{\lambda _{i}}(x)-f(x)} (f_{\lambda _{i}}+\textrm{I}_{{\text {dom}}f})(x),\text { for all }i. \end{aligned}$$
(13)

By the concavity-like assumption, there are associated indices \((t_{i} )_{i}\subset T\) such that \(f_{\lambda _{i}}\le f_{t_{i}}\) and so, using (6),

$$\begin{aligned} x_{i}^{*}\in \partial _{\varepsilon _{i}+f_{t_{i}}(x)-f(x)}(f_{t_{i} }+\textrm{I}_{{\text {dom}}f})(x),\text { for all }i. \end{aligned}$$

Moreover, the compactness and the upper semicontinuity assumptions allow us to suppose, without loss of generality, that \(t_{i}\rightarrow t_{0}\in T\) and \(\limsup _{i}f_{t_{i}}(y)\le f_{t_{0}}(y)\) for all \(y\in {\text {dom}}f.\) Therefore, taking limits on i in the last relation, we deduce that \(x^{*}\in \partial _{\varepsilon +f_{t_{0}}(x)-f(x)}(f_{t_{0}}+\textrm{I} _{{\text {dom}}f})(x)\). \(\square \)

The second situation where the closure is removed from (11) corresponds to the finite-dimensional framework. The following theorem along with a brief outline of its proof comes from [4].

Theorem 3

([4, Theorem 13](iii)) Given convex functions \(f_{t}:{\mathbb {R}}^{n}\rightarrow \overline{{\mathbb {R}}},\) \(t\in T,\) and \(f:=\sup _{t\in T}f_{t}\) satisfying (10), we assume that \(T\ \)is compact and the mappings \(t\longmapsto f_{t}(x),\) \(x\in {\text {dom}}f,\) are usc. Then, for all \(x\in {\text {dom}}f\) and \(\varepsilon \ge 0,\) we have that

$$\begin{aligned} \partial _{\varepsilon }f(x)= {\textstyle \bigcup \limits _{\lambda \in \Delta (T), \left| {\text {*}}{supp}\lambda \right| \le n+1}} \partial _{\varepsilon +f_{\lambda }(x)-f(x)}\left( f_{\lambda }+\textrm{I} _{{\text {dom}}f}\right) (x). \end{aligned}$$
(14)

Proof

Take \(x^{*}\in \partial _{\varepsilon }f(x)\) so that, by (11), there are nets \(0<\varepsilon _{i}\rightarrow \varepsilon \) (with \(\varepsilon _{i}=\varepsilon \) when \(\varepsilon >0\)), \(x_{i}^{*}\rightarrow x^{*}\) and \((\lambda _{i})_{i}\subset \Delta (T)\) such that

$$\begin{aligned} x_{i}^{*}\in \partial _{\varepsilon _{i}+f_{\lambda _{i}}(x)-f(x)} (f_{\lambda _{i}}+\textrm{I}_{{\text {dom}}f})(x),\text { for all }i. \end{aligned}$$

More precisely, thanks to the finite-dimensional setting, we can suppose that \(\left| {\text {*}}{supp}\lambda _{i}\right| \le n+1,\) for all \(i\in I\) (see [4, Theorem 13(i)]). So, taking subnets if necessary, we also may suppose that \(\lambda _{i}\rightarrow \lambda \in \Delta (T)\) such that \(\left| {\text {*}}{supp}\lambda \right| \le n+1.\) Therefore, taking limits in the last inclusion above, we deduce that \(x^{*}\in \partial _{\varepsilon +f_{\lambda }(x)-f(x)}(f_{\lambda }+\textrm{I} _{{\text {dom}}f})(x).\) This proves the nontrivial inclusion in (14). \(\square \)

The case of finite families of convex functions is covered by the last theorem, even without the requirement of the closedness condition (10) (see [4, Corollary 14] and [19, Corollary 2.8.11]). Specifically, when \(T:=\{1,\cdots ,m\}\) Theorem 3 is applied to the family \(\left\{ {\textstyle \sum \nolimits _{k=1,\cdots ,m}} \lambda _{k}f_{k}, \lambda \in \Delta _{m}\right\} ,\) which consistently fulfills condition (10) (as demonstrated in the proof of [3, Corollary 5.1.9]).

Later, in Example 2, we will show that formula (14) may not hold true beyond the finite-dimensional framework. In fact, the role of this condition in Theorem 3 is crucial since it implicitly enables the problem to be reduced to the supremum of a finite number of functions, owing to the following relation

$$\begin{aligned} \inf _{x\in X}\sup _{T}f_{t}=\max _{S\subset T, S\text { finite}}\inf _{x\in X}\max _{t\in S}f_{t}. \end{aligned}$$

4 General Representations

In this section, we provide general representations of the \(\varepsilon \)-subdifferential of \(f:=\sup _{t\in T}f_{t},\) by means of appropriate sums performed on the data functions. Since we do not assume any topological or algebraic structure on the index set T,  our formulas will also involve specific singular measures operating on our data functions, these are the price of eliminating the closure from the general formula given in (11). These additional measures will be replaced by related limits involving the data functions in the next section.

Given the convex functions \(f_{t}:X\rightarrow {\mathbb {R}}_{\infty },\) \(t\in T,\) and the associated supremum \(f:=\sup _{t\in T}f_{t},\) we shall assume that

$$\begin{aligned} \inf _{t\in T}f_{t}(x)>-\infty ,\text { for all }x\in {\text {dom}}f. \end{aligned}$$
(15)

Then we consider the function \(F:{\text {dom}}f\subset X\rightarrow {\mathbb {R}}^{T}\) given by

$$\begin{aligned} F(x):=((f_{t}(x))_{t\in T}), \end{aligned}$$
(16)

and, for each \(\beta \in \ell _{+}^{s}(T),\) we define the (convex) composition function \(\beta \circ F:X\rightarrow {\mathbb {R}}_{\infty }\) as

$$\begin{aligned} (\beta \circ F)(x):=\beta ((f_{t}(x))_{t\in T}), \end{aligned}$$
(17)

if \(x\in {\text {dom}}f,\) and \(+\infty \) otherwise.

Given \(\lambda \in \ell _{1}^{+}(T)\) with \({\text {*}}{supp}\lambda :=\{t_{k},\) \(k\ge 1\},\) we introduce the function \(f_{\lambda }:X\rightarrow \overline{{\mathbb {R}}}\ \)defined as

$$\begin{aligned} f_{\lambda }(x):= {\textstyle \sum _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}f_{t}(x)=\limsup _{n\rightarrow \infty } {\textstyle \sum _{1\le k\le n}} \lambda _{t_{k}}f_{t_{k}}(x). \end{aligned}$$
(18)

Moreover, if \(x\in X\) is such that \(\sup _{t\in {\text {*}}{supp}\lambda }f_{t}(x)<+\infty \) (hence, \((f_{t}(x))_{t\in T}\in \ell _{\infty }(T)\) by (15)), (18) reduces to

$$\begin{aligned} f_{\lambda }(x)=\lim _{k\rightarrow \infty } {\textstyle \sum _{1\le k\le n}} \lambda _{t_{k}}f_{t_{k}}(x)=\lim _{k\rightarrow \infty }\left\langle \chi _{\{t_{k}, k\le 1\}}\lambda ,(f_{t}(x))_{t\in T}\right\rangle . \end{aligned}$$

In particular, (18) and (9) define the same object when T is finite.

The following technical lemma shows that the extended family \(\{f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f},\) \(\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)\}\) produces the same supremum f (remember that \({\hat{\Delta }}(T)\) is defined in (4)).

Lemma 4

Given convex functions \(f_{t}:X\rightarrow {\mathbb {R}}_{\infty },\) \(t\in T,\) and \(f:=\sup _{t\in T}f_{t},\) we assume that (15) holds. Then

$$\begin{aligned} f=\sup _{\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)}(f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f}). \end{aligned}$$
(19)

Proof

Let us denote

$$\begin{aligned} \varphi _{\alpha }:=f_{\lambda }+\beta \circ F,\text { \ }\alpha :=\lambda +\beta \in {\hat{\Delta }}(T). \end{aligned}$$

If \(x\notin {\text {dom}}f,\) then both sides in (19) are equal to \(+\infty \) and we are obviously done. Otherwise, given any \(x\in {\text {dom}}f\ \)and \(\alpha :=\lambda +\beta \in {\hat{\Delta }}(T),\) we obtain

$$\begin{aligned} \varphi _{\alpha }(x)\!=\!f_{\lambda }(x)\!+\!\beta (F(x))\le & {\textstyle \sum \limits _{t\in T}} \lambda _{t}f(x)\!+\!\beta (1_{T})f(x)\\= & \left( {\textstyle \sum \limits _{t\in T}} \lambda _{t}\!+\!\beta (1_{T})\right) f(x)\!=\!f(x), \end{aligned}$$

and the inequality “\(\ge \)” in (19) follows. Conversely, assume that \(f(x)>\mu \) for some \(\mu \in {\mathbb {R}};\) hence, \(f_{t}(x)>\mu \) for some \(t\in T.\) Take \({\hat{\lambda }}:=\chi _{_{\{t\}} }\in \ell _{1}^{+}(T)\) and \({\hat{\beta }}:=0_{T}\in \ell _{1}^{+}(T)\) \((\subset \ell _{+}^{s}(T)).\) Then \({\hat{\alpha }}:={\hat{\lambda }}+{\hat{\beta }}\in {\hat{\Delta }}(T)\ \)and

$$\begin{aligned} \mu <f_{t}(x)=f_{{\hat{\lambda }}}(x)+{\hat{\beta }}(F(x))+\textrm{I} _{{\text {dom}}f}(x)\le \sup _{\alpha \in {\hat{\Delta }}(T)}\varphi _{\alpha }(x). \end{aligned}$$

Whence, the remaining inequality “\(\le \)” in (19) is derived when \(\mu \uparrow f(x).\) \(\square \)

We give the main result of this section, assuming condition (15):

$$\begin{aligned} \inf _{t\in T}f_{t}(x)>-\infty ,\text { for all }x\in {\text {dom}}f. \end{aligned}$$

Theorem 5

Given convex functions \(f_{t}:X\rightarrow {\mathbb {R}}_{\infty },\) \(t\in T\ \)and \(f:=\sup _{t\in T}f_{t},\) we assume that (15) holds. Then, for all \(x\in {\text {dom}}f\) and\(\ \varepsilon \ge 0,\) we have

$$\begin{aligned} \partial _{\varepsilon }f(x)= {\textstyle \bigcup \limits _{\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)}} \partial _{\varepsilon +f_{\lambda }(x)+\beta (F(x))-f(x)}\left( f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f}\right) (x). \end{aligned}$$
(20)

Proof

Fix \(x\in {\text {dom}}f\) and\(\ \varepsilon \ge 0.\) To prove the inclusion “\(\supset \)” we take \(\alpha :=\lambda +\beta \in {\hat{\Delta }}(T).\) Then, since \(f_{\lambda }+\beta \circ F\le f\) by (19), relation (7) entails

$$\begin{aligned} \partial _{\varepsilon +f_{\lambda }(x)+\beta (F(x))-f(x)}\left( f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f}\right) (x)\subset \partial _{\varepsilon }\left( f+\textrm{I}_{{\text {dom}}f}\right) (x)=\partial _{\varepsilon }f(x). \end{aligned}$$
(21)

To prove the opposite inclusion, we suppose that \(\partial _{\varepsilon }f(x)\not =\emptyset .\) Let us first assume that \(\theta \in \partial _{\varepsilon }f(x)\) so that, using (19),

$$\begin{aligned} \inf _{z\in {\text {dom}}f}\sup _{\alpha \in {\hat{\Delta }}(T)}\varphi _{\alpha }(z)=\inf _{z\in X}\sup _{t\in T}f_{t}(z)=\inf _{z\in X}f(z)\ge f(x)-\varepsilon . \end{aligned}$$
(22)

Observe that the functions \(\varphi _{\alpha },\) \(\alpha \in {\hat{\Delta }}(T),\) are all convex. Also, since \(F(z)\in \ell _{\infty }(T)\) for all \(z\in {\text {dom}}f,\) the mappings \(\alpha :=\lambda +\beta \in \hat{\Delta }(T)\longmapsto \varphi _{\alpha }(z)=f_{\lambda }(z)+\beta (F(z)),\) \(z\in {\text {dom}}f\), are \(w^{*}\)-usc and concave. At the same time, we have that \(\varphi _{\alpha }(z)\in {\mathbb {R}}\) for all \(\alpha \in {\hat{\Delta }}(T)\) and \(z\in {\text {dom}}f\) (due to (15)). Therefore, since the set \({\hat{\Delta }}(T)\) is convex and \(w^{*}\)-compact in \((\ell _{\infty }(T))^{*}\) by Lemma 15, the minimax theorem (8) yields some \(\alpha _{0}:=\lambda _{0}+\beta _{0}\in {\hat{\Delta }}(T)\) such that

$$\begin{aligned} \inf _{z\in {\text {dom}}f}\sup _{\alpha \in {\hat{\Delta }}(T)}\varphi _{\alpha }(z)=\max _{\alpha \in {\hat{\Delta }}(T)}\inf _{z\in {\text {dom}}f} \varphi _{\alpha }(z)=\inf _{z\in {\text {dom}}f}\varphi _{\alpha _{0}}(z); \end{aligned}$$

that is, by (22),

$$\begin{aligned} \varphi _{\alpha _{0}}(z)\ge f(x)-\varepsilon =\varphi _{\alpha _{0} }(x)-(\varepsilon +\varphi _{\alpha _{0}}(x)-f(x)),\text { for all } z\in {\text {dom}}f. \end{aligned}$$

Consequently,

$$\begin{aligned} \theta\in & \partial _{\varepsilon \!+\!\varphi _{\alpha _{0}}(x)\!-\!f(x)}(\varphi _{\alpha _{0}}\!+\!\textrm{I}_{{\text {dom}}f})(x)\\ = & \partial _{\varepsilon \!+\!f_{\lambda _{0}}(x)\!+\!\beta _{0}(F(x))\!-\!f(x)}(f_{\lambda _{0}}\!+\!\beta _{0}\circ F\!+\textrm{I}_{{\text {dom}}f})(x), \end{aligned}$$

and (20) follows in the present case when \(\theta \in \partial _{\varepsilon }f(x)\).

More generally, if \(x^{*}\in \partial _{\varepsilon }f(x),\) then

$$\begin{aligned} \theta \in \partial _{\varepsilon }\left( f-\left\langle x^{*},\cdot \right\rangle \right) (x)=\partial _{\varepsilon }{\tilde{f}}(x), \end{aligned}$$

where \({\tilde{f}}_{t}:=f_{t}-\left\langle x^{*},\cdot \right\rangle ,\) \(t\in T\) and \({\tilde{f}}:=\sup _{t\in T}{\tilde{f}}_{t}.\) Therefore, since \({\text {dom}}f={\text {dom}}{\tilde{f}}\) and the family \(\{{\tilde{f}}_{t},\) \(t\in T\}\) also satisfies (15), the paragraph above yields some \(\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)\) such that \(\theta \in \partial _{\varepsilon +{\tilde{f}}_{\lambda }(x)+\beta (\tilde{F}(x))-{\tilde{f}}(x)}\left( {\tilde{f}}_{\lambda }+\beta \circ \tilde{F}+\textrm{I}_{{\text {dom}}f}\right) (x),\) where \({\tilde{f}}_{\lambda }\) and \({\tilde{F}}\) are defined as

$$\begin{aligned} {\tilde{f}}_{\lambda }:= {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}{\tilde{f}}_{t}\text { and }{\tilde{F}}(z):=(({\tilde{f}}_{t}(z))_{t\in T}), z\in {\text {dom}}f; \end{aligned}$$

hence, for all \(z\in X,\)

$$\begin{aligned} {\tilde{f}}_{\lambda }(z)=f_{\lambda }(z)- {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}\left\langle x^{*},z\right\rangle \text { and }(\beta \circ {\tilde{F}})(z)=(\beta \circ {\tilde{F}})(z)-\beta \left( \left\langle x^{*},z\right\rangle 1_{T}\right) . \end{aligned}$$

Consequently, observing that

$$\begin{aligned} - {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}\left\langle x^{*},z\right\rangle -\beta \left( \left\langle x^{*},z\right\rangle 1_{T}\right) =-\left( {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}+\beta (\chi _{_{T}})\right) \left\langle x^{*},z\right\rangle =-\left\langle x^{*},z\right\rangle , \end{aligned}$$

we obtain that

$$\begin{aligned} \theta&\in \partial _{\varepsilon +f_{\lambda }(x)+\beta (F(x))-\left\langle x^{*},x\right\rangle -(f(x)-\left\langle x^{*},x\right\rangle )}\left( f_{\lambda }+\beta \circ F-\left\langle x^{*},\cdot \right\rangle +\textrm{I}_{{\text {dom}}f}\right) (x)\\&=\partial _{\varepsilon +f_{\lambda }(x)+\beta (F(x))-f(x)}\left( f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f}\right) (x)-x^{*}. \end{aligned}$$

In other words, we have that \(x^{*}\in \partial _{\varepsilon +f_{\lambda }(x)+\beta (F(x))-f(x)}\left( f_{\lambda }+\beta \circ F\right) (x),\) and the desired inclusion follows. \(\square \)

Remark 1

In particular, when \(\varepsilon =0\ \)the formula in (20) also reads

$$\begin{aligned} \partial _{\varepsilon }f(x)= {\textstyle \bigcup \limits _{\begin{array}{c} \alpha :=\lambda +\beta \in \hat{\Delta }(T)\\ f_{\lambda }(x)+\beta (F(x))=f(x) \end{array}}} \partial \left( f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}} f}\right) (x). \end{aligned}$$
(23)

The proof of this formula and Theorem 5 relies primarily on the minimax theorem (8). Alternatively, we could also use the subdifferential formula (12) established in [4, Theorem 13] for the family \(\varphi _{\alpha }:=f_{\lambda }+\beta \circ F,\)\(\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)\). By doing so, we only need to verify that this family satisfies condition (10); that is,

$$\begin{aligned} {\text {*}}{cl}f=\sup _{\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)} {\text {*}}{cl}\left( f_{\lambda }+\beta \circ F+\textrm{I} _{{\text {dom}}f}\right) . \end{aligned}$$

In fact, as in [4, Lemma 8], this closedness property can be accomplished using the minimax theorem.

The previous theorem is illustrated in the following example, where it is possible to find the exact form of the \(\beta \)’s used in Theorem 5. More “regular” representations are provided in Sect. 5.

Example 1

We are going to express the set \(\partial f(0)\) when \(f:{\mathbb {R}} \rightarrow {\mathbb {R}}_{\infty }\) is given by

$$\begin{aligned} f(x):=\sup \{a_{n}x-b_{n}:n\ge 1\}, \end{aligned}$$

with \((a_{n})_{n\ge 1},\) \((b_{n})_{n}\subset {\mathbb {R}}\) being such that \(\lim _{n\rightarrow \infty }a_{n}={\bar{a}},\) \(\lim _{n\rightarrow \infty }b_{n} ={\bar{b}}\in {\mathbb {R}}\) and

$$\begin{aligned} f(0):=\sup \{-b_{n}:n\ge 1\}=0; \end{aligned}$$

hence, \({\text {dom}}f={\mathbb {R}}.\) By Theorem 5 (or, equivalently, (23)), we have

$$\begin{aligned} \partial f(0)= & {\textstyle \bigcup \limits _{\begin{array}{c} \alpha :=\lambda +\beta \in \hat{\Delta }(T)\\ f_{\lambda }(0)+\beta (F(0))=0 \end{array}}} \partial \left( f_{\lambda }+\beta \circ F\right) (0)\nonumber \\= & {\textstyle \bigcup \limits _{\begin{array}{c} \alpha :=\lambda +\beta \in \hat{\Delta }(T)\\ \beta ((b_{n})_{n\ge 1})=-{\textstyle \sum _{n\ge 1}}\lambda _{n}b_{n} \end{array}}} \partial \left( f_{\lambda }+\beta \circ F\right) (0). \end{aligned}$$
(24)

Let \(\alpha :=\lambda +\beta \in {\hat{\Delta }}({\mathbb {N}})\) be as in the last expression; that is, \(\lambda \in \ell _{1}^{+}({\mathbb {N}}),\) \(\beta \in \ell _{+}^{s}({\mathbb {N}})\), \( {\textstyle \sum _{n\ge 1}} \lambda _{n}+\beta (\chi _{T})=1\) and \(\beta ((b_{n})_{n\ge 1})=- {\textstyle \sum _{n\ge 1}} \lambda _{n}b_{n}.\) Then, for every \(x\in {\mathbb {R}}\), we have

$$\begin{aligned}&f_{\lambda }(x):= {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}(a_{n}x-b_{n})= {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}a_{n}x- {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}b_{n},&\\&F(x):=(a_{n}x-b_{n})_{n\ge 1}\in \ell _{\infty }({\mathbb {N}}),&\end{aligned}$$

and the linearity of \(\beta \) together with the fact that \((a_{n})_{n\ge 1},\) \((b_{n})_{n}\subset \ell _{\infty }({\mathbb {N}})\) entails

$$\begin{aligned} (\beta \circ F)(x):\!=\!\beta ((a_{n}x\!-\!b_{n})_{n\ge 1})\!=\!x\beta ((a_{n})_{n} )\!-\!\beta ((b_{n})_{n\ge 1})\!=\!x\beta ((a_{n})_{n})\!+\! {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}b_{n}. \end{aligned}$$

Moreover, since \(\beta \in \ell _{+}^{s}({\mathbb {N}}),\) we obtain

$$\begin{aligned} \beta (\chi _{T})\lim _{n}a_{n}=\beta (\chi _{T})\liminf _{n\rightarrow \infty } a_{n}\le \beta ((a_{n})_{n})\le \beta (\chi _{T})\limsup _{n\rightarrow \infty }a_{n}=\beta (\chi _{T})\lim _{n}a_{n}, \end{aligned}$$

and so

$$\begin{aligned} \beta ((a_{n})_{n})=\beta (\chi _{T})\lim _{n}a_{n}=(1- {\textstyle \sum _{n\ge 1}} \lambda _{n}){\bar{a}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \beta ((b_{n})_{n})=\beta (\chi _{T})\lim _{n}b_{n}=(1- {\textstyle \sum _{n\ge 1}} \lambda _{n}){\bar{b}}, \end{aligned}$$

and the functional \(\beta \circ F\) is fully characterized in terms of \(\lambda \) as

$$\begin{aligned} (\beta \circ F)(x)=x\left( 1- {\textstyle \sum _{n\ge 1}} \lambda _{n}\right) {\bar{a}}+ {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}b_{n},\text { for all }x\in {\mathbb {R}}. \end{aligned}$$

Consequently, the corresponding function \(f_{\lambda }+\beta \circ F\) is given by

$$\begin{aligned} (f_{\lambda }+\beta \circ F)(x)&= {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}a_{n}x- {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}b_{n}+x\left( 1- {\textstyle \sum _{n\ge 1}} \lambda _{n}\right) {\bar{a}}+ {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}b_{n}\\&=\left( {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}a_{n}+\left( 1- {\textstyle \sum _{n\ge 1}} \lambda _{n}\right) {\bar{a}}\right) x, \end{aligned}$$

and formula (24) simplifies to

$$\begin{aligned} \partial f(0)= {\textstyle \bigcup \limits _{\lambda }} ~ {\textstyle \sum \limits _{n\ge 1}} \lambda _{n}a_{n}+\left( 1- {\textstyle \sum _{n\ge 1}} \lambda _{n}\right) {\bar{a}}, \end{aligned}$$

where the union is taken over all \(\lambda \in \ell _{1}^{+}({\mathbb {N}})\) such that \( {\textstyle \sum _{n\ge 1}} \lambda _{n}\le 1\) and \((1- {\textstyle \sum _{n\ge 1}} \lambda _{n}){\bar{b}}=- {\textstyle \sum _{n\ge 1}} \lambda _{n}b_{n}.\)

The following corollary shows that the construction of the exact subdifferential \(\partial f(x)\) mainly relies on the data functions that are almost active at the point \(x\in {\text {dom}}f\), namely those indexed in the sets

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

for \(\varepsilon \ge 0\) quite small. We also denote \(T(x):=T_{0}(x),\) the set of active indices at \(x\in {\text {dom}}f.\) As for the rest of the functions, they only have influence at the level of the normal cone to the domain of f. This aligns with previous results characterizing the subdifferential of pointwise convex suprema by means of (approximate) subdifferentials of the \(f_{t}\)’s indexed in \(T_{\varepsilon }(x),\) for sufficiently small \(\varepsilon ,\) together with the normal cone to finite-dimensional sections of \({\text {dom}}f.\) For additional insights on this subject we refer to [7, 11, 12] and references therein.

Corollary 6

Given convex functions \(f_{t}:X\rightarrow {\mathbb {R}}_{\infty },\) \(t\in T\ \)and \(f:=\sup _{t\in T}f_{t},\) we assume that (15) holds. Then, for all \(x\in {\text {dom}}f\) and \(\varepsilon >0,\) we have

$$\begin{aligned} \partial f(x)= {\textstyle \bigcup \limits _{\begin{array}{c} \alpha :=\lambda +\beta \in \hat{\Delta }(T)\\ \\ {\text {*}}{supp}\lambda \subset T(x), \beta (\chi _{_{T\setminus T_{\varepsilon }(x)}})=0 \end{array}}} \partial _{f_{\lambda }(x)+\beta (F(x))-f(x)}\left( f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f}\right) (x). \end{aligned}$$

Proof

Fix \(x\in {\text {dom}}f\) and \(x^{*}\in \partial f(x).\) Then, by Theorem 5, there exists some \(\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)\) such that \(x^{*}\in \partial _{f_{\lambda }(x)+\beta (F(x))-f(x)}\left( f_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f}\right) (x);\) hence, \(f_{\lambda }(x)+\beta (F(x))-f(x)\ge 0.\) Consequently, \(f_{\lambda }(x)+\beta (F(x))-f(x)=0\) and we deduce that

$$\begin{aligned} f_{t}(x)=f(x)\text {, for all }t\in {\text {*}}{supp}\lambda ,\text { and }\beta (F(x))=\beta (1_{T})f(x); \end{aligned}$$

that is, \({\text {*}}{supp}\lambda \subset T(x).\) Moreover, using the linearity of \(\beta ,\ \)the last equality also implies

$$\begin{aligned} \left[ \beta (\chi _{_{T_{\varepsilon }(x)}}\odot F(x))-\beta (\chi _{_{T_{\varepsilon }(x)}})f(x)\right] +\left[ \beta (\chi _{_{_{T\setminus T_{\varepsilon }(x)}}}\odot F(x))-\beta (\chi _{_{_{T\setminus T_{\varepsilon }(x)}}})f(x)\right] =0, \end{aligned}$$

where the symbol \(\odot \) refers to the elementwise/Hadamard product (for instance, the function \(\chi _{_{S}}\odot F(x)\) is defined on T as \((\chi _{_{S}}\odot F(x))(t):=\chi _{_{S}}(t)f_{t}(x),\) \(t\in T\) ). Hence, for each and \(\varepsilon >0,\) the non-positivity of the terms between brackets in the expression above together with the fact that \(\beta \in \ell _{+}^{s}(T)\) ensures that

$$\begin{aligned} \beta (\chi _{_{T\setminus T_{\varepsilon }(x)}})f(x)=\beta (\chi _{_{T\setminus T_{\varepsilon }(x)}}\odot F(x))\le (f(x)-\varepsilon )\beta (\chi _{_{T\setminus T_{\varepsilon }(x)}}); \end{aligned}$$

that is, \(\beta (\chi _{_{T\setminus T_{\varepsilon }(x)}})=0.\) We are done because the other inclusion “\(\supset \)” is directly obtained from Theorem 5. \(\square \)

Condition (15) it not very restrictive and can be circumvented by adjusting the representations of \(\partial _{\varepsilon }f(x)\) provided earlier. This fact is shown in the following corollary, considering the case \(\varepsilon =0.\)

Corollary 7

Given convex functions \(f_{t}:X\rightarrow {\mathbb {R}}_{\infty },\) \(t\in T\ \)and \(f:=\sup _{t\in T}f_{t},\) for all \(x\in {\text {dom}}f\ \)we have

$$\begin{aligned} \partial f(x)= {\textstyle \bigcup \limits _{\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)}} \partial _{g_{\lambda }(x)+\beta (G(x))-f(x)}\left( g_{\lambda }+\beta \circ G+\textrm{I}_{{\text {dom}}f}\right) (x), \end{aligned}$$

where \(g_{t}:=\max \{f_{t},f(x)-1\},\) \(t\in T\) and \(G:=((g_{t}(\cdot ))_{t\in T}).\)

Proof

Fix \(x\in {\text {dom}}f.\) To prove the nontrivial inclusion “\(\subset \)” we assume that \(\partial f(x)\not =\emptyset .\) Then the function f is lsc at x and we have that \(\partial f(x)=\partial g(x),\) where

$$\begin{aligned} g:=\max \{f,f(x)-1\}=\sup _{t\in T}g_{t}. \end{aligned}$$

Since \({\text {dom}}f={\text {dom}}g,\) \(f(x)=g(x)\) and the family \(\{g_{t},\) \(t\in T\}\) satisfies condition (15), the conclusion follows by Theorem 5. \(\square \)

Arguments comparable to those in the proof of Corollary 7 can be made for the general case \(\varepsilon \ge 0.\) For instance, if the \(f_{t}\)’s and f are as above and \(\partial _{\varepsilon }f(x)\not =\emptyset ,\) then for any fixed element \(x_{\varepsilon }^{*}\in \partial _{\varepsilon }f(x)\) and any number \(c\ge \varepsilon -f(x)+\left\langle x_{\varepsilon }^{*},x\right\rangle \) we have

$$\begin{aligned} f-\left\langle x_{\varepsilon }^{*},\cdot \right\rangle \ge f(x)-\left\langle x_{\varepsilon }^{*},x\right\rangle -\varepsilon \ge f(x)-c, \end{aligned}$$

so that

$$\begin{aligned} f=\max \{f-\left\langle x_{\varepsilon }^{*},\cdot \right\rangle ,f(x)-c\}+\left\langle x_{\varepsilon }^{*},\cdot \right\rangle =\left\langle x_{\varepsilon }^{*},\cdot \right\rangle +\sup _{t\in T}g_{t}, \end{aligned}$$

where

$$\begin{aligned} g_{t}:=\max \{f_{t}-\left\langle x_{\varepsilon }^{*},\cdot \right\rangle ,f(x)-c\}\ge f(x)-c>-\infty ,\text { for all }t\in T. \end{aligned}$$

Thus, the family of convex functions \(\{g_{t},\) \(t\in T\}\) satisfies condition (15), and Theorems 5 and 8 can be applied to the \(g_{t}\)’s.

5 Regular Representations

This section provides simplified representations of \(\partial _{\varepsilon }f(x)\ \)that do not involve the use of singular measures. Instead, we employ countable sums of the \(f_{t}\)’s, as previously done, along with new functions derived from limiting processes performed on these countable functions. For this purpose, we consider specific yet fairly broad structures on the lcs X,  which can be a reflexive Banach space or a separable normed space. In all these cases, we require the data functions \(f_{t},\) \(t\in T,\) to exhibit a lower semicontinuity-like behavior together with the lower boundedness condition (15).

Let us start with the following simple example showing that Theorem 3 is not necessarily true in infinite-dimensional (separable Hilbert) spaces.

Example 2

Given the Hilbert space \(\ell _{2},\) the space of square summable sequences, we take the set \(A:=\{e_{n},\) \(n\ge 1\}\) along with the supremum function

$$\begin{aligned} f:=\sigma _{A}:=\sup _{a\in A}\left\langle a,\cdot \right\rangle . \end{aligned}$$

As a result, we have \(\partial f(\theta )=\overline{{\text {*}}{co} }(A)=B_{\ell _{2}}\) (the closed unit ball in \(\ell _{2}\)). However, since A is bounded, we have \({\text {dom}}f=X\) and, thus,

$$\begin{aligned} {\textstyle \bigcup _{\lambda \in \Delta (A)}} \partial _{f_{\lambda }(x)-f(x)}\left( f_{\lambda }+\textrm{I} _{{\text {dom}}f}\right) (x)= {\textstyle \bigcup _{\lambda \in \Delta (A)}} {\textstyle \sum \limits _{a\in {\text {*}}{supp}\lambda }} \lambda _{a}a={\text {*}}{co}(A). \end{aligned}$$

Observe that although \(\theta \in \partial f(\theta )\), it cannot be part of the last union. If it were, then \(\theta = {\textstyle \sum _{a\in {\text {*}}{supp}\lambda }} \lambda _{a}a\) and this would lead us to the contradiction \(\lambda \equiv 0_{A}.\)

The above example shows that finite convex combinations alone are insufficient to give a complete description of \(\partial _{\varepsilon }f(x).\) In fact, the same example proposes incorporating limits of such convex combinations or, in other terms, employing infinite sums.

The following result replaces the singular measures \(\beta \) involved in Theorem 5 with data functions that are “at the boundary of T”, reflecting the singularity of such \(\beta \)’s. In fact, the role of these “boundary” functions could effectively be summarized in a unique function \(f_{\infty }:X\rightarrow {\mathbb {R}}_{\infty }\) defined as

$$\begin{aligned} f_{\infty }(x):=\inf _{S\subset T, \left| S\right| <+\infty } \sup _{t\in T\setminus S}f_{t}(x). \end{aligned}$$
(25)

In this scenario, assuming condition (15); that is, \(\inf _{t\in T}f_{t}(x)>-\infty \ \)for all \(x\in {\text {dom}}f,\) the characterization of \(\partial _{\varepsilon }f(x)\) is obtained using the functions \(f_{\lambda },\) as defined in (18), with (see (36))

$$\begin{aligned} \lambda \in \Delta \left[ T\cup \{\infty \}\right] :=\left\{ \lambda \in {\mathbb {R}}_{+}^{[T\cup \{\infty \}]}:\ {\textstyle \sum \limits _{t\in T}} \lambda _{t}+\lambda _{\infty }=1\right\} ; \end{aligned}$$

that is,

$$\begin{aligned} f_{\lambda }(x)=\left\{ \begin{array}{ll} {\textstyle \sum _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}f_{t}(x), & \text {if }{\text {*}}{supp}\lambda \subset T,\\ {\textstyle \sum _{t\in T\cap {\text {*}}{supp}\lambda }} \lambda _{t}f_{t}(x)+\lambda _{\infty }f_{\infty }(x), & \text {if not.} \end{array} \right. \end{aligned}$$

The following theorem could also be obtained from [10, Theorem 1, chap. 6] (see the comments in the introduction).

Theorem 8

Given convex functions \(f_{t}:X\rightarrow {\mathbb {R}}_{\infty },\) \(t\in T\ \)and \(f:=\sup _{t\in T}f_{t},\) we suppose that (15) holds. Then, for all \(x\in {\text {dom}}f\) and\(\ \varepsilon \ge 0,\)

$$\begin{aligned} \partial _{\varepsilon }f(x)= {\textstyle \bigcup \limits _{\lambda \in \Delta \left[ T\cup \{\infty \}\right] }} \partial _{\varepsilon +f_{\lambda }(x)-f(x)}\left( f_{\lambda }+\textrm{I} _{{\text {dom}}f}\right) (x). \end{aligned}$$
(26)

Proof

Fix \(x\in {\text {dom}}f\),\(\ \varepsilon \ge 0\) and take \(x^{*} \in \partial _{\varepsilon }f(x).\) According to Theorem 5, there exists some \(\alpha :=\lambda +\beta \in {\hat{\Delta }}(T)\) such that

$$\begin{aligned} x^{*}\in \partial _{\varepsilon +g_{\lambda }(x)+\beta (F(x))-f(x)}\left( g_{\lambda }+\beta \circ F+\textrm{I}_{{\text {dom}}f}\right) (x), \end{aligned}$$

where \(g_{\lambda }:= {\textstyle \sum _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}f_{t}.\) Since \(\beta \in \ell _{+}^{s}(T),\) for every finite set \(S\subset T\ \)and\(\,y\in {\text {dom}}f\) we have

$$\begin{aligned} \beta (F(y))=\beta (\chi _{_{T\setminus S}}\odot F(y))\le \beta (\chi _{_{T\setminus S}})\sup _{t\in T\setminus S}f_{t}(y)=\beta (\chi _{_{T}} )\sup _{t\in T\setminus S}f_{t}(y). \end{aligned}$$

So, since \(S\subset T\) is any arbitrary set, \(\beta (F(y))\le \beta (T){\hat{f}}_{\infty }(y)\) and we deduce

$$\begin{aligned} g_{\lambda }(y)+\left( \beta \circ F\right) (y)\le g_{\lambda }(y)+\beta (\chi _{_{T}}){\hat{f}}_{\infty }(y). \end{aligned}$$

Therefore, using (7),

$$\begin{aligned} x^{*}&\in \partial _{\varepsilon +g_{\lambda }(x)+\beta (\chi )\hat{f}_{\infty }(x)-f(x)}\left( g_{\lambda }+\beta (\chi _{_{T}}){\hat{f}}_{\infty }+\textrm{I}_{{\text {dom}}f}\right) (x)\\&=\partial _{\varepsilon +f_{\lambda }(x)-f(x)}\left( f_{\lambda } +\textrm{I}_{{\text {dom}}f}\right) (x), \end{aligned}$$

and the inclusion “\(\subset \)” in (26) follows. This finishes the proof since the opposite inclusion is straightforward, due to the inequality \(f_{\lambda }\le f,\) for all \(\lambda \in \Delta \left[ T\cup \{\infty \}\right] .\) \(\square \)

The conclusion in Theorem 8 involves the function \({\hat{f}}_{\infty },\) which is also a supremum function. This is why we give next another result which makes the previous result more operational in the context of separable or reflexive Banach spaces. We will use the family of countable sets \(\omega \subset T\) which are represented as total order sets of the form \(\omega =(t_{n})_{n},\) together with the associated (convex) functions \(f_{\omega }:X\rightarrow {\mathbb {R}}_{\infty }\) defined as

$$\begin{aligned} f_{\omega }:=\limsup _{n\rightarrow \infty }f_{t_{n}}. \end{aligned}$$

For each set \(\omega \subset T,\) also regarded as a new index that is denoted by \(\{\omega \}\), we consider the augmented index set \(\omega \cup \{\omega \}\) and the associated canonical simplex (as defined in (36))

$$\begin{aligned} \Delta \left[ \omega \cup \{\omega \}\right] =\left\{ \alpha :=(\lambda ,\lambda _{\omega }):\lambda \in \ell _{1}^{+}(\omega ),\lambda _{\omega } \ge 0, {\textstyle \sum _{t\in \omega }} \lambda _{t}+\lambda _{\omega }=1\right\} . \end{aligned}$$

We need a couple of lemmas; the first one emphasizes the role under condition (10) of data functions indexed in the set \(T^{p}:=\{t\in T:{\bar{f}} _{t}>-\infty \}\).

Lemma 9

Let \(\{f_{t}:X\rightarrow \overline{{\mathbb {R}}},\) \(t\in T\}\) be a family of convex functions such that (10) holds; that is, \({\bar{f}}:=\sup _{t\in T}{\bar{f}}_{t}.\) If \(T^{p}\ne \emptyset ,\) then

$$\begin{aligned} \sup _{t\in T}{\bar{f}}_{t}=\sup _{t\in T^{p}}{\bar{f}}_{t}+\textrm{I}_{D_{0}}, \end{aligned}$$

where

$$\begin{aligned} D_{0}:= {\textstyle \bigcap \limits _{t\in T\setminus T^{p}}} {\text {*}}{cl}({\text {dom}}f_{t}). \end{aligned}$$
(27)

Proof

Since \({\bar{f}}_{t}\le {\bar{f}}\), for all \(t\in T,\) we have that \({\text {dom}}{\bar{f}}\subset D_{0}\) and condition (10) entails

$$\begin{aligned} \sup _{t\in T^{p}}{\bar{f}}_{t}+\textrm{I}_{D_{0}}\le \sup _{t\in T}{\bar{f}} _{t}+\textrm{I}_{{\text {dom}}{\bar{f}}}={\bar{f}}+\textrm{I} _{{\text {dom}}{\bar{f}}}={\bar{f}}. \end{aligned}$$

In particular, the functions \(\sup _{t\in T^{p}}{\bar{f}}_{t}+\textrm{I}_{D_{0}}\) and \({\bar{f}}\) coincide outside the set \(D_{0}.\) Moreover, if \(x\in D_{0},\) then \({\bar{f}}_{t}(x)=-\infty \) for all \(t\in T{\setminus } T^{p}\) and so, again by (10),

$$\begin{aligned} \sup _{t\in T^{p}}\left( {\bar{f}}_{t}+\textrm{I}_{D_{0}}\right) (x)=\sup _{t\in T}\left( {\bar{f}}_{t}+\textrm{I}_{D_{0}}\right) (x)\ge \sup _{t\in T}{\bar{f}}_{t}(x)={\bar{f}}(x). \end{aligned}$$

Thus, \(\sup _{t\in T^{p}}{\bar{f}}_{t}+\textrm{I}_{D_{0}}\ge {\bar{f}}\) and the conclusion holds. \(\square \)

The following lemma proposes a useful reduction of \(\partial _{\varepsilon }f(x)\) to subfamilies listed in a countable subset of T.

Lemma 10

Given functions \(f_{t}\in \Gamma _{0}(X),\) \(t\in T,\ \) and \(f:=\sup _{t\in T}f_{t},\) we assume that X is a reflexive Banach space or a separable normed space. Then, for every \(x\in {\text {dom}}f,\) \(\varepsilon \ge 0\) and every \(x^{*}\in \partial _{\varepsilon }f(x),\) there exists a countable set \(S\subset T\) such that

$$\begin{aligned} x^{*}\in \partial _{\varepsilon +\sup _{t\in S}f_{t}(x)-f(x)}\left( \sup _{t\in S}f_{t}\right) (x). \end{aligned}$$

Proof

We assume without loss of generality that \(x^{*}=\theta \) so that, for each integer number \(m\ge 1,\)

$$\begin{aligned} X= {\textstyle \bigcup _{r\ge 1}} rB_{X}\subset [f\ge f(x)-\varepsilon ] & \subset [f>f(x)-m^{-1} -\varepsilon ]\nonumber \\ & \subset {\textstyle \bigcup _{t\in T}} [f_{t}>f(x)-m^{-1}-\varepsilon ]. \end{aligned}$$
(28)

Observe that each of the sets \([f_{t}>f(x)-m^{-1}-\varepsilon ]\) is weakly open; hence, also open. If X is a reflexive Banach space, for each integer number \(r\ge 1,\) the ball \(rB_{X}\) is weakly compact and we can find a finite set \(S_{r,m}\subset T\) such that

$$\begin{aligned} rB_{X}\subset {\textstyle \bigcup _{t\in S_{r,m}}} [f_{t}>f(x)-m^{-1}-\varepsilon ]\subset {\textstyle \bigcup _{t\in S}} [f_{t}>f(x)-m^{-1}-\varepsilon ], \end{aligned}$$

where \(S:=\cup _{r,m\ge 1}S_{r,m}.\) Consequently, \(X\subset [\sup _{t\in S}f_{t}>f(x)-m^{-1}-\varepsilon ],\) and the arbitrariness of \(m\ge 1\) ensures that \(X\subset [\sup _{t\in S}f_{t}\ge f(x)-\varepsilon ].\) In other words, \(\theta \in \partial _{\varepsilon +\sup _{t\in S}f_{t}(x)-f(x)}\left( \sup _{t\in S}f_{t}\right) (x)\ \)and we are done.

If X is a (normed) separable space, then X is Lindelöf (that is, every open cover has a countable subcover [15]). Thus, since (28) entails

$$\begin{aligned} X\subset {\textstyle \bigcup _{t\in T}} [f_{t}>f(x)-m^{-1}-\varepsilon ], \end{aligned}$$

we find countable sets \(S_{m}\subset T\) and \(S:=\cup _{m\ge 1}S_{m}\) such that \(X\subset [\sup _{t\in S}f_{t}>f(x)-m^{-1}-\varepsilon ].\) Thus, as above, we deduce that \(\theta \in \partial _{\varepsilon +\sup _{t\in S}f_{t} (x)-f(x)}(\sup _{t\in S}f_{t})(x).\) \(\square \)

The following result uses conditions (10) and (15); that are, respectively,

$$\begin{aligned} {\bar{f}}=\sup _{t\in T}{\bar{f}}_{t}\text { and }\inf _{t\in T}f_{t}(x)>-\infty ,\ \text {\ for all }x\in {\text {dom}}f. \end{aligned}$$

Theorem 11

Suppose that X is a reflexive Banach space or a separable normed space. Given convex functions \(f_{t}:X\rightarrow {\mathbb {R}}_{\infty },\) \(t\in T,\ \)and \(f:=\sup _{t\in T}f_{t},\) we assume that (10) and (15) hold.Then, for all \(x\in {\text {dom}}f\) and\(\ \varepsilon \ge 0,\) we have

$$\begin{aligned} \partial _{\varepsilon }f(x)= {\textstyle \bigcup \limits _{\begin{array}{c} \lambda \in {\hat{\Delta }}\left[ \omega \cup \{\omega \}\right] \\ \\ \omega \subset T, \omega \text { countable} \end{array}}} \partial _{\varepsilon +f_{\lambda }(x)-f(x)}\left( f_{\lambda }+\textrm{I} _{{\text {dom}}f}\right) (x). \end{aligned}$$

Proof

Assuming \(x\in {\text {dom}}f\),\(\ \varepsilon \ge 0\) and \(x^{*} \in \partial _{\varepsilon }f(x),\) we initially consider that \(f_{t}\in \Gamma _{0}(X),\) for all \(t\in T.\) According to Lemma 10, there exists a countable set \(\omega \subset T,\) represented as a sequence \(\omega =(t_{n})_{n},\) such that

$$\begin{aligned} x^{*}\in \partial _{\varepsilon +\sup _{t\in \omega }f_{t}(x)-f(x)}\left( \sup _{t\in \omega }f_{t}\right) (x). \end{aligned}$$

Hence, by using Theorem 8 we conclude that there exists a \(\lambda \in \Delta \left[ \omega \cup \{\omega \}\right] \) such that \(x^{*} \in \partial _{\varepsilon +g_{\lambda }(x)-f(x)}\left( g_{\lambda } +\textrm{I}_{{\text {dom}}f}\right) (x),\) where

$$\begin{aligned} g_{\lambda }:= {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}f_{t}\text { if }{\text {*}}{supp}\lambda \subset \omega ,g_{\lambda }:= {\textstyle \sum \limits _{t\in \omega \cap {\text {*}}{supp}\lambda }} \lambda _{t}f_{t}+\lambda _{\omega }g_{\omega }\text { otherwise,} \end{aligned}$$
(29)

and \(g_{\omega }(x):=\inf _{S\subset \omega ,\left| S\right| <+\infty }\) \(\sup _{t\in \omega {\setminus } S}f_{t}.\) Consequently, since \(g_{\infty }\le f_{\omega },\) we deduce that \(g_{\lambda }\le f_{\lambda }\) and, as a result of (6), \(x^{*}\in \partial _{\varepsilon +f_{\lambda }(x)-f(x)}\left( f_{\lambda }+\textrm{I}_{{\text {dom}} f}\right) (x).\) This proves the nontrivial inclusion in the theorem’s statement.

In this second step, we consider the general case where condition (10) is satisfied. Then, due to (6), we have that

$$\begin{aligned} x^{*}\in \partial _{\varepsilon }f(x)\subset \partial _{\varepsilon +\bar{f}(x)-f(x)}{\bar{f}}(x)=\partial _{\varepsilon +{\bar{f}}(x)-f(x)}\left( \sup _{t\in T}{\bar{f}}_{t}\right) (x); \end{aligned}$$

hence, in particular, \(-\infty<f(x)-\varepsilon \le {\bar{f}}(x)<+\infty \) and the set \(T^{p}:=\{t\in T:{\bar{f}}_{t}>-\infty \}\) is nonempty. Thus, by Lemma 9,

$$\begin{aligned} {\bar{f}}=\sup _{t\in T}{\bar{f}}_{t}=\sup _{t\in T^{p}}g_{t}, \end{aligned}$$
(30)

where \(g_{t}:={\bar{f}}_{t}+\textrm{I}_{D_{0}}\in \Gamma _{0}(X)\) and \(D_{0}\) is the closed convex set defined in (27); that is, \(D_{0}=\cap _{t\in T{\setminus } T^{p}}{\text {*}}{cl}({\text {dom}}f_{t})\). Consequently, by (30) we obtain

$$\begin{aligned} x^{*}\in \partial _{\varepsilon +{\bar{f}}(x)-f(x)}(\sup \nolimits _{t\in T^{p} }g_{t})(x). \end{aligned}$$

Moreover, the arguments of the first part of the proof give rise to a countable set \(\omega :=(t_{n})_{n\ge 1}\subset T^{p}\) and \(\lambda \in \Delta \left[ \omega \cup \{\omega \}\right] \) such that (taking into account the fact that \({\text {dom}}f\subset {\text {dom}}{\bar{f}}\subset D_{0}),\)

$$\begin{aligned}&x^{*}\in \partial _{(\varepsilon +{\bar{f}}(x)-f(x))+{\hat{g}}_{\lambda } (x)-{\bar{f}}(x)}\left( {\hat{g}}_{\lambda }+\textrm{I}_{{\text {dom}} f}\right)&(x) \nonumber \\ &=\partial _{\varepsilon +{\hat{g}}_{\lambda }(x)-f(x)}\left( \hat{g}_{\lambda }+\textrm{I}_{{\text {dom}}f}\right) (x), \end{aligned}$$
(31)

where \({\hat{g}}_{\omega }:=\limsup _{n\rightarrow \infty }{\bar{f}}_{t_{n}}\) \((\le \limsup _{n\rightarrow \infty }f_{t_{n}})\),

$$\begin{aligned} {\hat{g}}_{\lambda }:= {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}{\bar{f}}_{t}\text { if }{\text {*}}{supp}\lambda \subset \omega ,\text { and }g_{\lambda }:= {\textstyle \sum \limits _{t\in \omega \cap {\text {*}}{supp}\lambda }} \lambda _{t}{\bar{f}}_{t}+\lambda _{\omega }{\hat{g}}_{\omega }\text { otherwise.} \end{aligned}$$

Observe that \({\hat{g}}_{\lambda }\le f_{\lambda }\) and so, again by (6), (31) yields

$$\begin{aligned} x^{*}\in \partial _{\varepsilon +f_{\lambda }(x)-f(x)}\left( f_{\lambda }+\textrm{I}_{{\text {dom}}f}\right) (x); \end{aligned}$$

that is, the nontrivial inclusion in the theorem statement also holds in the current case. \(\square \)

6 Application

We close this paper with an application to the duality in infinite convex optimization. We consider the optimization problem (P) posed in a lcs X as

$$\begin{aligned} \mu :=\inf _{f_{t}(x)\le 0,t\in T}f_{0}(x), \end{aligned}$$

where \(f_{0}\) and \(f_{t},\) \(t\in T\ \)(with \(0\notin T\)), are proper convex functions satisfying, for the sake of brevity, \(\mu \in {\mathbb {R}}\) and

$$\begin{aligned} \inf _{t\in T}f_{t}(x)>-\infty ,\ \text {for all }x\in {\text {dom}} f_{0}\cap {\text {dom}}f. \end{aligned}$$

Associated with (P),  we consider the dual conic dual (D) defined as

$$\begin{aligned} \nu :=\max _{\lambda \in \ell _{1}^{+}(T), \beta \in \ell _{+}^{s}(T)} \inf _{x\in {\text {dom}}f}f_{0}(x)+ {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda }} \lambda _{t}f_{t}(x)+\beta ((f_{t}(x))_{t\in T}). \end{aligned}$$

We refer to [5] and the references therein for more information on the duality of problem (P). The following corollary provides a strong duality relationship between (P) and its dual problem (D) under the usual (strong) Slater condition, requiring some \(x_{0}\in {\text {dom}}f_{0}\) such that

$$\begin{aligned} \sup _{t\in T}f_{t}(x_{0})<0. \end{aligned}$$
(32)

Theorem 12

With the notation above, we assume that (32) holds. Then \(\mu =\nu \) and (D) is solvable.

Proof

Since \(\mu \in {\mathbb {R}}\), we have that \(0=\inf _{x\in X}\max \{f_{0}-\mu ,\) \(\sup \nolimits _{t\in T}f_{t}\}\) or, equivalently,

$$\begin{aligned} \mu =\inf _{x\in X}~\max \{f_{0},\sup \nolimits _{t\in T}(f_{t}+\mu )\}. \end{aligned}$$
(33)

Let us consider the convex functions \(g_{t},\) \(t\in T^{0}:=T\cup \{0\}\) (assuming that \(0\notin T\)),defined as

$$\begin{aligned} g_{0}:=f_{0},g_{t}:=f_{t}+\mu \text {, }t\in T; \end{aligned}$$

hence, the family \(\{g_{t},\) \(t\in T\cup \{0\}\}\) satisfies condition (15). Therefore, by the minimax equality used in the proof of Theorem 5, there exists some \(\alpha ^{0}:=\lambda ^{0}+\beta ^{0}\in {\hat{\Delta }}(T^{0}),\) \(\lambda ^{0}\in \ell _{1}^{+}(T^{0})\) and \(\beta ^{0} \in \ell _{+}^{s}(T^{0})\) such that (33) reads

$$\begin{aligned} \mu =\inf _{x\in X}~\sup _{t\in T^{0}}g_{t}=\inf _{x\in {\text {dom}} f_{0}\cap {\text {dom}}f}~ {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda ^{0}}} \lambda _{t}^{0}g_{t}(x)+\beta ^{0}((g_{t}(x))_{t\in T^{0}}). \end{aligned}$$

Let us define \({\hat{\beta }}\in \ell _{+}^{s}(T)\) as \({\hat{\beta }}(u):=\beta ^{0}((\chi _{T}(t)u_{t})_{t\in T}),\) \(u\in \ell _{\infty }(T),\) so that the last equation above reads

$$\begin{aligned} \mu =\inf _{x\in {\text {dom}}f_{0}\cap {\text {dom}}f}~ {\textstyle \sum \limits _{t\in {\text {*}}{supp}\lambda ^{0}}} \lambda _{t}^{0}g_{t}(x)+{\hat{\beta }}((f_{t}(x))_{t\in T})+{\hat{\beta }} (\chi _{_{T}})\mu . \end{aligned}$$
(34)

Observe that \(\gamma _{0}:=\lambda ^{0}(0)>0\) because otherwise, due to (32), the last relation together with the singularity of \(\beta ^{0}\) would imply

$$\begin{aligned} \mu&\le {\textstyle \sum \limits _{t\in T\cap {\text {*}}{supp}\lambda ^{0}}} \lambda _{t}^{0}f_{t}(x_{0})+{\hat{\beta }}((f_{t}(x_{0}))_{t\in T})+\hat{\beta }(\chi _{_{T}})\mu \\&\le \left( {\textstyle \sum \limits _{t\in T}} \lambda _{t}^{0}+\beta (\chi _{_{T}})\right) \sup _{t\in T}f_{t}(x_{0} )+{\hat{\beta }}(\chi _{_{T}})\mu <{\hat{\beta }}(\chi _{_{T}})\mu =\beta ^{0} (\chi _{_{T^{0}}})\mu \le \mu . \end{aligned}$$

This yields the contradiction \(\mu <\mu .\) Consequently, (34) reads

$$\begin{aligned} \mu \!=\!\inf _{x\in {\text {dom}}f}\gamma _{0}f_{0}\!+\! {\textstyle \sum \limits _{t\in T\cap {\text {*}}{supp}\lambda ^{0}}} \lambda _{t}^{0}f_{t}(x)\!+\!(1\!-\!\gamma _{0}\!-\!{\hat{\beta }}(\chi _{_{T}}))\mu \!+\!\beta ((f_{t}(x))_{t\in T})\!+\!{\hat{\beta }}(\chi _{_{T}})\mu , \end{aligned}$$

and, dividing by \(\gamma _{0}\) \((>0),\) we obtain

$$\begin{aligned} \mu =\inf _{x\in {\text {dom}}f}f_{0}+ {\textstyle \sum \limits _{t\in T\cap {\text {*}}{supp}\lambda ^{0}}} \gamma _{0}^{-1}\lambda _{t}^{0}f_{t}(x)+\gamma _{0}^{-1}{\hat{\beta }} ((f_{t}(x))_{t\in T}). \end{aligned}$$

The conclusion follows by defining \(\lambda \in \ell _{1}^{+}(T)\) and \(\beta \in \ell _{+}^{s}(T)\) as \(\lambda _{t}:=\gamma _{0}^{-1}\lambda _{t}^{0},\) \(t\in T,\) and \(\beta :=\gamma _{0}^{-1}{\hat{\beta }}.\) \(\square \)

Remark 2

If T is finite in Theorem 12, then \(\beta \equiv 0\in \ell _{+} ^{s}(T)\) and we recover the classical duality of convex optimization problems satisfying the strong Slater condition.