New glimpses on convex infinite optimization duality

Abstract

Given a convex optimization problem $(P)$ in a locally convex topological vector space $X$ with an arbitrary number of constraints, we consider three possible dual problems of $(P),$ namely, the usual Lagrangian dual $(D)$, the perturbational dual $(Q)$, and the surrogate dual $(\Delta )$, the last one recently introduced in a previous paper of the authors (Goberna et al., J Convex Anal 21(4), 2014). As shown by simple examples, these dual problems may be all different. This paper provides conditions ensuring that $\inf (P)=\max (D)$, $\inf (P)=\max (Q),$ and $\inf (P)=\max (\Delta )$ (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing $\min (P)=\sup (Q)$ (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-infinite optimization problems (in which either the number of constraints or the dimension of $X$, but not both, is finite) and linear infinite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described.

1 Introduction

Given $m+1$, with $m\ge 1,$ convex lower semicontinuous (lsc) proper extended-real-defined functions $f,f_{1},\ldots ,f_{m}$ on a (real) separated locally convex topological vector space $X$ and a non-empty closed convex subset $C$ of $X,$ let us consider the convex semi-infinite problem (semi-infinite as the number of constraints is finite but the dimension of $X$ is infinite)

$$\begin{aligned} (P_{m}) \inf _{x}f(x), \hbox { s.t. } x\in C,\ f_{1}(x)\le 0,\ \ldots ,\ f_{m}(x)\le 0. \end{aligned}$$

Relaxing the inequality constraints, the Lagrangian dual of $(P_{m})$ is classically defined as

$$\begin{aligned} (P_{m}^{\prime }) \sup _{\lambda }\inf _{x\in C}\left( f(x)+\sum _{i=1}^{m}\lambda _{i}f_{i}(x)\right) , \hbox { s.t. } \lambda :=\left( \lambda _{1},\ldots ,\lambda _{m}\right) \in \mathbb {R}_{+}^{m}. \end{aligned}$$

Clearly, some care is necessary in order to give a precise sense to the expression $0\times (+\infty )$ that may appear in $(P_{m}^{\prime })$ formulation. Following Rockafellar [14, p.24], we may adopt the rule $0\times ( +\infty ) =0.$ Another possibility is to set $0\times ( +\infty ) =+\infty ,$ a choice made for instance by Zălinescu [15, p.39]. We shall denote by $(D_{m})$ and $(Q_{m})$ the corresponding versions of $(P_{m}^{\prime })$ associated with these rules. It holds that the corresponding optimal values of these problems satisfy

$$\begin{aligned} -\infty \le \sup (D_{m})\le \sup (Q_{m})\le \inf \left( P_{m}\right) \le +\infty . \end{aligned}$$

Given a non-empty closed convex subset $C$ of $X$ and a family $\{f_{t},\ t\in T\}$ of convex lsc proper functions on $X,$ where $T$ is a possibly infinite index set, let us consider now the general convex infinite problem

$$\begin{aligned} (P) \inf _{x}f(x), \hbox { s.t. } x\in C,\ f_{t}(x)\le 0,\ t\in T, \end{aligned}$$

whose feasible set is $F\cap C$ where

$$\begin{aligned} F:=\mathop {\textstyle \bigcap }\limits _{t\in T}\ \left[ f_{t}\le 0\right] =\left\{ x\in X:f_{t}(x)\le 0,\ t\in T\right\} . \end{aligned}$$

The associated Lagrange dual is classically defined as (see, e.g. [3, 5, 7], etc.),

$$\begin{aligned} (D) \sup _{\lambda }\inf _{x\in C}\left( f(x)+\sum _{t\in T}\lambda _{t}f_{t}(x)\right) , \hbox { s.t. } \lambda :=(\lambda _{t})_{t\in T}\in \mathbb {R}_{+}^{(T)}, \end{aligned}$$

with $\mathbb {R}_{+}^{(T)}$ denoting the positive cone of the space $\mathbb {R}^{(T)}$ of functions $\lambda :T\rightarrow \mathbb {R}$ whose support $\mathrm{supp}\,\lambda :=\{ t\in T:\lambda _{t}\ne 0\}$ is finite, and

$$\begin{aligned} \sum _{t\in T}\lambda _{t}f_{t}(x):=\left\{ \begin{array}{l@{\quad }l} 0, &{} \text {if }\lambda =0_{T}, \\ \sum \nolimits _{t\in \mathrm{supp}\,\lambda }\lambda _{t}f_{t}(x), &{} \text {if }\lambda \ne 0_{T}, \end{array} \right. \end{aligned}$$

where $0_{T}$ represents the null-function. It is worth noting that in the case of a finite number of constraints, that is $T=\{ 1,\ldots ,m\}$, the Lagrangian dual $(D)$ coincides with $(D_{m})$ while the generalization of $(Q_{m})$ is given by (e.g. [1, 7, 15])

$$\begin{aligned} (Q) \sup _{\lambda }\inf _{x\in C\cap M}\left( f(x)+\sum _{t\in T}\lambda _{t}f_{t}(x)\right) , \hbox { s.t. } \lambda \in \mathbb {R}_{+}^{(T)}, \end{aligned}$$

where $M:=\mathop {\displaystyle \bigcap }\nolimits _{t\in T}\mathrm{dom}\,f_{t}.$ Observe that if $M\supset C\cap \mathrm{dom}\,f,$ then $(D)\equiv (Q).$

Finally, replacing the set $\mathbb {R}_{+}^{(T)}$ by $\mathbb {P}(T):=\mathbb { R}_{+}^{(T)}\backslash \{ 0_{T} \}$ in the dual problem $(D),$ the following surrogate dual problem $(\Delta )$ was introduced in [7]:

$$\begin{aligned} (\Delta ) \sup _{\lambda }\inf _{x\in C}\left( f(x)+\sum _{t\in T}\lambda _{t}f_{t}(x)\right) , \hbox { s.t. } \lambda \in \mathbb {P}(T). \end{aligned}$$

One always has the following relations among the optimal value of these problems:

$$\begin{aligned} -\infty \le \sup (\Delta )\le \sup (D)\le \sup (Q)\le \inf \left( P\right) \le +\infty . \end{aligned}$$

(1.1)

The paper is organized as follows. Assuming that $\inf (P)<+\infty ,$ Section 2 is concerned with the characterization of the so-called strong duality property for the three pairs of dual problems, which respectively accounts for the relations $\inf (P)=\max (D),$ $\inf (P)=\max (Q),$ and $\inf (P)=\max (\Delta )$ (i.e., the optimal values coincide and the dual optimal values are attained) in terms of a property called $w^{*}$-closedness regarding to suitable sets (see [1, 13]). This is the purpose of Theorem 1, the main result in Sect. 2. Section 3 is devoted to the relation $\min (P)=\sup (\Delta )$ (i.e., we have again dual equality plus attainability of the primal optimal value). Theorem 2 provides sufficient conditions based on the notion of quasicontinuity and recession assumptions. This result improves the one obtained in [7, Theorem 4.7] in the sense that we do not assume that $\inf (P)<+\infty$ but only that $\sup (\Delta )<+\infty .$ It turns out that the use of this weakened assumption has important consequences. Section 4 shows applications of Theorem 2. In fact, Corollary 1 provides a new general form of the Clark–Duffin’s Theorem in terms of the finite intersection property (Corollary 2), while Corollaries  and 4 deal with the existence of solutions of convex infinite systems. Section 5 is concerned with the perturbations of the convex infinite problem $(P)$ (Corollary 5), leading us to the characterization of the property $\min (P)=\sup (Q)$ and its perturbational relatives in terms of $w^{*}$-closedness regarding to a set (Theorem 3 and Corollary 6). In this way, Theorems 2 and 3, and Corollaries 5 and 6 complete and improve the results obtained in Section 5 of [7]. In the last section we apply the previous results to linear infinite optimization problems. Corollaries 7 and provide our most important results in this field.

2 The inf-max property

We shall start this section with some necessary notation and preliminaries. Given a non-empty subset $A$ of a (real) separated locally convex tvs, we denote by $\mathrm{co}\,A, \, \mathrm{cone}\,A, \, \mathrm{aff}\,A, \, A^{+},$ and $A^{-}$, the convex hull of $A$, the convex cone generated by $A\cup \{0_{X}\}$, the smallest linear manifold containing $A,$ the positive polar cone of $A$, and the negative polar cone of $A$, respectively$.$ If $A\subset X^{*},$ where $X^{*}$ is the topological dual of $X$, it holds that $A^{++}=A^{--}=\mathrm{cl}\,^{w^{*}}\mathrm{cone}\,A.$ We denote by $C_{\infty }$ the recession cone of the non-empty closed convex set $C.$

Having a function $g:X\rightarrow \overline{\mathbb {R}}:=\mathbb {R\cup \{\pm \infty \}},$ we denote by $\mathrm{epi}\,g,$ $\mathrm{epi}\,_{s}g,$ and $g^{*}$ the epigraph, the strict epigraph, and the Legendre–Fenchel conjugate of $g$, respectively. The function $g$ is proper if $\mathrm{epi}\, g\ne \emptyset$ and never takes the value $-\infty$, it is convex if $\mathrm{epi}\,g$ is convex, and it is lower semicontinuous (lsc, in brief) if $\mathrm{epi}\,g$ is closed. We denote by $\Gamma (X)$ the class of lsc proper convex functions on $X$. The function $\mathrm{cl}\,\mathrm{co}\, g:X\longrightarrow \overline{\mathbb {R}}$ is the lsc convex function such that $\mathrm{epi}\,(\mathrm{cl}\,\mathrm{co}\,g)=\mathrm{cl}\,\mathrm{co}\,( \mathrm{epi}\,g).$

The indicator function of $A\subset X$ is represented by $\mathrm {i}_{A}$ (i.e. $\mathrm {i}_{A}(x)=0$ if $x\in A,$ and the $\mathrm {i}_{A}(x)=+\infty$ if $x\notin A$), and the support function of $A$ is the conjugate of its indicator, i.e. $\mathrm {i}_{A}^{*}.$ One has $\mathrm {i}_{A}^{*}=\mathrm {i}_{ \mathrm{co}\,A}^{*}=\mathrm {i}_{\mathrm{cl}\,(\mathrm{co}\,A)}^{*}.$

Given $g\in \Gamma (X)$, we denote by $g_{\infty }$ its recession function, i.e. the convex function whose epigraph is $(\mathrm{epi}\,g)_{\infty }$. One has $g_{\infty }:=\mathrm {i}_{\mathrm{dom}\, g^{*}}^{*}$ (e.g. [15, Exercise 2.35]), and

$$\begin{aligned} \left[ g_{\infty }\le 0\right] =\left( \mathrm{dom}\,g^{*}\right) ^{-}=\left( \mathrm{cone}\,\mathrm{dom}\,g^{*}\right) ^{-}, \end{aligned}$$

yielding

$$\begin{aligned} \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{dom}\,g^{*}=\left[ g_{\infty }\le 0\right] ^{-}. \end{aligned}$$

Moreover $[ g_{\infty }\le 0] = [ g\le \lambda ] _{\infty }$ for all $\lambda$ such that $[ g\le \lambda ] \ne \emptyset$.

Associated with the dual problems $(\Delta ),(D)$ and $(Q)$ we introduce the functions $h,k,\ell :X^{*}\rightarrow \overline{\mathbb {R}}$, respectively defined by

$$\begin{aligned} \begin{array}{l@{\quad }l} h:= &{} \inf _{\lambda \in \mathbb {P}(T)}\left( f_{C}+\sum \nolimits _{t\in T}\lambda _{t}f_{t}\right) ^{*},\\ k:= &{} \inf _{\lambda \in \mathbb {R}_{+}^{(T)}}\left( f_{C}+\sum \nolimits _{t\in T}\lambda _{t}f_{t}\right) ^{*},\\ \ell := &{} \inf _{\lambda \in \mathbb {R}_{+}^{(T)}}\left( f_{C\cap M}+\sum \nolimits _{t\in T}\lambda _{t}f_{t}\right) ^{*}, \end{array} \end{aligned}$$

(2.1)

where $f_{C}:=f+\mathrm {i}_{C}$ and $f_{C\cap M}=f+\mathrm {i}_{C\cap M}.$

The following properties can easily be proved following the same arguments that in [7, Lemmas 3.1 and 3.2] and taking into account the assumptions on $C$ and the functions $f,f_{t},t\in T:$

• (1) $\ell ,$ $k$ and $h$ are convex, and $\ell \le k\le h,$

• (2) $-\ell ( 0_{X^{*}}) =\sup (Q),$ $-k( 0_{X^{*}}) =\sup (D),$ and $-h( 0_{X^{*}}) =\sup (\Delta ),$

• (3) $\ell ^{*}=k^{*}=h^{*}=f_{C\cap F},$

• (4) $-\ell ^{**}( 0_{X^{*}}) =-k^{**}( 0_{X^{*}}) =-h^{**}( 0_{X^{*}}) =\inf ( P)$.

The functions $h,k$ and $\ell$ can be improper, possibility which was excluded in [7]. For instance, if $C\cap \mathrm{dom}\,f=\emptyset ,$ we obviously have $h=k=\ell \equiv -\infty$. In the following simple example, the functions $f_{C}+\sum _{t\in T}\lambda _{t}f_{t}$ are all proper:

Example 1

Let $X=C=\mathbb {R}^{2},$ $f( x) =x_{1},T=\{ 1\} ,$ and $f_{1}( x) =\exp (x_{2})$. We have $F=\emptyset ,$ and so $\inf (P) =\inf \{ x_{1}:\exp (x_{2}) \le 0\} =+\infty .$ Moreover

$$\begin{aligned} \sup (\Delta )=\sup _{\lambda >0}\inf _{x\in \mathbb {R}^{2}}(x_{1}+\lambda \exp \left( x_{2}\right) )=-\infty \end{aligned}$$

and

$$\begin{aligned} \sup (D)=\sup (Q)=\sup _{\lambda \ge 0}\inf _{x\in \mathbb {R} ^{2}}(x_{1}+\lambda \exp \left( x_{2}\right) )=-\infty . \end{aligned}$$

For $\lambda >0,$ Theorem 2.3.1 [(v),(viii)] in [15] allows us to write

$$\begin{aligned} \left( f+\lambda f_{1}\right) ^{*}\left( x_{1}^{*},x_{2}^{*}\right) =\mathrm {i}_{\{1\}}(x_{1}^{*})+\lambda \exp ^{*}(\lambda ^{-1}x_{2}^{*}), \end{aligned}$$

where we denote by $\exp ^{*}$ the conjugate of the exponential function $\exp$ , i.e.

$$\begin{aligned} \exp ^{*}\left( u\right) =\left\{ \begin{array}{l@{\quad }l} +\infty , &{} u<0, \\ 0, &{} u=0, \\ u\ln u-u, &{} u>0. \end{array} \right. \end{aligned}$$

Therefore

$$\begin{aligned} \left( f+\lambda f_{1}\right) ^{*}\left( x_{1}^{*},x_{2}^{*}\right) =\left\{ \begin{array}{l@{\quad }l} +\infty , &{} x_{1}^{*}\ne 1\hbox { or }x_{2}^{*}<0, \\ 0, &{} x_{1}^{*}=1\hbox { and }x_{2}^{*}=0, \\ x_{2}^{*}\ln x_{2}^{*}-x_{2}^{*}-x_{2}^{*}\ln \lambda , &{} x_{1}^{*}=1\hbox { and }x_{2}^{*}>0, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} h\left( x_{1}^{*},x_{2}^{*}\right) =\inf _{\lambda >0}\left( f+\lambda f_{1}\right) ^{*}(x_{1}^{*},x_{2}^{*})=\left\{ \begin{array}{l@{\quad }l} +\infty , &{} x_{1}^{*}\ne 1\text { or }x_{2}^{*}<0, \\ 0, &{} x_{1}^{*}=1\text { and }x_{2}^{*}=0, \\ -\infty , &{} x_{1}^{*}=1\text { and }x_{2}^{*}>0. \end{array} \right. \end{aligned}$$

We clearly have $h=k=\ell$ and $h^{*}=k^{*}=\ell ^{*}=+\infty =f+ i_{C\cap F}.$ Observe that $h,k,\ell$ are convex but neither proper nor lsc.

We also introduce the sets

$$\begin{aligned} \begin{array}{c@{\quad }c} \mathfrak {A}:= &{} \mathop {\textstyle \bigcup }\limits _{\lambda \in \mathbb {P}(T)}\mathrm{epi}\, \left( f_{C}+\sum \nolimits _{t\in T}\lambda _{t}f_{t}\right) ^{*},\\ \mathfrak {B}:= &{} \mathop {\textstyle \bigcup }\limits _{\lambda \in \mathbb {R}_{+}^{(T)}}\mathrm{ epi}\,\left( f_{C}+\sum \nolimits _{t\in T}\lambda _{t}f_{t}\right) ^{*},\\ \mathfrak {C}:= &{} \mathop {\textstyle \bigcup }\limits _{\lambda \in \mathbb {R}_{+}^{(T)}}\mathrm{ epi}\,\left( f_{C\cap M}+\sum \nolimits _{t\in T}\lambda _{t}f_{t}\right) ^{*}. \end{array} \end{aligned}$$

It holds that

$$\begin{aligned} \mathrm{epi}\,_{s}h\subset \mathfrak {A}\subset \mathrm{epi}\,h,\quad \mathrm{epi}\,_{s}k\subset \mathfrak {B}\subset \mathrm{epi}\,k,\quad \mathrm{epi}\,_{s}\ell \subset \mathfrak {C}\subset \mathrm{epi}\,\ell , \end{aligned}$$

and denoting by $\overline{h},\overline{k}$ and $\overline{\ell }$ the $w^{*}$-lsc hull of $h,k$ and $\ell ,$ respectively, we have

$$\begin{aligned} \mathrm{epi}\,\overline{h}=\mathrm{cl}\,^{w^{*}}\mathfrak {A} ,\quad \mathrm{epi}\,\overline{k}=\mathrm{cl}\,^{w^{*}}\mathfrak {B },\quad \mathrm{epi}\,\overline{\ell }=\mathrm{cl}\,^{w^{*}} \mathfrak {C.} \end{aligned}$$

(2.2)

Assuming that $C\cap F\cap \mathrm{dom}\,f\ne \emptyset$ one has, by the convexity of $h,k$ and $\ell$ and (3) above,

$$\begin{aligned} \overline{h}=\overline{k}=\overline{\ell }=(f_{C\cap F})^{*}=h^{**}=k^{**}=\ell ^{**}. \end{aligned}$$

(2.3)

We will need the following notion ([1], see also [13]).

Definition 1

Given two subsets $A,B$ of a topological space, $A$ is said to be closed regarding to $B$ if $B\cap \mathrm{cl}\,A=B\cap A.$

We are now in a position to state the main result of this section.

Theorem 1

Assume that $\inf (P) <+\infty .$ The following assertions are equivalent:

• (i) $\mathfrak {A}$ (resp. $\mathfrak {B},$ resp. $\mathfrak {C}$ ) is $w^{*}$-closed regarding to the set $\{ 0_{X^{*}}\}\times \mathbb {R}.$

• (ii) $\inf (P) =\max (\Delta )$ (resp. $\inf (P) =\max (D),$ resp. $\inf (P) =\max (Q)$ ), including the value $-\infty$.

Proof

We only give the proof relative to $(\Delta ),$ the two other ones being similar.

Since $\inf (P) <+\infty ,$ one has $C\cap F\cap \mathrm{dom}\,f\ne \emptyset$ and, by (2.3), $\overline{h}=(f_{C\cap F})^{*}.$

Assume first that $\inf (P) =-\infty$. By (1.1) we have

$$\begin{aligned} \inf _{C}\left( f+\sum _{t\in T}\lambda _{t}f_{t}\right) =-\infty \text { for any }\lambda \in \mathbb {P}(T), \end{aligned}$$

and so, $\inf (P) =-\infty =\max (\Delta ).$ On the other hand, $\overline{h}(0_{X^{*}}) =-\inf (P) =+\infty$ and, by (2.2),

$$\begin{aligned} \left( \left\{ 0_{X^{*}}\right\} \times \mathbb {R}\right) \cap \mathrm{ cl}\,^{w^{*}}\mathfrak {A=}\left( \left\{ 0_{X^{*}}\right\} \times \mathbb {R}\right) \cap \mathrm{epi}\,\overline{h}=\emptyset , \end{aligned}$$

implying that $\mathfrak {A}$ is $w^{*}$-closed regarding to $\{ 0_{X^{*}}\} \times \mathbb {R}.$ So, in the case that $\inf (P) =-\infty ,$ we have proved that statements (i) and (ii) are simultaneously true.

Assume now that $\alpha :=\inf (P) \in \mathbb {R}.$ By (4), (2.2) and (2.3) we have

$$\begin{aligned} \left( 0_{X^{*}},-\alpha \right) \in \mathrm{epi}\,h^{**}= \mathrm{epi}\,\overline{h}=\mathrm{cl}\,^{w^{*}}\mathfrak {A.} \end{aligned}$$

Assuming that (i) holds we get $(0_{X^{*}},-\alpha ) \in \mathfrak {A,}$ and there exists $\overline{\lambda }\in \mathbb {P}(T)$ such that $(f_{C}+\sum _{t\in T}\overline{\lambda }_{t}f_{t}) ^{*} (0_{X^{*}}) \le -\alpha .$ This yields

$$\begin{aligned} \sup (\Delta )\le \inf \left( P\right) =\alpha \le \inf _{C}\left\{ f_{C}+\sum _{t\in T}\overline{\lambda }_{t}f_{t}\right\} \le \sup (\Delta ) \end{aligned}$$

and (ii) is proved.

Assume now that (ii) holds and let $(0_{X^{*}},r) \in \mathrm{cl}\,^{w^{*}}\mathfrak {A}.$ By (4), (2.2) and (2.3), one has $(0_{X^{*}},r) \in \mathrm{epi}\,h^{**}$ and $-\inf (P) =h^{**} (0_{X^{*}}) \le r.$ By (ii), there exists $\overline{\lambda }\in \mathbb {P}(T)$ such that $-\inf (P) = (f_{C}+\sum _{t\in T}\overline{\lambda } _{t}f_{t}) ^{*}(0_{X^{*}})$, and we have

$$\begin{aligned} \left( 0_{X^{*}},r\right) \in \mathrm{epi}\,\left( f_{C}+\sum _{t\in T} \overline{\lambda }_{t}f_{t}\right) ^{*}\subset \mathfrak {A,} \end{aligned}$$

proving that (i) holds. $\square$

The next examples compare the characterizations of the inf-max property provided by Theorem 1 with the so-called Slater condition:

$$\begin{aligned} \exists \overline{x}\in C\cap \mathrm{dom}\,f\text { such that }f_{t}\left( \overline{x}\right) <0 \forall t\in T. \end{aligned}$$

When $T$ is finite, it is known that $-\infty \le \inf (P) =\max (Q)<+\infty$ whenever the above Slater condition holds [15, Theorem 2.9.3].

Example 2

Let $X=C=\mathbb {R}^{2}, \, f (x) =\exp (x_{2}), \, T=\{ 1\} ,$ and $f_{1} (x) =x_{1}+\mathrm {i}_{\mathbb {R\times R}_{+}} (x)$. We have $\inf (P) =\inf \{ \exp (x_{2}) :x_{1}\le 0,x_{2}\ge 0\} =1.$ As the minimum is achieved, we may write $\min (P) =1,$ with primal optimal set $S (P) =\mathbb {R}_{-}\mathbb {\times }\{ 0 \}.$ In order to check the conditions of Theorem 1, we must compute the functions $(f+\lambda f_{1}) ^{*}$ for all $\lambda \ge 0.$ If $\lambda >0,$ then

$$\begin{aligned} \left( f+\lambda f_{1}\right) ^{*}\left( x^{*}\right) =\left\{ \begin{array}{l@{\quad }l} x_{2}^{*}\ln x_{2}^{*}-x_{2}^{*}, &{} x_{1}^{*}=\lambda ,\ x_{2}^{*}>1, \\ -1, &{} x_{1}^{*}=\lambda ,\ x_{2}^{*}\le 1, \\ +\infty , &{} {\hbox {otherwise.}} \end{array} \right. \end{aligned}$$

The above equation remains valid for $\lambda =0$ under the rule $0\times (+\infty ) =+\infty$ (as in $(Q)$), but not under the rule $0\times (+\infty ) =0$ (as in $(D)$), in which case

$$\begin{aligned} \left( f+0f_{1}\right) ^{*}\left( x^{*}\right) =\left\{ \begin{array}{l@{\quad }l} x_{2}^{*}\ln x_{2}^{*}-x_{2}^{*}, &{} x_{1}^{*}=0,\ x_{2}^{*}>0, \\ 0, &{} x_{1}^{*}=x_{2}^{*}=0, \\ +\infty , &{} {\hbox {otherwise.}} \end{array} \right. \end{aligned}$$

Using again the symbol $\exp ^{*}$ for the conjugate of the exponential function $\exp$ we have

$$\begin{aligned} \mathfrak {A}&= \mathbb {R}_{++}\times \left( \mathrm{epi}\,(\exp ^{*})+ \mathbb {R}_{+}\left( -1,0\right) \right) ,\\ \mathfrak {B}&= \mathfrak {A}\cup (\left\{ 0\right\} \times \mathrm{epi}\, (\exp ^{*})),\\ \mathfrak {C}&= \mathbb {R}_{+}\times \left( \mathrm{epi}\,(\exp ^{*})+ \mathbb {R}_{+}\left( -1,0\right) \right) =\mathrm{cl}\,^{w^{*}} \mathfrak {A}. \end{aligned}$$

The closedness of $\mathfrak {C}$ entails its closedness regarding $\{(0,0)\} \times \mathbb {R},$ while $\mathfrak {A}$ and $\mathfrak {B}$ do not enjoy this property as $\mathfrak {A}\cap (\{ (0,0)\} \times \mathbb {R)}=\emptyset , \, \mathfrak {B}\cap (\{(0,0)\} \times \mathbb {R)}= \{(0,0,r):r\ge 0\}$, and

$$\begin{aligned} (\mathrm{cl}\,^{w^{*}}\mathfrak {A})\cap (\left\{ \left( 0,0\right) \right\} \times \mathbb {R)}=(\mathrm{cl}\,^{w^{*}} \mathfrak {B})\mathfrak {\cap }(\left\{ \left( 0,0\right) \right\} \times \mathbb {R)}=\left\{ \left( 0,0,r\right) :r\ge -1\right\} . \end{aligned}$$

Thus, by Theorem 1, $\inf ( P) =\max (Q)$ holds while both $\inf (P) =\max (\Delta )$ and $\inf (P) =\max (D)$ fail. Indeed, $\inf _{\mathbb {R}^{2}}\{ f+\lambda f_{1}\} =-\infty$ for all $\lambda >0,$ and

$$\begin{aligned} \inf \nolimits _{\mathbb {R}^{2}}\left\{ f+0f_{1}\right\} =\left\{ \begin{array}{c@{\quad }c} 0, &{} \hbox {for } \left( D\right) , \\ 1, &{} \hbox {for } \left( Q\right) . \end{array} \right. \end{aligned}$$

So, $\inf (P) =\max (Q)=1$ (attained for $\lambda =0$) while $\sup (D) =\max (D)=0$ (attained for $\lambda =0$) and $\sup (\Delta ) =-\infty .$ Hence, the Slater condition does not guarantee the relation $\inf (P) =\max (D),$ neither $\sup (D) =\sup (Q)$ nor $\sup (D) =\sup (\Delta )$.

Example 3

Let $X=C=\mathbb {R}, \, f(x) =\exp (x), \, T= \{1\}$, and $f_{1}(x) =x.$ Then, the primal problem is

$$\begin{aligned} (P) \inf _{x}\exp \left( x\right) , \hbox { s.t. } x\le 0, \end{aligned}$$

with associated dual problems

$$\begin{aligned} (\Delta ) \sup _{\lambda }\inf _{x\in \mathbb {R}}\left( \exp \left( x\right) +\lambda x)\right) , \hbox { s.t. } \lambda >0, \end{aligned}$$

and

$$\begin{aligned} (D)\equiv (Q) \sup _{\lambda }\inf _{x\in \mathbb {R}}\left( \exp \left( x\right) +\lambda x)\right) , \hbox { s.t. } \lambda \ge 0. \end{aligned}$$

One has

$$\begin{aligned} -\infty =\sup (\Delta )<0=\max (D)=\max (Q)=\inf \left( P\right) . \end{aligned}$$

Observe that, for any $\lambda >0,$ one has by [15, Theorem 2.3.1(vii)]

$$\begin{aligned} \left( f+\lambda f_{1}\right) ^{*}\left( x^{*}\right) =f^{*}(x^{*}-\lambda ), \end{aligned}$$

so that $\mathrm{epi}\,( f+\lambda f_{1}) ^{*}=\mathrm{ epi}\,(\exp ^{*})+(\lambda ,0)$. Thus,

$$\begin{aligned} \begin{array}{l@{\quad }l} \mathfrak {A}&=\mathop {\textstyle \bigcup }\limits _{\lambda >0}\mathrm{epi}\,\left( f+\lambda f_{1}\right) ^{*}=\mathrm{epi}\,(\exp ^{*})+(]0,+\infty [\times \left\{ 0\right\} ), \end{array} \end{aligned}$$

and, analogously, $\mathfrak {B}=\mathfrak {C}=\mathrm{epi}\,(\exp ^{*})+(\mathbb {R}_{+}\times \{0\} ).$ Since

$$\begin{aligned} \mathfrak {A}\cap \left( \left\{ 0\right\} \times \mathbb {R}\right) =\mathbb { \emptyset }\ne \left\{ 0\right\} \times \mathbb {R}_{+}=(\mathrm{cl}\, ^{w^{*}}\mathfrak {A})\cap \left( \left\{ 0\right\} \times \mathbb {R}\right) , \end{aligned}$$

$\mathfrak {A}$ is not closed regarding $\{0\} \times \mathbb {R}$ while $\mathfrak {B}=\mathfrak {C}$ is closed and, a fortiori, closed regarding $\{0\} \times \mathbb {R}.$ Observe that, once again in this case, Slater condition holds and, however, $\sup (\Delta )\ne \sup (D).$

Example 4

Let $X=\mathbb {R}, \, C=[-1,1], \, f(x) =-x, \, T=\{1\} ,$ and $f_{1}(x) =x$ if $x\ge 0, \, f_{1}(x) =0$ if $x<0.$ Now we have

$$\begin{aligned} (P) \inf _{x}\{-x, \hbox { s.t. } x\in [-1,1],\ x\le 0\}, \end{aligned}$$

with associated dual problems

$$\begin{aligned} (D)\equiv (Q) \sup _{\lambda }\inf _{-1\le x\le 1}\left( -x+\lambda f_{1}\left( x\right) \right) , \hbox { s.t. } \lambda \ge 0, \end{aligned}$$

$$\begin{aligned} (\Delta ) \sup _{\lambda }\inf _{-1\le x\le 1}\left( -x+\lambda f_{1}\left( x\right) \right) , \hbox { s.t. } \lambda >0. \end{aligned}$$

One has $\inf _{-1\le x\le 1}( -x+\lambda f_{1}(x)) =0=\inf (P)$ for any $\lambda \ge 1.$ Consequently,

$$\begin{aligned} \max (\Delta )=\max (D)=\max (Q)=\min \left( P\right) =0. \end{aligned}$$

In fact, for any $\lambda \ge 0,$ one has

$$\begin{aligned} \left( f+\lambda f_{1}\right) ^{*}\left( x^{*}\right) =\left\{ \begin{array}{l@{\quad }l} 0, &{} -1\le x^{*}\le \lambda -1, \\ +\infty , &{} {\hbox {otherwise,}} \end{array} \right. \end{aligned}$$

and so $\mathfrak {A}=\mathfrak {B}=\mathfrak {C}=[ -1,+\infty [ \times \mathbb {R}_{+}$ is closed. However, Slater condition is not satisfied, and this shows that it is sufficient, but not necessary, for having $\inf (P) =\max (Q)<+\infty$.

Example 5

Let $X=C=\mathbb {R}, \, f(x) =x^{2}, \, T=\{1\} ,$ and $f_{1}(x) =x_{+}-1.$ Thus, Slater condition holds and we have

$$\begin{aligned}&\displaystyle (P) \inf _{x}x^{2}, \hbox { s.t. } x_{+}-1\le 0,\\&\displaystyle (\Delta ) \sup _{\lambda }\inf _{x\in \mathbb {R}}\left\{ x^{2}+\lambda \left( x_{+}-1\right) \right\} , \hbox { s.t. } \lambda >0, \end{aligned}$$

and

$$\begin{aligned} (D)\equiv (Q) \sup _{\lambda }\inf _{x\in \mathbb {R}}\left\{ x^{2}+\lambda \left( x_{+}-1\right) \right\} , \hbox { s.t. } \lambda \ge 0. \end{aligned}$$

By the Moreau–Rockafellar Theorem (see, for instance, [1, Theorem 7.6])

$$\begin{aligned} \mathrm{epi}\,\left( f+\lambda f_{1}\right) ^{*}=\mathrm{epi}\,f^{*}+ \mathrm{epi}\,\left( \lambda f_{1}\right) ^{*}=\mathrm{epi}\,f^{*}+\lambda \mathrm{epi}\,f_{1}^{*} \end{aligned}$$

for any $\lambda >0.$ Setting $\mathrm{pos}\,(x) =x_{+}, \, x\in \mathbb {R},$ one has $f_{1}=\mathrm{pos}\,(\cdot )-1, \,f_{1}^{*}=\mathrm{pos}\,^{*}(\cdot )+1=\mathrm {i}_{[ 0,1]}+1,$ and so $\mathrm{epi}\,f_{1}^{*}=[0,1] \times [ 1,+\infty [.$ Thus,

$$\begin{aligned} \begin{array}{l@{\quad }l} \mathfrak {A} &{} =\mathop {\textstyle \bigcup }\limits _{\lambda >0}\mathrm{epi}\,\left( f+\lambda f_{1}\right) ^{*} \\ &{} =\mathrm{epi}\,f^{*}+ \mathop {\textstyle \bigcup }\limits _{\lambda >0}\left[ 0,\lambda \right] \times [ \lambda ,+\infty [ \\ &{} =\left\{ \left( x^{*},r\right) :\frac{\left( x^{*}\right) ^{2}}{4} \le r\right\} +\left\{ \left( x^{*},r\right) :\left( x^{*},r\right) \ne \left( 0,0\right) ,\ 0\le x^{*}\le r\right\} \\ &{} =\left\{ \left( x^{*},r\right) :x^{*}\le 2,\frac{\left( x^{*}\right) ^{2}}{4}<r\right\} \cup \left\{ \left( x^{*},r\right) :0<x^{*}-2\le r\right\} \end{array} \end{aligned}$$

while

$$\begin{aligned} \begin{array}{l@{\quad }l} \mathfrak {B}=\mathfrak {C} &{} =\mathfrak {A}\cup \mathrm{epi}\,f^{*}\\ &{} =\left\{ \left( x^{*},r\right) :x^{*}\le 2,\frac{\left( x^{*}\right) ^{2}}{4}\le r\right\} \cup \left\{ \left( x^{*},r\right) :0\le x^{*}-2\le r\right\} . \end{array} \end{aligned}$$

So, $\mathfrak {B}=\mathfrak {C}$ is closed and equal to $\mathrm{epi}\,(f+\mathrm {i}_{] -\infty ,1]})^{*}= \, \mathrm{cl}\,^{w^{*}}\mathfrak {A}.$ Since

$$\begin{aligned} \mathfrak {A}\cap \left( \left\{ 0\right\} \times \mathbb {R}\right) =\left\{ 0\right\} \mathbb {\times } ] 0,+\infty [ \ne \left\{ 0\right\} \times \mathbb {R}_{+}=(\mathrm{cl}\,^{w^{*}}\mathfrak {A})\cap \left( \left\{ 0\right\} \times \mathbb {R}\right) , \end{aligned}$$

$\mathfrak {A}$ is not closed regarding to $\{0\} \times \mathbb {R}.$ This is the reason why $\sup (\Delta )$ is not attained while $\sup (D)=\sup (Q)$ is attained (Fig. 1).

Fig. 1

The set $\mathfrak {A}$ in Example 5

3 The min-sup property

With each convex infinite problem

$$\begin{aligned} (P) \inf _{x}f(x), \hbox { s.t. } x\in C,\ f_{t}(x)\le 0,\ t\in T, \end{aligned}$$

we associate the closed convex cone

$$\begin{aligned} \mathrm{rec}\,(P):=\left[ f_{\infty }\le 0\right] \cap C_{\infty }\cap \left( \mathop {\textstyle \bigcap }\limits _{t\in T}\left[ \left( f_{t}\right) _{\infty }\le 0 \right] \right) . \end{aligned}$$

Obviously, $\mathrm{rec}\,(P)=\{0_{X}\}$ if and only if there is no common direction of recession to all the data of $(P),$ namely: $f,C,f_{t},t\in T$, and it is a linear space if and only if any direction of recession, say $d,$ which is common to all the data of $(P),$ if any, is equilibrated in the sense that the opposite direction $-d$ is also common to all the data of $(P).$

With the convex infinite system formed by the constraints of $(P),$

$$\begin{aligned} \sigma :=\left\{ f_{t}(x)\le 0,\ t\in T;\ x\in C\right\} , \end{aligned}$$

is associated the so-called characteristic cone ([2, 3, 6], etc.)

$$\begin{aligned} K:=\mathrm{cone}\,\left\{ \mathrm{epi}\,(\mathrm {i}_{C}^{*})\cup \left( \mathop {\textstyle \bigcup }\limits _{t\in T}\mathrm{epi}\,f_{t}^{*}\right) \right\} =\mathrm{ epi}\,(\mathrm {i}_{C}^{*})+\mathrm{cone}\,\left( \mathop {\textstyle \bigcup }\limits _{t\in T} \mathrm{epi}\,f_{t}^{*}\right) . \end{aligned}$$

Now we will make precise some links between $K$ and the epigraph of the function $h$ defined in (2.1). To this end we will just assume that (compare with [5] and [7])

$$\begin{aligned} f_{C}+\sum _{t\in T}\lambda _{t}f_{t}\text { is proper for any}\lambda \in \mathbb {P}(T). \end{aligned}$$

(3.1)

Given $\lambda \in \mathbb {P}(T)$ we denote by $\square _{t\in T}(\lambda _{t}f_{t})^{*}$ the infimal convolution of the functions $(\lambda _{t}f_{t})^{*}, \, t\in \mathrm{supp}\,\lambda ,$ i.e.

$$\begin{aligned} \left( \square _{t\in T}\left( \lambda _{t}f_{t}\right) ^{*}\right) \left( x^{*}\right) =\inf \left\{ \mathop {\textstyle \sum }\limits _{t\in \mathrm{supp}\, \lambda }\left( \lambda _{t}f_{t}\right) ^{*}\left( x_{t}^{*}\right) :\mathop {\textstyle \sum }\limits _{t\in \mathrm{supp}\,\lambda }x_{t}^{*}=x^{*}\right\} . \end{aligned}$$

Then, by [15, Theorem 2.8.7],

$$\begin{aligned} \left( f_{C}+\sum _{t\in T}\lambda _{t}f_{t}\right) ^{*}=\mathrm{cl}\,^{w^{*}}\left( f^{*}\square \mathrm {i}_{C}^{*}\square \left( \square _{t\in T}\left( \lambda _{t}f_{t}\right) ^{*}\right) \right) . \end{aligned}$$

Consequently,

$$\begin{aligned} \mathrm{epi}\,\left( f_{C}+\sum _{t\in T}\lambda _{t}f_{t}\right) ^{*}= \mathrm{cl}\,^{w^{*}}\left( \mathrm{epi}\,f^{*}+\mathrm{epi}\,( \mathrm {i}_{C}^{*})+\sum _{t\in T}\lambda _{t}\mathrm{epi}\,f_{t}^{*}\right) , \end{aligned}$$

so that, by (2.2),

$$\begin{aligned} \begin{array}{l@{\quad }l} \mathrm{cl}\,^{w^{*}}\mathrm{epi}\,h &{} =\mathrm{cl}\,^{w^{*}}\left\{ \mathop {\textstyle \bigcup }\limits _{\lambda \in \mathbb {P}(T)} \mathrm{cl}\,^{w^{*}}\left( \mathrm{epi}\,f^{*}+\mathrm{epi}\,( \mathrm {i}_{C}^{*})+\sum \nolimits _{t\in T}\lambda _{t}\mathrm{epi}\,f_{t}^{*}\right) \right\} \\ &{} =\mathrm{cl}\,^{w^{*}}\left\{ \mathrm{epi}\,f^{*}+\mathrm{ epi}\,(\mathrm {i}_{C}^{*})+\mathop {\textstyle \bigcup }\limits _{\lambda \in \mathbb {P} (T)}\left( \sum \nolimits _{t\in T}\lambda _{t}\mathrm{epi}\,f_{t}^{*}\right) \right\} \\ &{} =\mathrm{cl}\,^{w^{*}}\left\{ \mathrm{epi}\,f^{*}+\mathrm{ epi}\,(\mathrm {i}_{C}^{*})+\mathop {\textstyle \bigcup }\limits _{\lambda \in \mathbb {R} _{+}^{(T)}}\left( \sum \nolimits _{t\in T}\lambda _{t}\mathrm{epi}\,f_{t}^{*}\right) \right\} \\ &{} =\mathrm{cl}\,^{w^{*}}\left( \mathrm{epi}\,f^{*}+K\right) . \end{array} \end{aligned}$$

We thus have

$$\begin{aligned} \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{epi}\,h=\mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\left( \mathrm{cl}\,^{w^{*}} \mathrm{epi}\,h\right) =\mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\left( \mathrm{epi}\,f^{*}+K\right) \end{aligned}$$

and, finally,

$$\begin{aligned} \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{epi}\,h=\mathrm{cl}\,^{w^{*}}\left( K+\mathrm{cone}\,\mathrm{epi}\,f^{*}\right) . \end{aligned}$$

(3.2)

Denoting by $\Pi$ the projection of $X^{*}\times \mathbb {R}$ onto $X^{*}$ one has, according to (3.2),

$$\begin{aligned} \begin{array}{c} \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{dom}\,h=\mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\Pi \left( \mathrm{epi}\,h\right) = \mathrm{cl}\,^{w^{*}}\Pi \left( \mathrm{cone}\,\mathrm{epi}\, h\right) \\ =\mathrm{cl}\,^{w^{*}}\Pi \left( \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{epi}\,h\right) =\mathrm{cl}\,^{w^{*}}\Pi \left( K+\mathrm{cone}\,\mathrm{epi}\,f^{*}\right) . \end{array} \end{aligned}$$

Using the definition of $K$ we get the key relation

$$\begin{aligned} \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{dom}\,h=\mathrm{cl}\,^{w^{*}}\left( b\left( C\right) +\mathrm{cone}\,\left( \mathop {\textstyle \bigcup }\limits _{t\in T}\mathrm{dom}\,f_{t}^{*}\right) +\mathrm{cone}\, \mathrm{dom}\,f^{*}\right) , \end{aligned}$$

(3.3)

where $b(C) :=\mathrm{dom}\,(\mathrm {i}_{C}^{*})$ denotes the barrier cone of $C.$

Since the condition

$$\begin{aligned} \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{dom}\,h\text { is a linear space} \end{aligned}$$

(3.4)

will be of crucial importance in the sequel, we summarize below some equivalent reformulations of (3.4). To this aim we need the following equivalence whose simple proof is omitted: Having a linear space $U$ and a function $g:U\rightarrow \overline{\mathbb {R}}$ it holds that

$$\begin{aligned} (\mathrm{dom}\,g)\times \mathbb {R=(}\mathrm{epi}\,g)-\left\{ 0_{U}\right\} \times \mathbb {R}_{+}. \end{aligned}$$

(3.5)

Proposition 1

Assume that (3.1) holds. Then, each of the following statements is equivalent to (3.4):

• (i) $\mathrm{rec}\,(P)$ is a linear space.

• (ii) $\mathrm{cl}\,^{w^{*}}(b(C) +\mathrm{ cone}\,( \mathop {\textstyle \bigcup }\nolimits _{t\in T}\mathrm{dom}\,f_{t}^{*}) + \mathrm{cone}\,\mathrm{dom}\,f^{*})$ is a linear space.

• (iii) $\mathrm{cl}\,^{w^{*}} (K+\mathrm{cone}\,\mathrm{epi }\,f^{*}- \{ 0_{X^{*}}\} \times \mathbb {R}_{+})$ is a linear space.

• (iv) $\mathrm{cl}\,^{w^{*}} (K\cup \mathrm{epi}\,f^{*}\cup \{(0_{X^{*}},-1)\})$ is a linear space.

• (v) $\mathrm{cl}\,^{w^{*}} (b (C) \times \mathbb {R}+\mathrm{cone}\, (\mathop {\textstyle \bigcup }\nolimits _{t\in T}\mathrm{epi}\, f_{t}^{*}) +\mathrm{cone}\,\mathrm{epi}\,f^{*})$ is a linear space.

Proof

By taking the negative polar cone we obtain that $\mathrm{(i)}\Leftrightarrow \mathrm{(ii)}$. By (3.2) and (3.5) one has

$$\begin{aligned} \begin{array}{l@{\quad }l} \left( \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{dom}\,h\right) \times \mathbb {R} &{} =\mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\left( \mathrm{epi}\,h-\left\{ 0_{X^{*}}\right\} \times \mathbb {R}_{+}\right) \\ &{} =\mathrm{cl}\,^{w^{*}}\left( \mathrm{cl}\,^{w^{*}} \mathrm{cone}\,\mathrm{epi}\,h-\left\{ 0_{X^{*}}\right\} \times \mathbb {R}_{+}\right) \\ &{} =\mathrm{cl}\,^{w^{*}}\left( K+\mathrm{cone}\,\mathrm{epi}\, f^{*}-\left\{ 0_{X^{*}}\right\} \times \mathbb {R}_{+}\right) . \end{array} \end{aligned}$$

It follows that (3.4) $\Leftrightarrow$ (iii). Since $K$ is a cone, one has

$$\begin{aligned} K+\mathrm{cone}\,\mathrm{epi}\,f^{*}-\left\{ 0_{X^{*}}\right\} \times \mathbb {R}_{+}=\mathrm{cone}\,\left( K\cup \mathrm{epi}\,f^{*}\cup \left\{ \left( 0_{X^{*}},-1\right) \right\} \right) . \end{aligned}$$

We thus have $\mathrm{(iii)}\Leftrightarrow \mathrm{(iv)}$. By (3.5) one has $\mathrm{epi }\,(\mathrm {i}_{C}^{*})-\{ 0_{X^{*}}\} \times \mathbb {R} _{+}=b (C)\times \mathbb {R}.$ From the very definition of $K,$ it follows that $\mathrm{(iii)}\Leftrightarrow \mathrm{(v)}$. $\square$

3.1 Quasicontinuity and subdifferentiability

We denote by $w$ (respectively, $\tau ^{*}$) the weak topology on $X$ (respectively, the Mackey topology on $X^{*}$). Following [9] and [10], a convex function $g:X^{*}\rightarrow \overline{\mathbb { R}}$ is said to be $\tau ^{*}$-quasicontinuous when the affine hull of $\mathrm{dom}\,g,$ $\mathrm{aff}\,\mathrm{dom}\,g,$ is $w^{*}$-closed and of finite codimension, and the restriction of $g$ to the relative interior of $\mathrm{dom}\,g,$ say $\mathrm{ri}\,^{\tau ^{*}} \mathrm{dom}\,g$, is continuous with respect to the topology induced by $\tau ^{*}.$

If $g$ is $w^{*}$-lsc and proper, one has [11, Theorem 7.7.6]:

$$\begin{aligned} g\text { is }\tau ^{*}\text {-quasicontinuous}\Leftrightarrow g^{*} \text { is }w\text {-inf-locally-compact,} \end{aligned}$$

meaning that for each $r\in \mathbb {R}$, the sublevel set $[g^{*}\le r]$ is $w$-locally-compact.

Any extended real-valued convex function which is majorized by a $\tau ^{*}$-quasicontinuous convex function is $\tau ^{*}$-quasicontinuous too [12, Theorem 2.4]. Accordingly, the convex function $h$ defined in (2.1) is $\tau ^{*}$-quasicontinuous whenever there exists $\overline{\lambda }\in \mathbb {P}(T)$ such that $f_{C}+\sum \nolimits _{t\in T} \overline{\lambda }_{t}f_{t}$ is $w$-inf-locally-compact (this fact is observed in [7, p.11]). Such a condition is in particular fulfilled when $C$ is $w$-locally-compact, e.g. when $X$ is finite dimensional.

We will use the following subdifferentiability criterion [12, Theorem 3.3].

Lemma 1

Let $g:X^{*}\rightarrow \overline{\mathbb {R}}$ be convex and $\tau ^{*}-$quasicontinuous. Assume that $g(0_{X^{*}}) >-\infty$ and $\mathrm{cl}\,^{w^{*}}\mathrm{cone}\, \mathrm{dom}\,g$ is a linear space. Then, $\partial g(0_{X^{*}})$ is the sum of a non-empty $w$-compact convex set and a finite dimensional linear space.

3.2 The main result

Remember that by $S(P)$ we denote the optimal solution set of the convex infinite problem

$$\begin{aligned} (P) \inf _{x}f(x), \hbox { s.t. } x\in C,\ f_{t}(x)\le 0,\ t\in T, \end{aligned}$$

and recall also the formulation of the surrogate dual $(\Delta )$ of $(P):$

$$\begin{aligned} (\Delta ) \sup _{\lambda }\inf _{C}\left( f+\sum _{t\in T}\lambda _{t}f_{t}\right) , \hbox { s.t. } \lambda \in \mathbb {P}(T). \end{aligned}$$

Theorem 2

Assume that the following assumptions are fulfilled:

$$\begin{aligned} \sup (\Delta )<+\infty , \end{aligned}$$

(3.6)

$$\begin{aligned} \exists \overline{\lambda }\in \mathbb {R}_{+}^{\left( T\right) }\text { such that }f_{C}+\sum \limits _{t\in T}\overline{\lambda }_{t}f_{t}\text { is }w \text {-inf-locally-compact,} \end{aligned}$$

(3.7)

and

$$\begin{aligned} \mathrm{rec}\,\left( P\right) \text { is a linear space.} \end{aligned}$$

(3.8)

Then, $\min (P)=\sup (\Delta )\in \mathbb {R}$, and $S(P)$ is the sum of a non-empty $w$-compact convex set and a finite dimensional linear space.

Proof

Let us apply Lemma 1 to $g=h.$ By (3.6) one has $h(0_{X^{*}}) >-\infty .$ By (3.7), $h$ is $\tau ^{*}$-quasicontinuous and, by (3.3), (3.8) and the equivalence $\mathrm{(i)} \Leftrightarrow \mathrm{(ii)}$ in Proposition 1, $\mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{dom}\,h$ is a linear space. By Lemma 1, $\partial h( 0_{X^{*}})$ is the sum of a non-empty $w$-compact convex set and a finite dimensional linear space. Now $x\in \partial h(0_{X^{*}})$ means that $-h(0_{X^{*}}) =h^{*}(x) =f_{C\cap F}(x) \in \mathbb {R }.$ In other words, $x$ is feasible for $(P)$ and

$$\begin{aligned} \inf \left( P\right) \ge \sup (\Delta )=h^{*}\left( x\right) =f\left( x\right) \ge \inf \left( P\right) . \end{aligned}$$

We thus have $\min (P)=\sup (\Delta )\in \mathbb {R}$ and $\partial h(0_{X^{*}}) \subset S(P)$. To complete the proof, take $\overline{x}\in S(P)$ and write

$$\begin{aligned} +\infty >\sup (\Delta )=-h^{*}(0_{X^{*}})=\min (P)=f(\overline{x} )=f_{C\cap F}(\overline{x})=h^{*}(\overline{x}), \end{aligned}$$

i.e., $h^{*}(\overline{x})+h(0_{X^{*}})=0=\langle 0_{X^{*}}, \overline{x}\rangle$, entailing $\overline{x}\in \partial h(0_{X^{*}})$. $\square$

Let us revisit the examples of Sect. 2, where $X$ is finite dimensional and $\sup (\Delta )<+\infty ,$ so that Theorem 2 applies whenever $\mathrm{rec}\,(P)$ is a linear space. This is the case of Examples 4 and 5, where $\mathrm{rec}\,(P) =\{0\}$, with $\sup (\Delta )$ attained in Example 4 but not in Example 5. Observe that, in Example 2, $\mathrm{rec}\,(P) =\mathbb {R}_{-}\times \{0\} ,$ with $\inf (P)=1\ne -\infty =\sup (\Delta ),$ while, in Example 3, $\mathrm{rec}\,(P) =\mathbb {R}_{-},$ with $\inf (P)=0\ne -\infty =\sup (\Delta ).$

Remark 1

The same conclusion is obtained in [7, Theorems 4.7 and 4.8] replacing condition (3.6) by the stronger assumption that $\inf (P) <+\infty .$

Remark 2

In the case that $\sup (\Delta )=+\infty ,$ all the problems $(P), \, (D)$ and $(Q)$ share the same value.

We now provide a new version of the famous Clark–Duffin Theorem for semi-infinite optimization with $T$ finite. We are concerned with the problems

$$\begin{aligned}&(P_{m}) \inf _{x}f(x), \hbox { s.t. } x\in C,\ f_{1}(x)\le 0,\ \ldots ,f_{m}(x)\le 0,\\&(Q_{m}) \sup _{\lambda }\inf _{C}\left( f+\sum _{i=1}^{m}\lambda _{i}f_{i}\right) , \hbox { s.t. } \left( \lambda _{1},\ldots ,\lambda _{m}\right) \in \mathbb {R}_{+}^{m}, \end{aligned}$$

with the rule $0\times (+\infty ) =+\infty ,$

$$\begin{aligned} (D_{m}) \sup _{\lambda }\inf _{C}\left( f+\sum _{i=1}^{m}\lambda _{i}f_{i}\right) , \hbox { s.t. } \left( \lambda _{1},\ldots ,\lambda _{m}\right) \in \mathbb {R}_{+}^{m}, \end{aligned}$$

with the rule $0\times (+\infty ) =0,$ and

$$\begin{aligned} (\Delta _{m}) \sup _{\lambda }\inf _{C}\left( f+\sum _{i=1}^{m}\lambda _{i}f_{i}\right) , \hbox { s.t. } \left( \lambda _{1},\ldots ,\lambda _{m}\right) \in \mathbb {R}_{+}^{m}\diagdown \left\{ 0_{ \mathbb {R}^{m}}\right\} , \end{aligned}$$

where $X$ is a locally convex separated tvs, $C$ a non-empty closed convex subset of $X$ and $f,f_{1},\ldots ,f_{m}\in \Gamma (X).$ The next result is to be compared with [8, Theorem 5.1] and [4, Theorem 3.1].

Corollary 1

Assume that $\sup (\Delta _{m})<+\infty ,$ that there exists $\overline{\lambda }\in \mathbb {R}_{+}^{m}$ such that $f_{C}+\sum _{i=1}^{m} \overline{\lambda }_{i}f_{i}$ is $w$-inf-locally-compact, with the rule $0\times (+\infty ) =0,$ and that $\mathrm{rec}\,(P_{m})$ is a linear space. Then,

$$\begin{aligned} \sup (\Delta _{m})=\sup (D_{m})=\sup (Q_{m})=\min (P_{m})\in \mathbb {R} \end{aligned}$$

and $S(P_{m})$ is the sum of a non-empty $w$-compact convex set and a finite dimensional linear space.

Remark 3

If $X$ is finite dimensional, the second assumption in the statement of Corollary 1 is superfluous.

4 Applications

4.1 The finite intersection property

Recall that a family $\{C_{t},\ t\in T\}$ of sets of a topological space is said to have the finite-intersection property if every finite subfamily has non-empty intersection. As a substitute of compactness we have the following result:

Corollary 2

Let $\{C_{t},\ t\in T\}$ be a family of closed convex subsets of a locally convex separated tvs having the finite-intersection property. Moreover, assume the existence of $t_{1},\ldots ,t_{m}\in T$ such that $\mathop {\textstyle \bigcap }\nolimits _{i=1}^{m}C_{t_{i}}$ is $w$-locally-compact and that $\mathop {\textstyle \bigcap }\nolimits _{t\in T}(C_{t})_{\infty }$ is a linear space. Then $\mathop {\textstyle \bigcap }\nolimits _{t\in T}C_{t}$ is the sum of a non-empty $w$-compact convex set and a finite dimensional linear space.

Proof

Apply Theorem 2 with $C=X,$ $f\equiv 0,$ and $f_{t}= \mathrm {i}_{C_{t}},$ $t\in T,$ observing that $S(P) =\mathop {\textstyle \bigcap }\nolimits _{t\in T}C_{t},$ $\mathrm{rec}\,(P) =\mathop {\textstyle \bigcap }\nolimits _{t\in T} (C_{t})_{\infty },$ and $\sup (\Delta )<+\infty$ amounts to say that the family $\{C_{t},\ t\in T\}$ has the finite-intersection property. $\square$

Remark 4

Taking $C=X=\mathbb {R}, \, f\equiv 0,$ and $f_{t}= \mathrm {i}_{[t,+\infty [}, \, t>0,$ in Theorem 2, we get $M=\emptyset$ and, since the family $\{[t,+\infty [ ,\ t>0\}$ has the finite-intersection property, one gets

$$\begin{aligned} \max (\Delta )=\max (D)=0<+\infty =\sup (Q)=\inf \left( P\right) . \end{aligned}$$

Since $\mathrm{rec}\,(P) =[ 0,+\infty [$ is not a linear space, the assumption (3.8) in Theorem 2 is not satisfied.

4.2 Convex infinite systems

In this section we again apply Theorem 2 in the case that $f\equiv 0.$ We denote by $(P_{0})$ the corresponding convex infinite problem, and by

$$\begin{aligned} \sigma :=\left\{ f_{t}(x)\le 0,\ t\in T;\ x\in C\right\} , \end{aligned}$$

the general infinite convex system associated with the constraints of $(P_{0})$, whereas $K$ is the characteristic cone of $\sigma .$ The feasible set $C\cap F$ of $\sigma$ coincides with $S(P_{0})$. It may be empty even if we assume that $\sup (\Delta _{0})<+\infty$ (see Remark 4).

The function $h_{0}$ associated with $(P_{0})$ is

$$\begin{aligned} h_{0}=\inf _{\lambda \in \mathbb {P}(T)}\left( \mathrm {i}_{C}+\sum _{t\in T}\lambda _{t}f_{t}\right) ^{*}. \end{aligned}$$

Assuming that

$$\begin{aligned} \mathrm {i}_{C}+\sum _{t\in T}\lambda _{t}f_{t}\text { is proper for any } \lambda \in \mathbb {P}(T), \end{aligned}$$

(4.1)

which is the counterpart of (3.1) and it is weaker than $\sup (\Delta _{0})<+\infty ,$ it holds that

$$\begin{aligned} \mathrm{cl}\,^{w^{*}}\mathrm{epi}\,h_{0}=\mathrm{cl}\,^{w^{*}}K \end{aligned}$$

and, recalling (3.3),

$$\begin{aligned} \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\mathrm{dom}\,h_{0}=\mathrm{cl }\,^{w^{*}}\left( b\left( C\right) +\mathrm{cone}\,\left( \mathop {\textstyle \bigcup }\limits _{t\in T}\mathrm{dom}\,f_{t}^{*}\right) \right) . \end{aligned}$$

Let us define the recession cone associated with $\sigma$ by

$$\begin{aligned} \mathrm{rec}\,\left( \sigma \right) :=\mathrm{rec}\,\left( P_{0}\right) =C_{\infty }\cap \left( \mathop {\textstyle \bigcap }\limits _{t\in T}\left[ \left( f_{t}\right) _{\infty }\le 0\right] \right) . \end{aligned}$$

Assuming that (4.1) holds, the following assertions are equivalent (see Proposition 1):

$$\begin{aligned} \begin{array}{l} (\mathrm{i}_{0})\quad \mathrm{rec}\,\left( \sigma \right) \text { is a linear space,} \\ (\mathrm{ii}_{0})\quad \mathrm{cl}\,^{w^{*}}\left( b\left( C\right) + \mathrm{cone}\,\left( \mathop {\textstyle \bigcup }\limits _{t\in T}\mathrm{dom}\,f_{t}^{*}\right) \right) \text { is a linear space,} \\ (\mathrm{iii}_{0})\quad \mathrm{cl}\,^{w^{*}}\left( K-\left\{ 0_{X^{*}}\right\} \times \mathbb {R}_{+}\right) \text { is a linear space,}\\ (\mathrm{iv}_{0})\quad \mathrm{cl}\,^{w^{*}}\mathrm{cone}\,\left( K\cup \left\{ \left( 0_{X^{*}},-1\right) \right\} \right) \text { is a linear space,} \\ (\mathrm{v}_{0})\quad \mathrm{cl}\,^{w^{*}}\left( b\left( C\right) \times \mathbb {R}+\mathrm{cone}\,\left( \mathop {\textstyle \bigcup }\limits _{t\in T}\mathrm{epi}\, f_{t}^{*}\right) \right) \text { is a linear space.} \end{array} \end{aligned}$$

We are now in a position to state a generalization of Fan’s Theorem in general locally convex separated tvs:

Corollary 3

Assume that

$$\begin{aligned} \exists \overline{\lambda }\in \mathbb {R}_{+}^{m}\text { such that }\mathrm {i} _{C}+\sum _{t\in T}\overline{\lambda }_{t}f_{t}\text { is }w\text { -inf-locally-compact,} \end{aligned}$$

(4.2)

and that

$$\begin{aligned} \mathrm{rec}\,\left( \sigma \right) \text { is a linear space.} \end{aligned}$$

(4.3)

Then, the infinite convex system $\sigma$ is consistent if and only if

$$\begin{aligned} \inf _{C}\sum _{t\in T}\lambda _{t}f_{t}\le 0\text { for any }\lambda \in \mathbb {P}(T). \end{aligned}$$

(4.4)

Proof

Necessity is obvious. Sufficiency comes from Theorem 2 by taking $f\equiv 0.$ $\square$

Remark 5

With the same assumptions, statement (4.4) in Corollary 3 is equivalent to

$$\begin{aligned} \forall \lambda \in \mathbb {R}_{+}^{\left( T\right) }, \exists x_{\lambda }\in C\text { such that }\sum _{t\in T}\lambda _{t}f_{t}\left( x_{\lambda }\right) \le 0 \end{aligned}$$

that appears in [2, Theorem 3.5].

In [2, Theorem 3.5] it is assumed that either $K$ is $w^{*}$ -closed or $K$ is solid if $X$ is infinite dimensional, and $\mathrm{rec}\, (\sigma ) = \{ 0_{X^{*}}\}.$ We now provide an example where none of these two conditions is satisfied while Corollary 3 does work.

Example 6

Let $X$ be a reflexive Banach space whose open (respectively, closed) unit dual ball is represented by $\mathbb {B} ^{*}$ (resp., $\overline{\mathbb {B}}^{*}$). Notice that the topology $\tau ^{*}$ coincides with the dual norm topology. Given $a\in X, \, a\ne 0_{X},$ let us set $H:=\{a\} ^{\bot }$ and consider

$$\begin{aligned} D:=H\cap \overline{\mathbb {B}}^{*}. \end{aligned}$$

It holds that $\mathrm{cone}\,D=\mathrm{aff}\,D=H,$ a closed hyperplane, and $0_{X^{*}}\in \mathrm{ri}\,D=H\cap \mathbb {B}^{*}.$ Setting $f_{t}:=\mathrm {i}_{D}^{*}-\frac{1}{t}, \, t>0,$ we get a family of functions in $\Gamma (X)$ having the same recession cone, namely,

$$\begin{aligned} \left[ \left( f_{t}\right) _{\infty }\le 0\right] =\left[ \mathrm {i} _{D}^{*}\le 0\right] =H^{\bot }=\mathbb {R\{}a\}{\hbox {, for all } } t>0. \end{aligned}$$

Since $f_{t}^{*}=\mathrm {i}_{D}+\frac{1}{t}$ is $\tau ^{*}$-quasicontinuous, any $f_{t}$ is $w$-inf-locally-compact. Consequently, the system

$$\begin{aligned} \sigma :=\left\{ f_{t}(x)\le 0,\ t>0\right\} \end{aligned}$$

satisfies the assumptions of our Corollary 3. However,

$$\begin{aligned} K=\mathrm{cone}\,\left( \mathop {\textstyle \bigcup }\limits _{t>0}\mathrm{epi}\,f_{t}^{*}\right) =\left( H\times \left] 0,+\infty \right[ \right) \cup \left\{ \left( 0_{X^{*}},0\right) \right\} \end{aligned}$$

is not $w^{*}$ -closed, $K\subset H\times \mathbb {R}$ is not solid, and $\mathrm{rec}\,(\sigma ) =\mathbb {R\{}a\}$ is not $\{ ( 0_{X^{*}},0)\}$. Consequently, the assumptions of [2, Theorem 3.5] are not satisfied.

Given $m\ge 1, \, t_{1},\ldots ,t_{m}\in T,$ and $\varepsilon >0,$ let us consider the system

$$\begin{aligned} \sigma \left( t_{1},\ldots ,t_{m},\varepsilon \right) :=\left\{ f_{t_{i}}(x)\le \varepsilon ,\ i=1,\ldots ,m,\ x\in C\right\} . \end{aligned}$$

Corollary 4

Assume that (4.2) and (4.3) hold. Then the convex infinite system $\sigma$ is consistent if and only if all the semi-infinite systems $\sigma (t_{1},\ldots ,t_{m},\varepsilon ), \, m\ge 1, \, t_{1},\ldots ,t_{m}\in T, \, \varepsilon >0,$ are consistent.

Proof

Necessity is obvious; now we show the sufficiency. Applying Corollary 3, we have just to verify that (4.4) holds. So, let $\lambda \in \mathbb {P}(T)$ and $\mathrm{supp}\, \lambda = \{ t_{1},\ldots ,t_{m}\}$. For any $\alpha >0$ there exists $\overline{x}\in C$ such that

$$\begin{aligned} f_{t_{i}}(\overline{x})\le \frac{\alpha }{\mathop {\textstyle \sum }\nolimits _{j=1}^{m}\lambda _{j}},\ i=1,\ldots ,m. \end{aligned}$$

We thus have

$$\begin{aligned} \sum _{t\in T}\lambda _{t}f_{t}(\overline{x})=\mathop {\textstyle \sum }\limits _{i=1}^{m}\lambda _{t_{i}}f_{t_{i}}(\overline{x})\le \alpha . \end{aligned}$$

Since $\alpha >0$ is arbitrary, we have that (4.4) holds. $\square$

Remark 6

Every time, when the conditions of Corollaries 3 and 4 are fulfilled, then the solution set of the convex infinite system $\sigma$ is the sum of a non-empty $w$ -compact convex set and a finite dimensional linear space.

5 Perturbational approach

Having $\mu =(\mu _{t}) _{t\in T}\in \mathbb {R}^{T},$ we consider the parametric convex infinite problem

$$\begin{aligned} (P^{\mu }) \inf _{x}f(x), \hbox { s.t. } x\in C,f_{t}(x)\le -\mu _{t},\ t\in T, \end{aligned}$$

where $f,f_{t},t\in T,$ are proper convex functions defined on the locally convex separated tvs $X,$ and $C\subset X$ is a non-empty convex set. Let us observe that all these problems have the same recession cone:

$$\begin{aligned} \mathrm{rec}\,\left( P^{\mu }\right) =\mathrm{rec}\,\left( P^{0_{T}}\right) = \mathrm{rec}\,\left( P\right) . \end{aligned}$$

Considering the associated dual problems

$$\begin{aligned}&(D^{\mu }) \sup _{\lambda }\left\{ \sum _{t\in T}\lambda _{t}\mu _{t}+\inf _{C}\left( f+\sum _{t\in T}\lambda _{t}f_{t}\right) \right\} , \hbox { s.t. } \lambda \in \mathbb {R}_{+}^{(T)},\\&(\Delta ^{\mu }) \sup _{\lambda }\left\{ \sum _{t\in T}\lambda _{t}\mu _{t}+\inf _{C}\left( f+\sum _{t\in T}\lambda _{t}f_{t}\right) \right\} , \hbox { s.t. } \lambda \in \mathbb {P}(T), \end{aligned}$$

we can thus state, applying Theorem 2:

Corollary 5

Assume that (3.7) and (3.8) hold. For any $\mu \in \mathbb {R}^{T}$ we have either

$$\begin{aligned} \min \left( P^{\mu }\right) =\sup (D^{\mu })=\sup (\Delta ^{\mu })\in \mathbb {R}, \end{aligned}$$

or

$$\begin{aligned} \inf \left( P^{\mu }\right) =\sup (D^{\mu })=\sup (\Delta ^{\mu })=+\infty . \end{aligned}$$

By using the value function $v:\mathbb {R}^{T}\rightarrow \overline{\mathbb {R}},$

$$\begin{aligned} v\left( \mu \right) :=\inf \left( P^{\mu }\right) , \end{aligned}$$

we can develop in a natural way the classical perturbational duality theory for convex infinite problems (see, e.g. [1, 15]) by computing the conjugate of $v,$ namely,

$$\begin{aligned} -v^{*}\left( \lambda \right) =\left\{ \begin{array}{l@{\quad }l} \inf _{C\cap M}\left( f+\sum _{t\in T}\lambda _{t}f_{t}\right) , &{} \text {if } \lambda \in \mathbb {R}_{+}^{(T)}, \\ -\infty , &{} \text {if }\lambda \in \mathbb {R}^{(T)}\diagdown \mathbb {R} _{+}^{(T)}, \end{array} \right. \end{aligned}$$

(5.1)

and defining the perturbational dual of $(P^{\mu })$ as

$$\begin{aligned} \left( Q^{\mu }\right) \sup _{\lambda }\left\{ \sum _{t\in T}\lambda _{t}\mu _{t}+\inf _{C\cap M}\left( f+\sum _{t\in T}\lambda _{t}f_{t}\right) \right\} , \hbox { s.t. } \lambda \in \mathbb {R}_{+}^{(T)}. \end{aligned}$$

We observe that $(Q^{0_{T}})$ coincides with the problem $(Q)$ defined in Sect. 1.

One has, in general, the following well-known properties:

• (a) $-\infty \le \sup (\Delta ^{\mu })\le \sup (D^{\mu })\le \sup (Q^{\mu })=v^{**}(\mu ) \le v(\mu ) =\inf (P^{\mu }) \le +\infty ,$

• (b) $E:=\mathop {\textstyle \bigcup }\nolimits _{x\in C\cap M\cap \mathrm{dom}\,f}\{ ((f_{t}(x))_{t\in T},f(x))\} +\mathbb {R}_{+}^{T}\times \mathbb {R}_{+}$ is convex,

• (c) $v$ is convex,

• (d) $\mathrm{epi}\,_{s}v\subset \widehat{E}:=\{( \mu ,r) \in \mathbb {R}^{T}\times \mathbb {R}:(-\mu ,r) \in E\} \subset \mathrm{epi}\,v,$ and

• (e) $\mathrm{epi}\,\overline{v}=\mathrm{cl}\,\mathrm{epi}\,v=\mathrm{cl}\, \widehat{E}.$

Observe that all these properties are true just assuming the convexity of the data of $(P): f,C,f_{t},t\in T$.

Theorem 3

Assume that $f,f_{t}:X\rightarrow \mathbb {R\cup }\{ +\infty \}$ are proper convex and $C$ is a non-empty convex subset of the locally convex tvs $X$ such that

$$\begin{aligned} \exists \overline{\lambda }\in \mathbb {R}_{+}^{(T)}\text { such that } \inf _{C\cap M}\left( f+\sum _{t\in T}\overline{\lambda }_{t}f_{t}\right) \ne -\infty . \end{aligned}$$

(5.2)

Then, for any $\mu \in \mathbb {R}^{T},$ the following statements are equivalent:

• (i) $\min (P^{\mu }) =\sup (Q^{\mu })\in \mathbb {R}$ or $\sup (Q^{\mu })=+\infty .$

• (ii) $E$ is closed regarding to $\{-\mu \} \times \mathbb {R}.$

Proof

By (5.1) and (5.2) one has $v^{*}(\overline{ \lambda })<+\infty$ and so, $\mathrm{dom}\,v^{*}\ne \emptyset$. Since $v$ is convex, $\overline{v}=v^{**}$ (either $v$ is proper or $+\infty =v^{**}=\overline{v}=v$ ).

Let us begin with the case that $\sup (Q^{\mu })=+\infty .$ Then $\overline{v}(\mu ) =+\infty$ and

$$\begin{aligned} \emptyset =\left( \left\{ \mu \right\} \times \mathbb {R}\right) \cap \mathrm{epi}\,\overline{v}=\left( \left\{ \mu \right\} \times \mathbb {R} \right) \cap \mathrm{cl}\,\widehat{E}. \end{aligned}$$

So, $\widehat{E}$ is closed regarding to $\{ \mu \} \times \mathbb {R}$ and, equivalently, $E$ is closed regarding to $\{ -\mu \} \times \mathbb {R}.$ Thus, if $\sup (Q^{\mu })=+\infty ,$ the statements (i) and (ii) are simultaneously satisfied.

Assume now that $\beta :=\sup (Q^{\mu })<+\infty .$ By (5.2) we have $\beta \in \mathbb {R}$ and so $(\mu ,\beta ) \in \mathrm{cl}\, \mathrm{epi}\,v=\mathrm{cl}\,\widehat{E},$ that is

$$\begin{aligned} \left( -\mu ,\beta \right) \in \mathrm{cl}\,E. \end{aligned}$$

(5.3)

Assume that (i) holds and let $( -\mu ,r) \in \mathrm{cl}\,E,$ so that $\overline{v}(\mu ) =\beta \le r.$ Taking $\overline{x} \in S( P^{\mu })$ we get $\overline{x}\in C\cap M\cap \mathrm{ dom}\,f,$ $f_{t}(\overline{x}) \le -\mu _{t},$ $t\in T,$ and $f(\overline{x}) =\beta \le r.$ So,

$$\begin{aligned} \left( -\mu ,r\right) \in \left\{ \left( (f_{t}\left( \overline{x}\right) )_{t\in T},f\left( \overline{x}\right) \right) \right\} +\mathbb {R} _{+}^{(T)}\times \mathbb {R}_{+}\subset E, \end{aligned}$$

and (ii) holds.

Conversely, assume that (ii) holds. By (5.3) we thus have $(-\mu ,r) \in E,$ and there exists $\overline{x}\in C\cap M\cap \mathrm{dom}\,f$ such that

$$\begin{aligned} f_{t}\left( \overline{x}\right) \le -\mu _{t},\ t\in T, f\left( \overline{x}\right) \le \beta \le \inf \left( P^{\mu }\right) . \end{aligned}$$

Since $\overline{x}$ is feasible for $(P^{\mu })$, we obtain (i). $\square$

This section ends with an application of Theorem 3 to the convex system

$$\begin{aligned} \sigma :=\left\{ f_{t}(x)\le 0,\ t\in T;\ x\in C\right\} , \end{aligned}$$

where $f_{t}:X\rightarrow \mathbb {R\cup }\{+\infty \} ,$ $t\in T,$ are proper convex and $C$ is a non-empty convex subset of $X.$ We have (compare with Corollary 3):

Corollary 6

Let $\sigma$ be as above and assume that

$$\begin{aligned} \inf _{C\cap M}\left( \sum _{t\in T}\lambda _{t}f_{t}\right) \le 0\text { for any }\lambda \in \mathbb {R}_{+}^{\left( T\right) }. \end{aligned}$$

(5.4)

Then $\sigma$ is consistent if and only if

$$\begin{aligned} \mathop {\textstyle \bigcup }\limits _{x\in C\cap M}\left\{ (\left( f_{t}\left( x\right) )_{t\in T},0\right) \right\} +\mathbb {R}_{+}^{T}\times \mathbb {R}_{+} \end{aligned}$$

is closed regarding $\{0_{T}\} \times \mathbb {R}.$

Proof

Apply Theorem 3 with $f\equiv 0$ and $\mu =0_{T}.$ Observe that (5.2) is satisfied (with $\overline{\lambda }=0_{T}$) and that (5.3) amounts to $\sup (Q^{\mu })=0.$ Then it suffices to notice that $\min (P^{\mu }) =0$ amounts to say that $\sigma$ is consistent. $\square$

6 Linear infinite problems

In this section we will apply the previous results, essentially Theorems , 2 and 3, to the linear infinite problem

$$\begin{aligned} (P) \inf _{x}\left\langle c^{*},x\right\rangle , \hbox { s.t. } x\in C,\ \left\langle x_{t}^{*},x\right\rangle \le r_{t},\ t\in T, \end{aligned}$$

where $(x_{t}^{*},r_{t}) \in X^{*}\times \mathbb {R}, \, t\in T, \, c^{*}\in X^{*},$ and $C$ is a closed convex cone in the locally convex separate tvs $X.$ One has straightforwardly,

$$\begin{aligned} (D)\equiv \left( Q\right) \sup _{\lambda }-\left( \mathrm {i} _{C^{+}}\left( c^{*}+\sum _{t\in T}\lambda _{t}x_{t}^{*}\right) +\sum _{t\in T}\lambda _{t}r_{t}\right) , \hbox { s.t. } \lambda \in \mathbb {R}_{+}^{(T)}. \end{aligned}$$

Modifying the feasible set (but not the value) of $(D)$ we get a classical Haar dual-type problem

$$\begin{aligned} (D^{\#}) \sup _{\lambda }-\sum _{t\in T}\lambda _{t}r_{t}, \hbox { s.t. } \lambda \in \mathbb {R}_{+}^{(T)},\sum _{t\in T}\lambda _{t}x_{t}^{*}\in C^{+}-c^{*}. \end{aligned}$$

In order to apply Theorem 1 to the present situation, let us introduce the $w^{*}$-continuous linear mapping

$$\begin{aligned} \Lambda :\mathbb {R}^{(T)}\rightarrow X^{*}\times \mathbb {R},\ \Lambda \left( \lambda \right) =\sum _{t\in T}\lambda _{t}\left( x_{t}^{*},r_{t}\right) . \end{aligned}$$

Denoting by $K$ the characteristic cone of $\sigma :=\{ \langle x_{t}^{*},x \rangle \le r_{t},\ t\in T,\ x\in C\}$, one has

$$\begin{aligned} \begin{array}{l@{\quad }l} K &{} =\mathrm{epi}\,(\mathrm {i}_{C}^{*})+\mathrm{cone}\,\left( \mathop {\textstyle \bigcup }\limits _{t\in T}\mathrm{epi}\,\left( x_{t}^{*}-r_{t}\right) ^{*}\right) \\ &{} =C^{-}\times \mathbb {R}_{+}+\mathrm{cone}\,\left( \mathop {\textstyle \bigcup }\limits _{t\in T} \mathrm{epi}\,(\mathrm {i}_{\{x_{t}^{*}\}}+r_{t})\right) \\ &{} =C^{-}\times \mathbb {R}_{+}+\Lambda \left( \mathbb {R}_{+}^{(T)}\right) +\left\{ 0_{X^{*}}\right\} \times \mathbb {R}_{+} \\ &{} =C^{-}\times \mathbb {R}_{+}+\Lambda \left( \mathbb {R}_{+}^{(T)}\right) . \end{array} \end{aligned}$$

Corollary 7

Assume that $(P)$ is consistent. Then, the following statements are equivalent

• (i) $\sup (D^{\#})=-\infty$ or $\inf (P) =\max (D^{\#})\in \mathbb {R}.$

• (ii) $K$ is $w^{*}$-closed regarding to $\{ -c^{*}\} \times \mathbb {R}.$

Proof

Theorem 1 establishes that (i) holds if and only if $\mathfrak {B}$ is $w^{*}$-closed with respect to $\{0_{X^{*}}\}\times \mathbb {R}$. In this linear setting, we get straightforwardly, for any $\lambda \in \mathbb {R}_{+}^{(T)},$

$$\begin{aligned} \mathrm{epi}\,\left( \mathrm {i}_{C}+c^{*}+\sum _{t\in T}\lambda _{t}\left( x_{t}^{*}-r_{t}\right) \right) ^{*}=\left( c^{*},0\right) +\Lambda \left( \lambda \right) +C^{-}\times \mathbb {R}_{+}. \end{aligned}$$

Consequently,

$$\begin{aligned} \mathfrak {B}=\left( c^{*},0\right) +\Lambda \left( \mathbb {R} _{+}^{(T)}\right) +C^{-}\times \mathbb {R}_{+}=\left( c^{*},0\right) +K, \end{aligned}$$

and $\mathfrak {B}$ is $w^{*}$-closed regarding to $\{ 0_{X^{*}}\} \times \mathbb {R}$ if and only if (ii) holds. $\square$

Remark 7

Whenever $(P)$ and $(D^{\#})$ are both consistent, condition (ii) in Corollary 7 characterizes the identity $\inf (P) =\max (D^{\#})$ with the common value in $\mathbb {R}.$

Remark 8

According to the assumptions of Theorem 3, the convex cone $C$ does not need to be closed in Corollary 7.

We will now apply Theorem 3 for $\mu =0_{T}$ to the linear infinite problem $(P)$. To this end, let us consider the continuous linear mapping

$$\begin{aligned} L:X\rightarrow \mathbb {R}^{T}\times \mathbb {R}, L\left( x\right) =\left( \left( \left\langle x_{t}^{*},x\right\rangle \right) _{t\in T},\left\langle c^{*},x\right\rangle \right) . \end{aligned}$$

We have (compare with [7, Theorem 5.5]):

Corollary 8

Assume that $c^{*}\in C^{+}-\mathrm{cone}\,\{ x_{t}^{*},t\in T\}$. Then, the following statements are equivalent:

• (i) $\sup (D^{\#})=+\infty$ or $\min (P) =\sup (D^{\#})\in \mathbb {R}.$

• (ii) $L(C) +\mathbb {R}_{+}^{T}\times \mathbb {R}_{+}$ is closed regarding to $\{(r_{t})_{t\in T}\} \times \mathbb {R}.$

Proof

Applying Theorem 3 we observe that (5.2) is equivalent to $c^{*}\in C^{+}-\mathrm{cone}\,\{ x_{t}^{*},t\in T\}$, and we have

$$\begin{aligned} E=L\left( C\right) +\mathbb {R}_{+}^{T}\times \mathbb {R}_{+}-\left\{ \left( r_{t}\right) _{t\in T}\right\} \times \left\{ 0\right\} . \end{aligned}$$

Consequently, $E$ is closed regarding to $\{ 0_{T}\} \times \mathbb {R}$ amounts to statement (ii) in Corollary 7, and we are done. $\square$