Some relationships among the constraint qualifications for Lagrangian dualities in DC infinite optimization problems

Abstract

In this paper, we establish some relationships among several constraint qualifications, which characterize strong Lagrangian dualities and total Lagrangian dualities for DC infinite optimization problems.

1 Introduction

Consider the following DC infinite optimization problem:

$$ (\mbox{P})\quad \begin{array}{@{}l@{\quad}l} \mbox{Min}&f(x)-g(x),\\ \mbox{s. t. }& f_{t}(x)-g_{t}(x)\le0,\quad t\in T, \\ & x\in C, \end{array} $$

(1.1)

where T is an arbitrary (possibly infinite) index set, C is a nonempty convex subset of a locally convex Hausdorff topological vector space X and $f, g,f_{t},g_{t}:X\rightarrow\overline{{\mathbb {R}}}:={\mathbb {R}}\cup\{ +\infty\}$, $t\in T$, are proper convex functions. This problem has been studied extensively by many researchers. For example, the authors in [1–11] studied Lagrange dualities, Farkas lemmas, and optimality condition in the case when $g=g_{t}=0$, $t\in T$ and the authors in [12] established the Fenchel-Lagrange duality in the case when $X={\mathbb {R}}^{n}$ and T is finite, and Sun et al. gave some dualities and Farkas-type results in [13, 14]. In particular, the authors in [15] defined the dual problem of (1.1) by

$$ (\mbox{D}) \quad \sup_{\lambda\in {\mathbb {R}}_{+}^{(T)}} \inf _{w^{\ast}\in H^{\ast}}L\bigl( w^{*}, \lambda\bigr), $$

(1.2)

where $H^{\ast}=\operatorname{dom}g^{\ast}\times \prod_{t\in T}\operatorname{dom}g_{t}^{\ast}$, and the Lagrange function $L: H^{*}\times {\mathbb {R}}_{+}^{(T)}\to\overline{{\mathbb {R}}}$ for (1.1) is defined by

$$ L\bigl( w^{*}, \lambda\bigr):=g^{\ast}\bigl(u^{\ast}\bigr)+\sum_{t\in T}\lambda_{t}g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr) - \biggl(f+\delta_{C}+\sum _{t\in T}\lambda_{t}f_{t} \biggr)^{\ast}\biggl(u^{\ast}+\sum_{t\in T} \lambda_{t}v_{t}^{\ast}\biggr) $$

(1.3)

for any $( w^{*},\lambda)\in H^{*}\times {\mathbb {R}}_{+}^{(T)}$ with $w^{*}=(u^{*}, (v_{t}^{*}))\in H^{*}$ and $\lambda=(\lambda_{t})\in {\mathbb {R}}_{+}^{(T)}$, and they established some Lagrangian dualities between (P) and (D).

Usually, the main interest for the above optimization problems is focused on two aspects: one is about strong Lagrangian duality and the other is about total Lagrangian duality. For the strong Lagrangian duality for problem (1.1), one seeks conditions ensuring

$$ \inf_{x\in A} \bigl\{ f(x)-g(x)\bigr\} =\max _{\lambda\in {\mathbb {R}}_{+}^{(T)}}\inf_{w^{\ast}\in H^{\ast}}L\bigl( w^{*}, \lambda\bigr); $$

(1.4)

and, for the problem of total Lagrangian duality, one seeks conditions ensuring the following equality holds:

$$ \min_{x\in A} \bigl\{ f(x)-g(x)\bigr\} =\max _{\lambda \in {\mathbb {R}}_{+}^{(T)}}\inf_{w^{\ast}\in H^{\ast}}L\bigl( w^{*}, \lambda\bigr), $$

(1.5)

where $A:=\{x\in C: f_{t}(x)-g_{t}(x)\le0, \mbox{for each } t\in T\}$. To establish the strong Lagrangian duality, the authors in [15] introduced the following constraint qualification (the conical $(WEHP)$):

$$\begin{aligned} \operatorname{epi}(f-g+\delta_{A})^{\ast}={}&\bigcup _{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap_{(u^{\ast},v^{\ast})\in H^{\ast}} \biggl( \operatorname{epi} \biggl(f+\delta_{C}+\sum _{t\in T}\lambda_{t}f_{t} \biggr)^{\ast}-\bigl(u^{\ast},g^{\ast}\bigl(u^{\ast}\bigr)\bigr)\\ &{}-\sum_{t\in T}\lambda_{t} \bigl(v_{t}^{\ast}, g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)\bigr) \biggr) \biggr), \end{aligned}$$

and to consider the total Lagrangian duality, the authors in [16] introduced two constraint qualifications: the quasi-$(WBCQ)$

$$\partial (f-g+\delta_{A}) (x)\subseteq\bigcup _{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap_{(u^{\ast},v^{\ast})\in\partial H(x)} \biggl( \partial\biggl(f+\delta_{C}+\sum_{t\in T(x)} \lambda_{t}f_{t}\biggr) (x)-u^{\ast}-\sum _{t\in T(x)}\lambda_{t}v_{t}^{\ast}\biggr) \biggr), $$

and the $(WBCQ)$

$$\partial (f-g+\delta_{A}) (x)\subseteq\bigcup _{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap_{(u^{\ast},v^{\ast})\in H^{\ast}} \biggl( \partial\biggl(f+\delta_{C}+\sum_{t\in T(x)} \lambda_{t}f_{t}\biggr) (x)-u^{\ast}-\sum _{t\in T(x)}\lambda_{t}v_{t}^{\ast}\biggr) \biggr), $$

where $\partial H(x):=\partial g(x )\times\prod_{t\in T}\partial g_{t}( x )$, for each $x\in X$ and $T(x):=\{t\in T: f_{t}(x)-g_{t}(x)=0\}$.

In this paper, we continuous to study the general case, that is, C is not necessarily closed and f, g, $f_{t}$, $g_{t}$, $t\in T$, are not necessarily lsc. Our main aim in the present paper is focused on the relationships among the conical $(WEHP)$, the quasi-$(WBCQ)$, and the $(WBCQ)$. The paper is organized as follows. The next section contains some necessary notations and preliminary results. In Section 3, some relationships among the conical $(WEHP)$, the quasi-$(WBCQ)$, and the $(WBCQ)$ are obtained and some examples illustrating the relationships are given.

2 Notations and preliminaries

The notations used in this paper are standard (cf. [17]). In particular, we assume throughout the whole paper that X is a real locally convex space and let $X^{*}$ denote the dual space of X. For $x\in X$ and $x^{*}\in X^{*}$, we write $\langle x^{*},x\rangle$ for the value of $x^{*}$ at x, that is, $\langle x^{*},x\rangle:=x^{\ast}(x)$. Let Z be a set in X. The closure of Z is denoted by clZ. If $W\subseteq X^{\ast}$, then clW denotes the weak^∗-closure of W. For the whole paper, we endow $X^{\ast}\times{\mathbb{R}}$ with the product topology of $w^{\ast}(X^{\ast},X)$ and the usual Euclidean topology.

The normal cone of Z at $z_{0}\in Z$ is denoted by $N_{Z}(z_{0})$ and is defined by

$$N_{Z}(z_{0})=\bigl\{ x^{\ast}\in X^{\ast}: \bigl\langle x^{\ast},z-z_{0} \bigr\rangle \le0 \mbox{ for all }z \in Z\bigr\} . $$

The indicator function $\delta_{Z}$ of Z is defined by

$$\delta_{Z}(x):= \left \{ \begin{array}{@{}l@{\quad}l} 0,&x\in Z,\\ +\infty, &\mbox{otherwise}. \end{array} \right . $$

Let f be a proper function defined on X. The effective domain, the conjugate function, and the epigraph of f are denoted by domf, $f^{*}$, and epif, respectively; they are defined by

$$\begin{aligned}& \operatorname{dom} f:=\bigl\{ x\in X: f(x)< +\infty\bigr\} , \\& f^{*}\bigl(x^{*}\bigr):=\sup\bigl\{ \bigl\langle x^{*},x\bigr\rangle -f(x): x\in X\bigr\} ,\quad \mbox{for each }x^{*}\in X^{*}, \end{aligned}$$

and

$$\operatorname{epi} f:=\bigl\{ (x,r)\in X\times {\mathbb {R}}: f(x)\le r\bigr\} . $$

It is well known and easy to verify that $\operatorname{epi} f^{*}$ is weak^∗-closed. The closure of f is denoted by clf, which is defined by

$$\operatorname{epi} (\operatorname{cl}f)=\operatorname{cl}(\operatorname{epi} f). $$

Then (cf. [17, Theorems 2.3.1]),

$$ f^{*}=(\operatorname{cl} f)^{*}. $$

(2.1)

By [17, Theorem 2.3.4], if clf is proper and convex, then the following equality holds:

$$ f^{\ast\ast}=\operatorname{cl} f. $$

(2.2)

Let $x\in X$. The subdifferential of f at x is defined by

$$ \partial f(x):=\bigl\{ x^{\ast}\in X^{\ast}: f(x)+ \bigl\langle x^{\ast },y-x\bigr\rangle \leq f(y), \mbox{for each }y\in X \bigr\} $$

(2.3)

if $x\in\operatorname{dom} f$, and $\partial f(x):=\emptyset$ otherwise. We also define

$$\operatorname{dom} \partial f=\bigl\{ x \in X : \partial f(x) \neq\emptyset\bigr\} , $$

and

$$\operatorname{Im} \partial f=\bigl\{ x^{\ast}\in X^{\ast}: x^{\ast}\in \partial f(x) \mbox{ for some }x\in X\bigr\} . $$

By [17, Theorems 2.3.1 and 2.4.2(iii)], the Young-Fenchel inequality below holds:

$$ f(x)+f^{\ast}\bigl(x^{\ast}\bigr)\ge\bigl\langle x, x^{\ast}\bigr\rangle , \quad\mbox{for each pair } \bigl(x, x^{\ast}\bigr) \in X\times X^{\ast}, $$

(2.4)

and the Young equality holds:

$$ f(x)+f^{*}\bigl(x^{*}\bigr)=\bigl\langle x^{*},x\bigr\rangle \quad\mbox{if and only if}\quad x^{*}\in\partial f(x). $$

(2.5)

Furthermore, if g, h are proper functions, then

$$\begin{aligned}& \operatorname{epi} g^{\ast}+\operatorname{epi} h^{\ast}\subseteq\operatorname{epi} (g+h)^{\ast}, \end{aligned}$$

(2.6)

$$\begin{aligned}& g\le h \quad\Rightarrow\quad g^{\ast}\ge h^{\ast}\quad\Leftrightarrow\quad\operatorname{epi} g^{\ast}\subseteq\operatorname{epi} h^{\ast}, \end{aligned}$$

(2.7)

and

$$ \partial g(a)+\partial h(a)\subseteq\partial(g+h) (a), \quad\mbox{for each }a\in\operatorname{dom} g\cap\operatorname{dom} h. $$

(2.8)

We end this section with the remark that an element $p\in X^{*}$ can be naturally regarded as a function on X in such way that

$$ p(x):=\langle p,x\rangle, \quad\mbox{for each }x\in X. $$

(2.9)

Thus the following fact is clear for any $a\in {\mathbb {R}}$ and real-valued proper function f:

$$ \operatorname{epi}(f+p+a)^{*}=\operatorname{epi}f^{*}+(p,-a). $$

(2.10)

3 Relationships among constraint qualifications

Let X be a real locally convex Hausdorff vector space, and $C\subseteq X$ be a convex set. Let T be an index set and let f, g, $f_{t}$, $g_{t}$, $t\in T$ be proper convex functions such that $f-g$ and $f_{t}-g_{t}$, $t\in T$, are proper functions. Here and throughout the whole paper, following [17, p.39], we adapt the convention that $(+\infty)+(-\infty)=(+\infty)-(+\infty)=+\infty$, $0\cdot(+\infty)=+\infty$, and $0\cdot(-\infty)=0$. Then

$$ \emptyset\neq \operatorname{dom} f\subseteq\operatorname{dom} g \quad\mbox{and}\quad \emptyset\neq \operatorname{dom} f_{t}\subseteq \operatorname{dom} g_{t}. $$

(3.1)

Let $A\neq\emptyset$ be the solution set of the following system with the assumption that $A\cap \operatorname{dom} (f-g)$ is nonempty:

$$ x\in C;\quad f_{t}(x)-g_{t}(x)\le0, \quad\mbox{for each } t\in T, $$

(3.2)

and let $A^{\operatorname{cl}}$ be the solution set of the following system:

$$ x\in C;\quad f_{t}(x)-\operatorname{cl}g_{t}(x) \le0, \quad\mbox{for each } t\in T. $$

(3.3)

Then $A^{\operatorname{cl}}\subseteq A$. Following [18], we use ${\mathbb {R}}^{(T)}$ to denote the space of real tuples $\lambda=(\lambda_{t})$ with only finitely many $\lambda_{t}\neq0$, and let ${\mathbb {R}}_{+}^{(T)}$ denote the nonnegative cone in ${\mathbb {R}}^{(T)}$, that is,

$${\mathbb {R}}_{+}^{(T)}:=\bigl\{ \lambda=(\lambda_{t})\in {\mathbb {R}}^{(T)}: \lambda_{t}\ge 0, \mbox{for each } t\in T \bigr\} . $$

For simplicity, we denote

$$H^{\ast}:=\operatorname{dom}g^{\ast}\times\prod _{t\in T}\operatorname{dom} g_{t}^{\ast}$$

and

$$\partial H(x ):=\partial g(x )\times\prod_{t\in T} \partial g_{t}( x ), \quad\mbox{for each } x\in X. $$

To make the dual problem considered here well defined, we further assume that clg and $\operatorname{cl} g_{t}$, $t\in T$, are proper. Then $H^{\ast}\neq\emptyset$. For the whole paper, any elements $\lambda\in {\mathbb {R}}^{(T)}$ and $v^{\ast}\in \prod_{t\in T}\operatorname{dom} g_{t}^{\ast}$ are understood as $\lambda=(\lambda_{t})\in {\mathbb {R}}^{(T)}$ and $v^{\ast}=(v_{t}^{\ast})\in\prod_{t\in T}\operatorname{dom} g_{t}^{\ast}$, respectively. Following [15], we define the characteristic set K for the DC optimization problem (1.1) by

$$\begin{aligned} K:= \bigcup_{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap _{(u^{\ast},v^{\ast})\in H^{\ast}} \biggl(\operatorname{epi} \biggl(f+ \delta_{C}+\sum_{t\in T}\lambda_{t}f_{t} \biggr)^{\ast}-\bigl(u^{\ast},g^{\ast}\bigl(u^{\ast}\bigr)\bigr)-\sum_{t\in T}\lambda_{t} \bigl(v_{t}^{\ast}, g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)\bigr) \biggr) \biggr), \end{aligned}$$

(3.4)

where we adopt the convention that $\bigcap_{t\in\emptyset}S_{t}=X$ (see [17, p.2]). Below we will make use of the subdifferential $\partial h(x)$ for a general proper function (not necessarily convex) $h:X\to\overline{{\mathbb {R}}}$; see (2.3). Clearly, the following equivalence holds:

$$ x_{0}\mbox{ is a minimizer of }h\mbox{ if and only if }0 \in\partial h(x_{0}). $$

(3.5)

For each $x\in X$, let $T(x)$ be the active index set of system (3.2), that is,

$$T(x):=\bigl\{ t\in T: f_{t}(x)-g_{t}(x)=0\bigr\} . $$

Define $N^{\prime}(x)$ by

$$N^{\prime}(x):=\bigcup_{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap_{(u^{\ast},v^{\ast})\in H^{\ast}} \biggl(\partial\biggl(f+ \delta_{C}+\sum_{t\in T(x)}\lambda _{t}f_{t}\biggr) (x)-u^{\ast}-\sum _{t\in T(x)}\lambda_{t}v_{t}^{\ast}\biggr) \biggr) $$

(3.6)

and define $N_{0}^{\prime}(x)$ by

$$ N_{0}^{\prime}(x):=\bigcup _{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap_{(u^{\ast},v^{\ast})\in\partial H(x)} \biggl( \partial\biggl(f+\delta_{C}+\sum_{t\in T(x)} \lambda_{t}f_{t}\biggr) (x)-u^{\ast}-\sum _{t\in T(x)}\lambda_{t}v_{t}^{\ast}\biggr) \biggr). $$

(3.7)

Then, for each $x\in X$,

$$N^{\prime}(x)\subseteq N_{0}^{\prime}(x). $$

Definition 3.1

The family $\{f,g,\delta_{C}; f_{t},g_{t}:t\in T\}$ is said to satisfy

• (a) the lower semi-continuity closure ($(LSC)$) if

  $$ \operatorname{epi} (f -g+\delta_{A})^{\ast}= \operatorname{epi} (f -\operatorname{cl}g+\delta_{A^{\operatorname{cl}}})^{\ast}; $$

  (3.8)

• (b) the conical weak epigraph hull property ($(WEHP)$) if

  $$ \operatorname{epi} (f-g+\delta_{A})^{\ast}=K; $$

  (3.9)

• (c) the quasi-weakly basic constraint qualification (the quasi-$(WBCQ)$) at $x\in A$ if

  $$ \partial (f-g+\delta_{A}) (x)\subseteq N_{0}^{\prime}(x); $$

  (3.10)

• (d) the weakly basic constraint qualification (the $(WBCQ)$) at $x\in A$ if

  $$ \partial (f-g+\delta_{A}) (x)\subseteq N^{\prime}(x). $$

  (3.11)

It is said that the family $\{f,g,\delta_{C}; f_{t},g_{t}:t\in T\}$ satisfies the quasi-$(WBCQ)$ (resp. the $(WBCQ)$) if it satisfies the quasi-$(WBCQ)$ (resp. the $(WBCQ)$) at each point $x \in A$.

Remark 3.1

• (a) The notions of $(LSC)$ and the conical $(WEHP)$ were introduced in [15] and the quasi-$(WBCQ)$ and the $(WBCQ)$ were taken from [16].

• (b) Recall from [3, 4] that the family $\{\delta _{C}; f_{t}: t\in T\}$ has the conical $(WEHP)_{f}$ if

  $$ \operatorname{epi} (f+\delta_{A})^{\ast}= \bigcup_{\lambda\in R_{+}^{(T)}}\operatorname{epi} \biggl(f+ \delta_{C}+\sum_{t\in T}\lambda_{t}f_{t} \biggr) ^{\ast}$$

  (3.12)

  and has the $(WBCQ)_{f}$ at $x\in\operatorname{dom} f\cap A$ if

  $$ \partial (f+\delta_{A}) (x)= \mathop{\bigcup_{\lambda\in R_{+}^{(T)}}} _{\sum_{t\in T}\lambda_{t}f_{t}(x)=0} \partial \biggl(f+\delta_{C}+\sum _{t\in T}\lambda_{t}f_{t} \biggr) (x). $$

  (3.13)

Thus, in the special case when $g=g_{t}=0$, $t\in T$, the conical $(WEHP)$ coincides with the conical $(WEHP)_{f}$ for the family $\{\delta _{C}; f_{t}: t\in T\}$ and the quasi-$(WBCQ)$ and $(WBCQ)$ are reduced to the $(WBCQ)_{f}$ for the family $\{\delta_{C}; f_{t}: t\in T\}$.

Theorems 3.1 and 3.2 characterize the relationships among the quasi-$(WBCQ)$, the $(WBCQ)$, and the conical $(WEHP)$.

Theorem 3.1

The following implication holds:

$$ \bigl[\operatorname{epi}(f-g+\delta_{A})^{\ast}\subseteq K\bigr]\quad\Longrightarrow\quad\textit{the quasi-}(WBCQ). $$

(3.14)

Consequently,

$$ \textit{the conical } (WEHP) \quad\Longrightarrow\quad\textit{the quasi-} (WBCQ). $$

(3.15)

Proof

Suppose that $\operatorname{epi}(f-g+\delta_{A})^{\ast}\subseteq K$. To show the quasi-$(WBCQ)$, let $x_{0}\in A$ and let $x^{\ast}\in\partial(f-g+\delta _{A})(x_{0})$. Then, by (2.5),

$$\bigl\langle x^{\ast},x_{0}\bigr\rangle -(f-g+ \delta_{A}) (x_{0})=(f-g+\delta_{A})^{\ast}\bigl(x^{\ast}\bigr). $$

This implies that

$$\bigl(x^{\ast}, \bigl\langle x^{\ast},x_{0}\bigr\rangle -(f-g+\delta_{A}) (x_{0})\bigr)\in \operatorname{epi}(f-g+\delta_{A})^{\ast}\subseteq K. $$

Hence, there exists $\lambda\in {\mathbb {R}}_{+}^{(T)}$ such that, for each $(u^{\ast},v^{\ast})\in\partial H(x_{0})$,

$$\bigl(x^{\ast}, \bigl\langle x^{\ast},x_{0}\bigr\rangle -(f-g+\delta_{A}) (x_{0})\bigr)\in\operatorname{epi} \biggl(f+\delta_{C}+\sum_{t\in T} \lambda_{t}f_{t}\biggr)^{\ast}-\bigl(u^{\ast},g^{\ast}\bigl(u^{\ast}\bigr)\bigr)-\sum_{t\in T} \lambda_{t}\bigl(v_{t}^{\ast}, g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)\bigr). $$

Let $(u^{\ast},v^{\ast})\in\partial H(x_{0})$. There exists $(x_{1}^{\ast},r_{1})\in \operatorname{epi} (f+\delta_{C}+\sum_{t\in J}\lambda_{t}f_{t})^{\ast}$ such that

$$ x^{\ast}=x_{1}^{\ast}-u^{\ast}- \sum_{t\in J}\lambda_{t} v_{t}^{\ast}$$

(3.16)

and

$$ \bigl\langle x^{\ast},x_{0}\bigr\rangle -(f-g+ \delta_{A}) (x_{0})=r_{1}- g^{\ast}\bigl(u^{\ast}\bigr) -\sum_{t\in J} \lambda_{t} g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr), $$

(3.17)

where $J:=\{t\in T:\lambda_{t}\neq0\}$ is a finite subset of T. Below we only need to show that $x_{1}^{\ast}\in\partial(f+\delta_{C}+\sum_{t\in J}\lambda_{t} f_{t})(x_{0})$ and $J\subseteq T(x_{0})$. To do this, note by the definition of epigraph, one has

$$ \biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t}f_{t}\biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr)\le r_{1}. $$

(3.18)

Note that $(u^{\ast},v^{\ast})\in\partial H(x_{0})$, it follows from (2.5) that

$$ g(x_{0})+g^{\ast}\bigl(u^{\ast}\bigr)= \bigl\langle u^{\ast},x_{0} \bigr\rangle \quad\mbox{and}\quad g_{t}(x_{0})+g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)=\bigl\langle v_{t}^{\ast},x_{0} \bigr\rangle , \quad\mbox{for each } t\in T. $$

(3.19)

This together with (3.16), (3.17), and (3.18) implies that

$$\begin{aligned} &\biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t}\biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr) \\ &\quad\le\bigl\langle x^{\ast},x_{0}\bigr\rangle -(f-g+ \delta_{A}) (x_{0})+g^{\ast}\bigl(u^{\ast}\bigr)+\sum_{t\in J}\lambda_{t} g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr) \\ &\quad\le \biggl\langle x_{1}^{\ast}-u^{\ast}-\sum _{t\in J}\lambda_{t} v_{t}^{\ast},x_{0}\biggr\rangle -\biggl(f-g+\delta_{C}+\sum _{t\in J}\lambda_{t}(f_{t} -g_{t} )\biggr) (x_{0})\\ &\qquad{}+g^{\ast}\bigl(u^{\ast}\bigr)+\sum _{t\in J}\lambda_{t} g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr) \\ &\quad\le \bigl\langle x_{1}^{\ast},x_{0}\bigr\rangle - \biggl(f +\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t} \biggr) (x_{0})+\bigl\{ g(x_{0})-\bigl\langle u^{\ast},x_{0}\bigr\rangle +g^{\ast}\bigl(u^{\ast}\bigr)\bigr\} \\ &\qquad{}+ \sum _{t\in J}\lambda_{t} \bigl\{ g_{t}(x_{0})- \bigl\langle v_{t}^{\ast},x_{0}\bigr\rangle +g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr) \bigr\} \\ &\quad= \bigl\langle x_{1}^{\ast},x_{0}\bigr\rangle - \biggl(f +\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t} \biggr) (x_{0}), \end{aligned}$$

where the second inequality holds because $x_{0}\in A$. Hence,

$$\biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t}\biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr)+\biggl(f +\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t} \biggr) (x_{0})=\bigl\langle x_{1}^{\ast},x_{0}\bigr\rangle $$

since

$$\biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t}\biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr)\ge \bigl\langle x_{1}^{\ast},x_{0}\bigr\rangle -\biggl(f +\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t} \biggr) (x_{0}) $$

holds automatically by the Fenchel-Young inequality (2.4). Therefore, by (2.5), $x^{\ast}\in\partial( f+\delta_{C}+\sum_{t\in J}\lambda_{t} f_{t})(x_{0})$. To show $J\subseteq T(x_{0})$, note that $x_{0}\in A$, then

$$\biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t}\biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr)\le\bigl\langle x^{\ast},x_{0}\bigr\rangle -f(x_{0})+g(x_{0})+g^{\ast}\bigl(u^{\ast}\bigr)+\sum_{t\in J}\lambda_{t} g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr) $$

and

$$\biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t}\biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr)\ge \bigl\langle x_{1}^{\ast},x_{0}\bigr\rangle - f(x_{0})- \sum_{t\in J} \lambda_{t} f_{t} (x_{0}). $$

Thus, by (3.16) and (3.19), we have

$$\begin{aligned} f(x_{0})-g(x_{0})-\bigl\langle x^{\ast},x_{0} \bigr\rangle \le& g^{\ast}\bigl(u^{\ast}\bigr)+\sum _{t\in J}\lambda_{t} g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)-\biggl(f+\delta_{C}+\sum _{t\in J}\lambda_{t} f_{t} \biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr) \\ \le& g^{\ast}\bigl(u^{\ast}\bigr)+\sum _{t\in J}\lambda_{t} g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)- \bigl\langle x_{1}^{\ast},x_{0} \bigr\rangle + f(x_{0})+ \sum_{t\in J} \lambda_{t} f_{t} (x_{0}) \\ =&f(x_{0})-g(x_{0})-\bigl\langle x^{\ast},x_{0} \bigr\rangle +\sum_{t\in J}\lambda _{t} \bigl(f_{t}(x_{0})-g_{t}(x_{0})\bigr) \\ \le& f(x_{0})-g(x_{0})-\bigl\langle x^{\ast},x_{0} \bigr\rangle . \end{aligned}$$

Since $\lambda_{t}>0$ and $f_{t}(x_{0})-g_{t}(x_{0})\le0$, for each $t\in J$, it follows that $\lambda_{t}(f_{t}(x_{0})-g_{t}(x_{0}))=0$, that is, $f_{t}(x_{0})-g_{t}(x_{0})=0$, for each $t\in J$. Thus, $J\subseteq T(x_{0})$ and hence the quasi-$(WBCQ)$ holds. □

Theorem 3.2

If $\operatorname{dom}(f-g+\delta_{A})^{\ast}\subseteq\operatorname{im} \partial (f-g+\delta_{A})$, then

$$\textit{the } (WBCQ) \quad\Longrightarrow\quad\bigl[ \operatorname{epi}(f-g+ \delta_{A})^{\ast}\subseteq K\bigr]. $$

(3.20)

Furthermore, if the $(LSC)$ holds, then

$$ \textit{the } (WBCQ)\Longrightarrow\textit{ the conical } (WEHP). $$

(3.21)

Proof

Suppose that $\operatorname{dom}(f-g+\delta_{A})^{\ast}\subseteq\operatorname{im} \partial (f-g+\delta_{A})$ and that the $(WBCQ)$ holds. To show $\operatorname{epi}(f-g+\delta_{A})^{\ast}\subseteq K$, let $(x^{\ast},\alpha)\in\operatorname{epi}(f-g+\delta_{A})^{\ast}$. Since $x^{\ast}\in\operatorname{dom}(f-g+\delta_{A})^{\ast}\subseteq\operatorname{im}\partial(f-g+\delta_{A})$, it follows that there exists $x_{0}\in\operatorname{dom}(f-g)\cap A$ such that $x^{\ast}\in\partial (f-g+\delta_{A})(x_{0})\subseteq N^{\prime}(x_{0})$, thanks to the assumed $(WBCQ)$. This means that there exists $\lambda\in {\mathbb {R}}_{+}^{(T)}$ such that, for each $(u^{\ast},v^{\ast})\in H^{\ast}$,

$$x^{\ast}\in\partial\biggl(f+\delta_{C}+\sum _{t\in J}\lambda_{t} f_{t}\biggr) (x_{0})-u^{\ast}-\sum_{t\in J} \lambda_{t}v_{t}^{\ast}$$

for some finite subset $J\subseteq T(x_{0})$ and $\{\lambda_{t}\}\subseteq {\mathbb {R}}$ with $\lambda_{t}\ge0$, for each $t\in J$. Let $(u^{\ast},v^{\ast})\in H^{\ast}$. Then there exists $x_{1}^{\ast}\in\partial(f+\delta_{C}+\sum_{t\in J}\lambda_{t} f_{t})(x_{0})$ such that

$$ x^{\ast}=x_{1}^{\ast}-u^{\ast}- \sum_{t\in J}\lambda_{t}v_{t}^{\ast}. $$

(3.22)

By the Young equality (2.5), we have

$$ \bigl\langle x_{1}^{\ast},x_{0}\bigr\rangle = \biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t}\biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr)+\biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t}\biggr) (x_{0}) $$

(3.23)

and

$$ \bigl\langle x^{\ast},x_{0}\bigr\rangle = (f-g+ \delta_{A})^{\ast}\bigl(x^{\ast}\bigr)+(f-g+\delta _{A}) (x_{0})\le\alpha+f(x_{0})-g(x_{0}), $$

(3.24)

where the last inequality holds because of $(x^{\ast},\alpha)\in\operatorname{epi}(f-g+\delta_{A})^{\ast}$ and $x_{0}\in A$. This together with (3.22) and (3.23) implies that

$$\begin{aligned} \biggl(f+\delta_{C}+\sum_{t\in J} \lambda_{t} f_{t}\biggr)^{\ast}\bigl(x_{1}^{\ast}\bigr) \le&\bigl\langle u^{\ast},x_{0}\bigr\rangle +\sum _{t\in J}\lambda_{t}\bigl\langle v_{t}^{\ast},x_{0} \bigr\rangle +\alpha-g(x_{0})-\sum_{t\in J} \lambda_{t} f_{t}(x_{0}) \\ \le&\alpha+g^{\ast}\bigl(u^{\ast}\bigr)+\sum _{t\in J}g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)-\sum_{t\in J}\lambda_{t} \bigl(f_{t}(x_{0})-g_{t}(x_{0})\bigr) \\ =& \alpha+g^{\ast}\bigl(u^{\ast}\bigr)+\sum _{t\in J}g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr), \end{aligned}$$

where the second inequality holds by the Fenchel-Young inequality and the last equality holds because $J\subseteq T(x_{0})$. This means that

$$\biggl(x_{1}^{\ast}, \alpha+g^{\ast}\bigl(u^{\ast}\bigr)+\sum_{t\in J}g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)\biggr)\in \operatorname{epi}\biggl(f+ \delta_{C}+\sum_{t\in J}\lambda_{t} f_{t}\biggr)^{\ast}. $$

Hence,

$$\begin{aligned} \bigl(x^{\ast},\alpha\bigr) =&\biggl(x_{1}^{\ast}, \alpha+g^{\ast}\bigl(u^{\ast}\bigr)+\sum _{t\in J}g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)\biggr)-\bigl(u^{\ast},g^{\ast}\bigl(u^{\ast}\bigr) \bigr)-\sum_{t\in J}\lambda_{t} \bigl(v_{t}^{\ast},g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)\bigr) \\ \in& \operatorname{epi}\biggl(f+\delta_{C}+\sum _{t\in J}\lambda_{t} f_{t} \biggr)^{\ast}-\bigl(u^{\ast},g^{\ast}\bigl(u^{\ast}\bigr)\bigr)-\sum_{t\in J}\lambda_{t} \bigl(v_{t}^{\ast},g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)\bigr) \end{aligned}$$

and so $(x^{\ast},\alpha)\in K$ by the arbitrary of $(u^{\ast},v^{\ast})\in H^{\ast}$. Therefore,

$$ \operatorname{epi}(f-g+\delta_{A})^{\ast}\subseteq K. $$

(3.25)

Furthermore, we assume that the $(LSC)$ holds. Then (3.8) holds. By [15, Lemma 3.1], we see that

$$ K=\bigcup_{\lambda\in {\mathbb {R}}_{+}^{(T)}}\biggl(f- \operatorname{cl}g+\delta_{C}+\sum_{t\in T} \lambda_{t} (f_{t}-\operatorname{cl}g_{t}) \biggr)^{\ast}; $$

(3.26)

while by [3, (3.5)],

$$ \bigcup_{\lambda\in {\mathbb {R}}_{+}^{(T)}}\biggl(f- \operatorname{cl}g+\delta_{C}+\sum_{t\in T} \lambda_{t} (f_{t}-\operatorname{cl}g_{t}) \biggr)^{\ast}\subseteq\operatorname{epi}(f-\operatorname{cl}g+ \delta_{A^{\operatorname{cl}}})^{\ast}. $$

(3.27)

Combining (3.26), (3.27) with (3.8), we have

$$ K\subseteq\operatorname{epi}(f- g+\delta_{A})^{\ast}. $$

(3.28)

Hence, by (3.25), the conical $(WEHP)$ holds and the proof is complete. □

Remark 3.2

By [16, Remark 3.2], we see that

$$\mbox{the }(WBCQ)\quad\Longrightarrow\quad\mbox{the quasi-}(WBCQ) $$

and by Theorems 3.1 and 3.2, we get

$$\begin{aligned} &\bigl[\mbox{the } (WBCQ)\ \&\ \operatorname{dom}(f-g+\delta _{A})^{\ast}\subseteq\operatorname{im} \partial (f-g+\delta_{A}) \ \&\ \mbox{the } (LSC)\bigr] \\ &\quad\Longrightarrow\quad \mbox{the conical } (WEHP) \quad\Longrightarrow\quad \mbox{the quasi-} (WBCQ). \end{aligned}$$

By Theorems 3.1 and 3.2, we get the following corollary directly, which was given in [4, Proposition 3.1]. Note that the conical $(WEHP)_{f}$ and the $(WBCQ)_{f}$ for the family $\{\delta _{C}; f_{t}: t\in T\}$ were introduced in [3, 4]; see also Remark 3.1(ii).

Corollary 3.1

For the family $\{\delta_{C}; f_{t}: t\in T\}$, the following implication holds:

$$\textit{the conical } (WEHP)_{f} \quad\Longrightarrow\quad\textit{the quasi-} (WBCQ)_{f} $$

and

$$\textit{the conical } (WEHP)_{f} \quad\Longleftrightarrow\quad\textit{the quasi-} (WBCQ)_{f} $$

if $\operatorname{dom}(f+\delta_{A})^{\ast}\subseteq\operatorname{im} \partial (f+\delta_{A})$.

The following example illustrates (3.14) and shows that the quasi-$(WBCQ)$ in (3.14) cannot be replaced by the $(WBCQ)$.

Example 3.1

Let $X=C:={\mathbb {R}}$ and let $T=\{1\}$. Define $f,g,f_{1},g_{1}:{\mathbb {R}}\to\overline{{\mathbb {R}}}$, respectively, by

$$f(x):= \left \{ \begin{array}{@{}l@{\quad}l} x, &x\ge0,\\ +\infty, & x< 0, \end{array} \right .\qquad g(x):=\left \{ \begin{array}{@{}l@{\quad}l} 0, & x>0,\\ 1, & x= 0,\\ +\infty, &x<0, \end{array} \quad\mbox{for each } x\in {\mathbb {R}},\right . $$

$f_{1}:=\delta_{[0,+\infty)}$ and $g_{1}:=0$. Then f, g, $f_{1}$, and $g_{1}$ are proper convex functions and $A=[0,+\infty)$. Note that, for each $x\in {\mathbb {R}}$,

$$(f-g+\delta_{A}) (x )= \left \{ \begin{array}{@{}l@{\quad}l} x, & x>0,\\ -1,& x=0,\\ +\infty, &x< 0, \end{array} \right . $$

and $f+\delta_{C}+\lambda f_{1}=f$ holds, for each $\lambda\ge0$. Then, for each $x^{\ast}\in {\mathbb {R}}$, $g^{\ast}=\delta_{(-\infty,0]}$,

$$(f-g+\delta_{A})^{\ast}\bigl(x^{\ast}\bigr) =\left \{ \begin{array}{@{}l@{\quad}l} 1, & x^{\ast}\le1,\\ +\infty, &x^{\ast}>1, \end{array} \right . $$

and, for each $\lambda\ge0$,

$$(f+\delta_{C}+\lambda f_{1})^{\ast}\bigl(x^{\ast}\bigr)= \left \{ \begin{array}{@{}l@{\quad}l} 0, & x^{\ast}\le1,\\ +\infty, &x^{\ast}>1. \end{array} \right . $$

This means that $\operatorname{dom}g^{\ast}=(-\infty,0]$,

$$\operatorname{epi}(f-g+\delta_{A})^{\ast}=(-\infty,1] \times[1,+\infty) $$

and

$$\operatorname{epi}(f+\delta_{C}+\lambda f_{1})^{\ast}= (-\infty,1]\times[0,+\infty ), \quad\mbox{for each } \lambda\ge0. $$

Hence

$$K=\bigcup_{\lambda\ge0}\biggl(\bigcap _{u^{\ast}\in(-\infty,0]}\bigl(\operatorname{epi}(f+\delta_{C}+\lambda f_{1})^{\ast}-\bigl(u^{\ast},g^{\ast}\bigl(u^{\ast}\bigr)\bigr)\bigr)\biggr)=(-\infty ,1]\times[0,+\infty ). $$

This implies that $\operatorname{epi}(f-g+\delta_{A})^{\ast}\subseteq K$. Moreover, it is easy to see that, for each $x\in A$,

$$\partial g(x ) = \left \{ \begin{array}{@{}l@{\quad}l} \{0\},& x>0,\\ \emptyset, &x=0, \end{array} \right . $$

and, for each $\lambda\ge0$,

$$\partial(f-g+\delta_{A}) (x )=\partial(f+\delta_{C}+ \lambda f_{1}) (x) = \left \{ \begin{array}{@{}l@{\quad}l} 1, & x>0,\\ (-\infty,1],& x=0. \end{array} \right . $$

Hence, for each $x\in A$,

$$N_{0}^{\prime}(x)=\bigcup_{\lambda\ge0} \biggl(\bigcap_{u^{\ast}\in\partial g(x)}\bigl(\partial(f+ \delta_{C}+\lambda_{1}f_{1}) (x)-u^{\ast}\bigr)\biggr)= \left \{ \begin{array}{@{}l@{\quad}l} 1, & x>0,\\ {\mathbb {R}},& x=0, \end{array} \right . $$

and

$$N^{\prime}(x)=\bigcup_{\lambda\ge0}\biggl(\bigcap _{u^{\ast}\in\operatorname{dom}g^{\ast}}\bigl(\partial(f+\delta_{C}+ \lambda_{1}f_{1}) (x)-u^{\ast}\bigr)\biggr)= \left \{ \begin{array}{@{}l@{\quad}l} \emptyset, & x>0,\\ (-\infty,1],& x=0. \end{array} \right . $$

This means that $\partial(f-g+\delta_{A})(x)\subseteq N_{0}^{\prime}(x)$ but $\partial(f-g+\delta_{A})(x)\nsubseteq N^{\prime}(x)$, for each $x\in A$. Thus, the quasi-$(WBCQ)$ holds but not the $(WBCQ)$.

Example 3.2 illustrates Theorem 3.2 and Example 3.3 shows that the condition $(LSC)$ is essential for (3.21) to hold.

Example 3.2

Let $X=C:={\mathbb {R}}$. Define $f,g,f_{1},g_{1}:{\mathbb {R}}\rightarrow\overline{{\mathbb {R}}}$, respectively, by $f=f_{1}=g:= \delta_{(-\infty,0]}$, $g_{1}:=0$. Then f, g, $f_{1}$, and $g_{1}$ are proper convex functions. Consider the system (3.2) with $T:=\{1\}$. Then one sees that

$$A=\bigl\{ x\in {\mathbb {R}}:f_{1}(x)-g_{1}(x)\le0\bigr\} =(-\infty,0]. $$

It is easy to see that

$$ f-g+\delta_{A} =\delta_{A} \quad\mbox{and}\quad (f-g+ \delta_{A})^{\ast}=\delta _{[0,+\infty)}. $$

Hence,

$$\operatorname{dom}(f-g+\delta_{A})^{\ast}=[0,+\infty), $$

and, for each $x\in A$,

$$\partial(f-g+\delta_{A}) (x )=N_{A}(x) = \left \{ \begin{array}{@{}l@{\quad}l} \{0\},& x< 0,\\ {[}0,+\infty), & x=0. \end{array} \right . $$

This implies that $\operatorname{dom}(f-g+\delta_{A})^{\ast}\subseteq\operatorname{im} \partial (f-g+\delta_{A})$. Note that $g_{1}^{\ast}=\delta_{\{0\}}$, $g^{\ast}=\delta _{[0,+\infty)}$, and $(f+\lambda f_{1})^{\ast}=\delta_{[0,+\infty)}$, for each $\lambda\ge0$. It follows that, for each $x\in A$,

$$N^{\prime}(x)=\bigcup_{\lambda\ge0}\biggl(\bigcap _{u^{\ast}\in[0,+\infty )}\bigl(N_{A}(x)-u^{\ast}\bigr) \biggr)=\left \{ \begin{array}{@{}l@{\quad}l} \{0\},& x< 0,\\ {[}0,+\infty), &x=0. \end{array} \right . $$

Thus, $\partial(f-g+\delta_{A})(x)=N^{\prime}(x)$ and the $(WBCQ)$ holds. Therefore, by Theorem 3.1, we see that $\operatorname{epi}(f-g+\delta _{A})^{\ast}\subseteq K$. Moreover, since g is lsc, it follows that the $(LSC)$ holds. Therefore, by (3.21), one sees that the conical $(WEHP)$ holds. In fact, it is easy to see that

$$\operatorname{epi} (f-g+\delta_{A})^{\ast}=[0,+\infty) \times[0,+\infty) $$

and

$$K=\bigcup_{\lambda\ge0}\biggl(\bigcap _{u^{\ast}\in[0,+\infty)}\bigl( \operatorname{epi} (f+\lambda f_{1})^{\ast}- \bigl(u^{\ast},g^{\ast}\bigl(u^{\ast}\bigr)\bigr)\bigr) \biggr)=[0,+\infty)\times [0,+\infty). $$

Example 3.3

Let $X=C:={\mathbb {R}}$. Define $f,g,f_{1},g_{1}:{\mathbb {R}}\rightarrow\overline{{\mathbb {R}}}$ as in [15, Example 3.1], that is, $f=f_{1}:= \delta_{(-\infty,0]}$, $g_{1}:=0$ and, for each $x\in {\mathbb {R}}$,

$$g(x) :=\left \{ \begin{array}{@{}l@{\quad}l} 0, & x< 0,\\ 1, &x=0,\\ +\infty, &x>0. \end{array} \right . $$

Then f, g, $f_{1}$, and $g_{1}$ are proper convex functions. Consider the system (3.2) with $T:=\{1\}$. Then one sees that

$$A=\bigl\{ x\in {\mathbb {R}}:f_{1}(x)-g_{1}(x)\le0\bigr\} =(-\infty,0]. $$

It is easy to see that, for each $x\in {\mathbb {R}}$,

$$ (f-g+\delta_{A}) (x)= \left \{ \begin{array}{@{}l@{\quad}l} 0, &x< 0,\\ -1, & x=0,\\ +\infty, &x>0, \end{array} \right . $$

and, for each $x^{\ast}\in {\mathbb {R}}$,

$$ (f-g+\delta_{A})^{\ast}\bigl(x^{\ast}\bigr)= \left \{ \begin{array}{@{}l@{\quad}l} 1, & x^{\ast}\ge0,\\ +\infty,& x^{\ast}< 0. \end{array} \right . $$

Moreover, for each $x\in A$, we see that

$$\partial(f-g+\delta_{A}) (x)= \left \{ \begin{array}{@{}l@{\quad}l} \emptyset, &x< 0,\\ {[}0,+\infty), & x=0. \end{array} \right . $$

Thus, $\operatorname{dom}(f-g+\delta_{A})^{\ast}\subseteq\operatorname{im} \partial (f-g+\delta_{A})$. Note that $g_{1}^{\ast}=\delta_{\{0\}}$, $g^{\ast}=\delta_{[0,+\infty)}$, and $(f+\lambda f_{1})^{\ast}=\delta_{[0,+\infty)}$, for each $\lambda\ge0$. It follows that, for each $x\in A$,

$$N^{\prime}(x)=\bigcup_{\lambda\ge0}\biggl(\bigcap _{u^{\ast}\in[0,+\infty )}\bigl(N_{A}(x)-u^{\ast}\bigr) \biggr)=\left \{ \begin{array}{@{}l@{\quad}l} \{0\}, &x< 0,\\ {[}0,+\infty) & x=0. \end{array} \right . $$

Therefore, the $(WBCQ)$ holds. However, the conical $(WEHP)$ does not hold as shown in Example 3.1 in [15]. Actually, the family $\{f,g,\delta_{C}; f_{t},g_{t}:t\in T\}$ does not satisfy the $(LSC)$, since

$$\operatorname{epi} (f-g+\delta_{A})^{\ast}=[0,+\infty) \times[1,+\infty); $$

but

$$\operatorname{epi} (f-\operatorname{cl} g+\delta_{A})^{\ast}=[0,+ \infty)\times[0,+\infty). $$