Recall that a function \(\psi : Z \to \overline{\mathbb{R}}\) is said to be K-increasing if for any \(x, y \in Z\) such that \(y \le _{K} x\), we have \(\psi (y) \le \psi (x)\). A function \(\phi : X \to Y^{\bullet }\) is said to be S-convex if for all \(x,y \in \operatorname{dom} \phi := \{ x \in X : \phi (x) \in Y\}\) and all \(t \in [0,1]\),
$$ \phi \bigl(tx+(1-t)y\bigr) \le _{S} t\phi (x)+(1-t)\phi (y). $$
Throughout this paper, unless otherwise specified, \(C \subseteq X\) is a nonempty convex set, \(f:Z\rightarrow \overline{\mathbb{R}}\) is a proper convex K-increasing function, \(\varphi :X\rightarrow Z\) is a proper K-convex mapping, and \(h:X\rightarrow Y^{\bullet }\) is a proper S-convex mapping. Set
$$ (f\circ \varphi ) (x):= \textstyle\begin{cases} f(\varphi (x)) &\mbox{if } x\in \operatorname{dom} \varphi , \\ +\infty &\mbox{otherwise}. \end{cases} $$
Then \(f\circ \varphi \) is a proper convex function. For convenience, we write, for each \(\lambda \in S^{\oplus }\),
$$ (\lambda h) (x):= \textstyle\begin{cases} \langle \lambda ,h(x)\rangle &\mbox{if } x\in \operatorname{dom} h, \\ +\infty &\mbox{otherwise}. \end{cases} $$
It is easy to see that h is S-convex if and only if λh is convex for each \(\lambda \in S^{\oplus }\). Let \({A:=\{x\in C:h(x) \in -S\}}\). We always assume that \(A\cap \operatorname{dom}(f\circ \varphi -g)\neq \emptyset \). Let \(p\in X^{\ast }\). Consider the following primal problem with linear perturbation:
Define its dual Lagrange problem by
In the case where \(p=0\), problem (\(P_{p}\)) and its dual problem (\(D_{p}\)) reduce to problem (P) and problem (D), respectively. Let \(v(P_{p})\) and \(v(D_{p})\) denote optimal values of problems (\(P_{p}\)) and (\(D_{p}\)), respectively. As usual, we denote by \(S(P_{p})\) the solution set of problem (\(P_{p}\)), that is,
$$ S(P_{p}):=\{x_{0}\in A:f\bigl(\varphi (x_{0}) \bigr)-g(x_{0})-\langle p,x_{0} \rangle =\min _{x\in A}\bigl\{ f\bigl(\varphi (x)\bigr)-g(x)-\langle p,x\rangle \bigr\} . $$
As before, we use \(S(P)\) to denote \(S(P_{0})\). As is shown in [6, Example 3.2], the weak duality between (P) and (D) does not necessarily hold in general even in the case φ is an identity operator. To establish the weak duality and the stable weak duality between (P) and (D), the authors in [18] introduce the following condition:
$$ \operatorname{epi} (f\circ \varphi -g+\delta _{A})^{\ast }= \operatorname{epi} (f \circ \varphi -\operatorname{cl} g+\delta _{A})^{\ast }. $$
(9)
We will further need the following lemma, taken from [18, Proposition 4.2].
Lemma 3.1
Suppose that (9) holds. Then, for each
\(p\in X^{\ast }\), we have
$$ v(D_{p})\leq v(P_{p}), $$
(10)
that is, the stable weak duality holds between (P) and (D).
In particular, if (10) holds for \(p=0\), then we say that the weak duality holds between (P) and (D). Following [18], if (10) becomes an equality, then we say that the inf–sup-type stable zero duality gap property holds, and if \(v(P)=v(D)\), then we say the inf–sup-type zero duality gap property holds. In this section, we mainly study the min–sup-type zero duality gap property, that is, when does the inf–sup-type zero duality gap property holds between (P) and (D) (assuming that \(S(P)\neq \emptyset \)). We also study the min–sup-type stable zero duality gap property, that is, when does the following implication hold for any \(p\in X^{\ast }\):
$$ \mathrm{S}(P_{p})\neq \emptyset\quad \Rightarrow \quad \mathrm{v}(P_{p})=v(D_{p}). $$
To study the min–sup zero duality gap property of problem (P), we introduce some new constraint qualifications. For this purpose, we make use of the subdifferential \(\partial \varphi (x)\) for a general proper function (not necessarily convex) \(\varphi : X\rightarrow \overline{ \mathbb{R}}\); see (4). For simplicity, we denote
$$ {\varLambda }(x_{0})= \bigcap_{ \substack{ \epsilon >0 \\ u^{\ast }\in \operatorname{dom} g^{\ast } } } \biggl( \bigcup_{ \substack{ \lambda \in S^{\oplus },\beta \in \partial _{\epsilon }(f(\varphi (x_{0}))) \\ (\lambda h)(x_{0})\in [-\epsilon ,0]} } \bigl\{ \partial _{\epsilon }(\beta \varphi +\delta _{C}+\lambda h) (x _{0})-u^{\ast } \bigr\} \biggr) $$
and
$$ {\varLambda _{0}}(x_{0})= \bigcap _{ \substack{ \epsilon >0 \\ u^{\ast }\in \partial g(x_{0})} } \biggl(\bigcup_{ \substack{ \lambda \in S^{\oplus },\beta \in \partial _{\epsilon }(f(\varphi (x_{0}))) \\ (\lambda h)(x_{0})\in [-\epsilon ,0]} } \bigl\{ \partial _{\epsilon }(\beta \varphi +\delta _{C}+\lambda h) (x _{0})-u^{\ast }\bigr\} \biggr), $$
where \(x_{0}\in \operatorname{dom} (f\circ \varphi -g)\cap A\). By definitions we have \(\varLambda (x_{0})\subseteq \varLambda _{0}(x_{0})\).
Definition 3.2
We say that the family \(\{f,\varphi ,g;\delta _{A}\}\) satisfies
-
(i)
strong-\((ABCQ)\) at \(x_{0}\in \operatorname{dom}(f \circ \varphi -g)\cap A\) if
$$ \partial (f\circ \varphi -g+\delta _{A}) (x_{0})\subseteq {\varLambda }(x _{0}). $$
(11)
-
(ii)
\((ABCQ)\) at \(x_{0}\in \operatorname{dom}(f\circ \varphi -g)\cap A\) if
$$ {\partial }(f\circ \varphi -g+\delta _{A}) (x_{0})\subseteq {\varLambda } _{0}(x_{0}). $$
(12)
Moreover, we say the family \(\{f,\varphi ,g;\delta _{A}\}\) satisfies strong-\((ABCQ)\) (resp., the \((ABCQ)\)) if strong-\((ABCQ)\) (resp., \((ABCQ)\)) holds at each point \(x \in \operatorname{dom}(f\circ \varphi -g) \cap A\).
Remark 3.3
-
(a)
We have the following implication:
$$ \mbox{strong-}(ABCQ) \quad \Rightarrow \quad (ABCQ). $$
-
(b)
Note that, in the particular case \(g=0\), \(\operatorname{dom} g ^{\ast }=\partial g(x_{0})=\{0\}\), and hence strong-\((ABCQ)\) and \((ABCQ)\) turn into the following qualification condition:
$$ \partial (f\circ \varphi +\delta _{A}) (x)\subseteq \bigcap_{ \epsilon >0 } \biggl(\bigcup _{ \substack{ \lambda \in S^{\oplus },\beta \in \partial _{\epsilon }(f(\varphi (x))) \\ (\lambda h)(x)\in [-\epsilon ,0]} } \bigl\{ \partial _{\epsilon }(\beta \varphi +\delta _{C}+\lambda h) (x)\bigr\} \biggr), $$
(13)
which was introduced in [7] to study the zero duality gap property for problem (P).
-
(c)
In the case where \(\varphi =\mathrm{Id}_{X}\) and \(g=0\), strong-\((ABCQ)\) and \((ABCQ)\) collapse into
$$ \overline{(ABCQ)}\mbox{:}\quad \partial (f+\delta _{A}) (x) \subseteq \bigcap_{\epsilon >0} \biggl( \partial f(x) + \bigcup _{ \substack{ \lambda \in S^{\oplus } \\ (\lambda h)(x)\in [-\epsilon ,0]} } \bigl\{ \partial _{\epsilon }(\delta _{C} + \lambda h) (x) \bigr\} \biggr), $$
which was introduced in [7].
Given two proper functions \(h_{1}, h_{2}:X\rightarrow \overline{ \mathbb{R}}\), their infimal convolution is
$$ h_{1} \mathbin{\Box} h_{2}:X\rightarrow \mathbb{R}\cup \{ \pm \infty \},\qquad (h_{1} \mathbin{\Box} h_{2}) (x):=\inf _{z\in X}\bigl\{ h_{1}(z)+h_{2}(x-z)\bigr\} . $$
Recall that the authors in [18] introduced the qualification condition \((DCCQ)\)
$$ \operatorname{epi}(f\circ \varphi -g+\delta _{A})^{\ast }= \bigcap_{u^{\ast }\in \operatorname{dom} g^{\ast }}\operatorname{epi}\bigl(\digamma \mathbin{\Box} \delta ^{\ast }_{C} \mathbin{\Box} h^{\diamond } \bigr)-\bigl(u^{\ast },g^{\ast }\bigl(u ^{\ast }\bigr)\bigr) $$
(14)
to study the inf–sup-type zero duality gap property for problem (P), where the functions \(h^{\diamond }:X^{\ast }\rightarrow \overline{ \mathbb{R}}\) and \(\digamma :X^{\ast }\rightarrow \overline{\mathbb{R}}\) are defined respectively by
$$ h^{\diamond }\bigl(x^{\ast }\bigr)=\inf_{\lambda \in S^{\oplus }}(\lambda h)^{ \ast }\bigl(x^{\ast }\bigr) \quad \mbox{for } x^{\ast }\in X^{\ast } $$
and
$$ \digamma \bigl(x^{\ast }\bigr)=\inf_{\beta \in \operatorname{dom} f^{\ast }}{\bigl( \beta \varphi -f^{\ast }(\beta )\bigr)^{\ast }\bigl(x^{\ast }\bigr)} \quad \mbox{for } x^{\ast }\in X^{\ast }. $$
The following proposition on relationship between \((DCCQ)\) and \((ABCQ)\) is an extension of [7, Proposition 3.8] from \(g=0\) to the general g.
Proposition 3.4
We have the following implication:
$$ (DCCQ) \quad \Rightarrow \quad (ABCQ). $$
Proof
Suppose that condition \((DCCQ)\) holds. Let \(x_{0}\in \operatorname{dom}(f \circ \varphi -g)\cap A\) and \(p\in {\partial }(f\circ \varphi -g+ \delta _{A})(x_{0})\). By (6) we have
$$\begin{aligned} \bigl(p,\langle p,x_{0}\rangle -(f\circ \varphi -g+\delta _{A}) (x_{0})\bigr) &\in \operatorname{epi}(f\circ \varphi -g+\delta _{A})^{\ast } \\ &= \bigcap_{u^{\ast }\in \operatorname{dom} g^{\ast }}\operatorname{epi}\bigl( \digamma \mathbin{\Box} \delta ^{\ast }_{C}\mathbin{\Box} h^{\diamond } \bigr)-\bigl(u^{\ast },g^{ \ast }\bigl(u^{\ast }\bigr)\bigr). \end{aligned}$$
Let \(u^{\ast }\in \partial g(x_{0})\). Then
$$ \bigl(p+u^{\ast },\langle p,x_{0}\rangle -(f\circ \varphi -g+ \delta _{A}) (x _{0})+g^{\ast }\bigl(u^{\ast } \bigr)\bigr)\in \operatorname{epi}\bigl(\digamma \mathbin{\Box} \delta ^{\ast }_{C}\mathbin{\Box} h^{\diamond }\bigr). $$
It follows that
$$\begin{aligned} \bigl(\digamma \mathbin{\Box} \delta ^{\ast }_{C}\mathbin{\Box} h^{\diamond }\bigr) \bigl(p+u^{\ast }\bigr) & \leq \langle p,x_{0}\rangle -(f\circ \varphi -g+ \delta _{A}) (x_{0})+g ^{\ast }\bigl(u^{\ast }\bigr) \\ &=\langle p,x_{0}\rangle -(f\circ \varphi + \delta _{A}) (x_{0})+g^{ \ast }\bigl(u^{\ast }\bigr)+g(x_{0}) \\ &=\bigl\langle p+u^{\ast },x_{0}\bigr\rangle -(f\circ \varphi + \delta _{A}) (x _{0}), \end{aligned}$$
where the last equality holds by (7). This implies that
$$ \inf_{x^{\ast }_{1},x^{\ast }_{2}\in X^{\ast }}\bigl\{ \digamma \bigl(x^{\ast } _{1}\bigr)+\delta ^{\ast }_{C}\bigl(x^{\ast }_{2} \bigr)+h^{\diamond }\bigl(p+u^{\ast }-x ^{\ast }_{1}-x^{\ast }_{2} \bigr)\bigr\} \leq \bigl\langle p+u^{\ast },x_{0} \bigr\rangle -f \bigl(\varphi (x_{0})\bigr). $$
(15)
Let \(\epsilon >0\). Then by (15) there exist \(\overline{x}^{ \ast }_{1},\overline{x}^{\ast }_{2}\in X^{\ast }\) such that
$$ \digamma \bigl(\overline{x}^{\ast }_{1}\bigr)+ \delta ^{\ast }_{C}\bigl(\overline{x} ^{\ast }_{2} \bigr)+h^{\diamond }\bigl(p+u^{\ast }-\overline{x}^{\ast }_{1}- \overline{x}^{\ast }_{2}\bigr) \leq \bigl\langle p+u^{\ast },x_{0} \bigr\rangle -f\bigl( \varphi (x_{0})\bigr)+\frac{\epsilon }{3}, $$
(16)
whereas by definitions there exist \(\overline{\beta }\in \operatorname{dom}f ^{\ast }\) and \(\overline{\lambda }\in S^{\oplus }\) such that
$$ (\overline{\beta }\varphi )^{\ast }\bigl( \overline{x}^{\ast }_{1}\bigr)+f^{ \ast }(\overline{\beta }) \leq \digamma \bigl(\overline{x}^{\ast }_{1}\bigr)+ \frac{ \epsilon }{3} $$
(17)
and
$$ (\overline{\lambda }h)^{\ast }\bigl(p+u^{\ast }- \overline{x}^{\ast }_{1}- \overline{x}^{\ast }_{2} \bigr) \leq h^{\diamond }\bigl(p+u^{\ast }-\overline{x} ^{\ast }_{1}-\overline{x}^{\ast }_{2}\bigr)+ \frac{\epsilon }{3}. $$
Combining this with (16) and (17), we have
$$ (\overline{\beta }\varphi )^{\ast }\bigl( \overline{x}^{\ast }_{1}\bigr)+f^{ \ast }(\overline{\beta })+\delta ^{\ast }_{C}\bigl(\overline{x}^{\ast }_{2} \bigr) +(\overline{\lambda }h)^{\ast }\bigl(p+u^{\ast }- \overline{x}^{\ast }_{1}- \overline{x}^{\ast }_{2} \bigr)\leq \bigl\langle p+u^{\ast },x_{0} \bigr\rangle -f\bigl( \varphi (x_{0})\bigr)+\epsilon . $$
(18)
Noting \(\delta _{C}(x_{0})=0\) and \((\overline{\lambda }h)(x_{0})\leq 0\), it follows from (18) and the Young–Fenchel inequality (3) that
$$\begin{aligned} 0 & \leq f^{\ast }(\overline{\beta })+f\bigl(\varphi (x_{0}) \bigr)-(\overline{ \beta }\varphi ) (x_{0}) \\ &\leq \bigl\langle p+u^{\ast },x_{0} \bigr\rangle -(\overline{ \beta }\varphi )^{ \ast }\bigl(\overline{x}^{\ast }_{1} \bigr)-(\overline{\beta }\varphi ) (x_{0}) - \delta ^{\ast }_{C}\bigl(\overline{x}^{\ast }_{2} \bigr)-(\overline{\lambda }h)^{ \ast }\bigl(p+u^{\ast }- \overline{x}^{\ast }_{1}-\overline{x}^{\ast }_{2} \bigr)+ \epsilon \\ &\leq \bigl\langle p+u^{\ast },x_{0} \bigr\rangle -\bigl\langle \overline{x}^{ \ast }_{1},x_{0}\bigr\rangle -\bigl\langle \overline{x}^{\ast }_{2},x_{0}\bigr\rangle + \delta _{C}(x_{0})-\bigl\langle p+u^{\ast }- \overline{x}^{\ast }_{1}- \overline{x}^{\ast }_{2},x_{0} \bigr\rangle +(\overline{\lambda }h) (x_{0})+ \epsilon \\ &\leq \epsilon . \end{aligned}$$
This, together with (6), implies that \(\overline{\beta } \in \partial _{\epsilon }f(\varphi (x_{0}))\) and \((\overline{\lambda }h)(x _{0})\in [-\epsilon ,0]\). Moreover, by (18) and (3) we get that, for each \(x\in X\),
$$\begin{aligned} (\overline{\beta }\varphi ) (x_{0})+(\overline{\lambda }h) (x_{0})- \bigl\langle p+u^{\ast },x_{0}\bigr\rangle \leq &f^{\ast }(\overline{\beta })+f\bigl( \varphi (x_{0}) \bigr)+(\overline{\lambda }h) (x_{0})-\bigl\langle p+u^{\ast },x _{0} \bigr\rangle \\ \leq &-(\overline{\beta }\varphi )^{\ast }\bigl(\overline{x}^{\ast }_{1} \bigr)- \delta ^{\ast }_{C}\bigl(\overline{x}^{\ast }_{2} \bigr) -(\overline{\lambda }h)^{ \ast }\bigl(p+u^{\ast }- \overline{x}^{\ast }_{1}-\overline{x}^{\ast }_{2} \bigr) +\epsilon \\ \leq &(\overline{\beta }\varphi ) (x)+\delta _{C}(x)+(\overline{ \lambda }h) (x)-\bigl\langle p+u^{\ast },x\bigr\rangle +\epsilon . \end{aligned}$$
This yields \(p+u^{\ast }\in \partial _{\epsilon }(\overline{\beta } \varphi +\delta _{C}+\overline{\lambda }h)(x_{0})\), and hence \(p\in {\varLambda }_{0}(x_{0})\). Therefore the result holds, and the proof is complete. □
The following theorem gives a sufficient condition and a necessary condition to ensure the min–sup-type stable zero duality gap property for problem (P).
Theorem 3.5
Suppose that (9) holds. Let
\(x_{0}\in \operatorname{dom}(f\circ \varphi -g)\cap A\). Consider the following statements:
-
(i)
The family
\(\{ f,\varphi ,g;\delta _{A}\}\)
satisfies strong-\((ABCQ)\)
at
\(x_{0}\).
-
(ii)
For each
\(p\in X^{\ast }\)
such that
\(x_{0} \in S(P_{p})\), \(v(P_{p})=v(D_{p})\).
-
(iii)
The family
\(\{ f,\varphi ,g;\delta _{A}\}\)
satisfies
\((ABCQ)\)
at
\(x_{0}\).
Then
\((\mathrm{i})\Rightarrow (\mathrm{ii})\Rightarrow (\mathrm{iii})\).
Proof
\((\mathrm{i})\Rightarrow (\mathrm{ii})\). Suppose that (i) holds. Let \(p\in X^{\ast }\) be such that \(x_{0} \in S(P_{p})\), that is,
$$ f\bigl(\varphi (x_{0})\bigr)-g(x_{0})-\langle p,x_{0}\rangle =\inf_{x\in A}\bigl\{ f\bigl( \varphi (x) \bigr)-g(x)-\langle p,x\rangle \bigr\} . $$
Then by (8) \(p\in {\partial }(f\circ \varphi -g+\delta _{A})(x _{0})\), and hence \(p\in {\varLambda }(x_{0})\) by strong-\((ABCQ)\). Let \(\epsilon >0\) and \(u^{\ast }\in \operatorname{dom} g^{\ast }\). Then there exist \(\overline{\beta }\in \partial _{\epsilon }(f(\varphi (x_{0})))\) and \(\overline{\lambda }\in S^{\oplus }\) with \((\overline{\lambda }h)(x _{0})\in [-\epsilon ,0]\) such that \(p+u^{\ast }\in \partial _{\epsilon }(\overline{\beta }\varphi +\delta _{C}+\overline{\lambda }h)(x_{0})\). This implies that, for each \(x\in X\),
$$\begin{aligned} \bigl\langle p+u^{\ast },x\bigr\rangle &\leq (\overline{\beta }\varphi + \delta _{C}+\overline{\lambda }h) (x)-(\overline{\beta }\varphi +\delta _{C}+\overline{ \lambda }h) (x_{0})+\epsilon +\langle p,x_{0}\rangle +\bigl\langle u^{ \ast },x_{0}\bigr\rangle \\ &\leq (\overline{\beta }\varphi +\delta _{C}+\overline{\lambda }h) (x)-(\overline{ \beta }\varphi +\delta _{C}+\overline{\lambda }h-g) (x_{0}) +\langle p,x _{0}\rangle +g^{\ast } \bigl(u^{\ast }\bigr)+\epsilon , \end{aligned}$$
where the inequality holds since \(g^{\ast }(u^{\ast })+g(x_{0})\geq \langle u^{\ast }, x_{0}\rangle \) by (3). Thus, for each \(x\in C\),
$$ (\overline{\beta }\varphi ) (x_{0})+(\overline{\lambda }h) (x_{0})-g(x _{0}) -\langle p,x_{0}\rangle \leq (\overline{\beta }\varphi ) (x) +(\overline{ \lambda }h) (x)+g^{\ast } \bigl(u^{\ast }\bigr)- \bigl\langle p+u^{\ast },x\bigr\rangle + \epsilon . $$
(19)
Noting that \(\overline{\beta }\in \partial _{\epsilon }(f(\varphi (x _{0})))\), it follows from (6) that
$$ f\bigl(\varphi (x_{0})\bigr)+ f^{\ast }(\overline{\beta })\leq (\overline{ \beta }\varphi ) (x_{0})+\epsilon . $$
This, together with (19) and the fact \((\overline{\lambda }h)(x _{0})\in [-\epsilon ,0]\), implies that, for each \(x\in C\),
$$\begin{aligned} &f\bigl(\varphi (x_{0})\bigr)-g(x_{0})-\langle p,x_{0}\rangle \\ &\quad \leq (\overline{\beta }\varphi ) (x_{0})-f^{\ast }( \overline{\beta })-g(x _{0})-\langle p,x_{0}\rangle + \epsilon \\ &\quad \leq (\overline{\beta }\varphi ) (x) +(\overline{\lambda }h) (x)+g ^{\ast }\bigl(u^{\ast }\bigr)- \bigl\langle p+u^{\ast },x\bigr\rangle -f^{\ast }(\overline{ \beta })-(\overline{\lambda }h) (x_{0})+2\epsilon \\ &\quad \leq (\overline{\beta }\varphi ) (x)+(\overline{\lambda }h) (x)+g^{ \ast }\bigl(u^{\ast }\bigr)-\bigl\langle p+u^{\ast },x \bigr\rangle -f^{\ast }(\overline{ \beta })+3\epsilon . \end{aligned}$$
Consequently, we get that
$$\begin{aligned}& g^{\ast }\bigl(u^{\ast }\bigr)-(\overline{\beta }\varphi +\overline{\lambda } h+\delta _{C})^{\ast }\bigl(p+u^{\ast } \bigr)-f^{\ast }(\overline{\beta }) \\& \quad = \inf_{x\in C} \bigl\{ (\overline{\beta }\varphi ) (x)+( \overline{\lambda } h) (x)+g^{\ast }\bigl(u^{\ast } \bigr)-f^{\ast }(\overline{\beta })-\bigl\langle p+u ^{\ast },x\bigr\rangle \bigr\} \\& \quad \geq f\bigl(\varphi (x_{0})\bigr)-g(x_{0})-\langle p,x_{0}\rangle -3\epsilon . \end{aligned}$$
This means that
$$ v{(D_{p})}\geq f\bigl(\varphi (x_{0})\bigr)-g(x_{0})- \langle p,x_{0}\rangle -3 \epsilon . $$
Letting \(\epsilon \rightarrow 0\), we have that
$$ v{(D_{p})}\geq f\bigl(\varphi (x_{0}) \bigr)-g(x_{0})-\langle p,x_{0}\rangle =v {(P_{p})}. $$
(20)
This, together with Lemma 3.1, implies that \(v(D_{p})=v(P_{p})\).
\((\mathrm{ii})\Rightarrow (\mathrm{iii})\). Suppose that (ii) holds. Let \(p\in \partial (f\circ \varphi -g+\delta _{A})(x_{0})\). Then by (8) we see that \(x_{0}\in S(P_{p})\). This implies that
$$ f\bigl(\varphi (x_{0})\bigr)-g(x_{0})-\langle p,x_{0}\rangle =\min_{x\in A} \bigl\{ f\bigl( \varphi (x)\bigr)-g(x)-\langle p,x\rangle \bigr\} , $$
and hence, by (ii),
$$ f\bigl(\varphi (x_{0})\bigr)-g(x_{0})- \langle p,x_{0}\rangle =v{(D_{p})}. $$
(21)
We will further show that \(p\in \varLambda _{0}(x_{0})\). For this purpose, let \(\epsilon >0\) and \({u^{\ast }} \in \partial g(x_{0})\). It follows from (21) that there exists \((\overline{\lambda },\overline{ \beta }) \in S^{\oplus }\times \operatorname{dom} f^{\ast }\) such that, for each \(x\in X\),
$$\begin{aligned} f\bigl(\varphi (x_{0})\bigr)-g(x_{0})-\langle p,x_{0}\rangle &\leq g^{\ast }\bigl(u ^{\ast }\bigr)- ( \overline{\beta }\varphi +\overline{\lambda } h+\delta _{C})^{\ast } \bigl(p+u^{\ast }\bigr)-f^{\ast }(\overline{\beta })+\epsilon \\ &\leq (\overline{\beta }\varphi +\overline{\lambda } h+\delta _{C}) (x)- \bigl\langle p+{u^{\ast }},x\bigr\rangle +g^{\ast } \bigl({u^{\ast }}\bigr)-f^{\ast }(\overline{ \beta })+\epsilon , \end{aligned}$$
(22)
where the last inequality holds by (3). Note that \({u^{\ast }} \in \partial g(x_{0})\). It follows that \(g^{\ast }(u^{\ast })+g(x_{0})= \langle u^{\ast },x_{0}\rangle \). Combining this with (22), we have that, for each \(x\in C\),
$$ f\bigl(\varphi (x_{0})\bigr)-\langle p,x_{0}\rangle \leq ( \overline{\beta } \varphi +\overline{\lambda } h) (x)-\bigl\langle p+{u^{\ast }},x\bigr\rangle + \bigl\langle {u^{\ast }},x_{0} \bigr\rangle -f^{\ast }(\overline{\beta })+\epsilon, $$
that is,
$$ f\bigl(\varphi (x_{0})\bigr)-\bigl\langle p+{u^{\ast }},x_{0}\bigr\rangle \leq (\overline{ \beta }\varphi ) (x) +(\overline{\lambda }h) (x)- \bigl\langle p+{u^{\ast }},x \bigr\rangle -f^{\ast }(\overline{\beta }) +\epsilon . $$
(23)
Letting \(x=x_{0}\) and noting that \((\overline{\lambda }h)(x_{0}) \leq 0\), we see from (23) and (3) that
$$ 0\geq (\overline{\lambda }h) (x_{0})\geq f\bigl(\varphi (x_{0})\bigr)+f^{\ast }(\overline{ \beta })-(\overline{\beta } \varphi ) (x_{0})-\epsilon \geq -\epsilon . $$
This implies that \((\overline{\lambda }h)(x_{0})\in [-\epsilon ,0]\) and
$$ f\bigl(\varphi (x_{0})\bigr)+f^{\ast }(\overline{\beta })-( \overline{\beta } \varphi ) (x_{0})\leq \epsilon . $$
Thus by (6) \(\overline{\beta }\in \partial _{\epsilon }f( \varphi (x_{0}))\). Moreover, by (23) and (3) we can see that, for each \(x\in C\),
$$\begin{aligned} (\overline{\beta }\varphi ) (x_{0})+(\overline{\lambda }h) (x_{0})- \bigl\langle p+{u^{\ast }},x_{0}\bigr\rangle &\leq f\bigl(\varphi (x_{0})\bigr)+f^{\ast }(\overline{ \beta })+(\overline{\lambda }h) (x_{0})-\bigl\langle p+{u^{\ast }},x_{0} \bigr\rangle \\ &\leq (\overline{\beta }\varphi ) (x)+(\overline{\lambda }h) (x)- \bigl\langle p+{u^{\ast }},x\bigr\rangle +\epsilon . \end{aligned}$$
This, together with (6), implies that \(p+{u^{\ast }}\in \partial _{\epsilon }(\overline{\beta }\varphi +\delta _{C}+\overline{ \lambda }h)(x_{0}) \), and hence \(p\in {\varLambda }_{0}(x_{0})\). Therefore (12) holds, and the proof is complete. □
In [18, Theorem 4.5] the authors showed that condition \((DCCQ)\) implies that the inf–sup-type stable zero duality gap property holds for problem (P). Thus by definition we easily to see that the following corollary holds.
Corollary 3.6
Suppose that (9) holds. If the family
\(\{ f,\varphi ,g;\delta _{A}\}\)
satisfies condition
\((DCCQ)\), then for each
\(p\in X^{\ast }\)
such that
\(x_{0} \in S(P_{p})\), \(v(P_{p})=v(D_{p})\).
Remark 3.7
Let \(\phi :X\rightarrow [-\infty ,+\infty ]\) be an extended real-valued function. Recall from [16] (see also [15, p. 90]) that the Fréchet subdifferential of ϕ at a point \(x_{0}\) with \(|\phi (x_{0})|<\infty \) is defined by
$$ \widehat{\partial }\phi (x_{0}):=\biggl\{ x^{\ast }\in X^{\ast }: \liminf_{x\rightarrow x_{0}} \frac{{\phi (x)-\phi (x_{0})-\langle x ^{\ast },x-x_{0} \rangle }}{{\parallel x-x_{0}\parallel } }\geq 0 \biggr\} . $$
By using Fréchet subdifferential properties we can give a new constraint qualification
$$ \widehat{\partial }(f\circ \varphi -g+\delta _{A}) (x_{0})\subseteq \bigcap_{ \substack{ \epsilon >0 \\ u^{\ast }\in \operatorname{dom} g^{\ast }} } \biggl( \bigcup_{ \substack{\lambda \in S^{\oplus },\beta \in \partial _{\epsilon }(f(\varphi (x_{0}))) \\ (\lambda h)(x_{0})\in [-\epsilon ,0]} } \bigl\{ \partial _{\epsilon }(\beta \varphi +\delta _{C}+\lambda h) (x _{0})-u^{\ast } \bigr\} \biggr), $$
(24)
where \(x_{0}\in \operatorname{dom}(f\circ \varphi -g)\cap A\). Similarly to the proof of implication \((\mathrm{i})\Rightarrow (\mathrm{ii})\) in Theorem 3.5, we see the min–sup-type stable zero duality gap property also holds under condition (24).
Taking \(p=0\) in Theorem 3.5(ii), we get the following corollary.
Corollary 3.8
Suppose that (9) holds. Let
\(x_{0} \in \operatorname{dom} (f \circ \varphi -g)\cap A \cap S(P)\). If the family
\(\{ f,\varphi ,g; \delta _{A}\}\)
satisfies strong-\((ABCQ)\)
at
\(x_{0}\), then
$$ \min_{x\in A} \bigl\{ f\bigl(\varphi (x)\bigr)-g(x) \bigr\} = \inf_{u \in \operatorname{dom} g^{\ast }} \sup_{(\lambda ,\beta ) \in S^{\oplus }\times \operatorname{dom}f^{\ast }} \bigl\{ g^{\ast }\bigl(u^{\ast }\bigr)-(\beta \varphi +\delta _{C}+\lambda h)^{\ast }\bigl(u ^{\ast } \bigr)-f^{\ast }(\beta ) \bigr\} . $$