Abstract
In this paper, we present some strong and total Fenchel dualities for convex programming problems with data uncertainty within the framework of robust optimization in locally convex Hausdorff vector spaces. By using the properties of the epigraph of the conjugate functions, we give some new constraint qualifications, which characterizes completely the strong duality and the stable strong duality. Moreover, some sufficient and/or necessary conditions for the total duality and converse duality are also obtained.
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1 Introduction
Let X and Y be real locally convex Hausdorff topological vector spaces, whose dual spaces, \(X^{\ast}\) and \(Y^{\ast}\), are endowed with the weak∗-topologies \(w^{\ast}(X^{\ast}, X)\) and \(w^{\ast}(Y^{\ast}, Y)\), respectively. Let \(f:X\rightarrow\overline{{\mathbb {R}}}:={\mathbb {R}}\cup\{+\infty\} \), \(g:Y\rightarrow\overline{{\mathbb {R}}}\) be proper convex functions, and let \(A:X\rightarrow Y\) be a linear operator such that \(A(\operatorname{dom} f)\cap\operatorname{dom} g\neq\emptyset\). The classical form of convex optimization problem in the absence of data uncertainty is (see, for example, [1–7])
and its Fenchel dual problem is
where \(f^{\ast}\) and \(g^{\ast}\) are the Fenchel conjugates of f and g, respectively, and \(A^{\ast}:Y^{\ast}\rightarrow X^{\ast}\) is the adjoint operator of A.
It is well known that the optimal values of these problems, \(v(\mathcal {P})\) and \(v(\mathcal{D})\), respectively, satisfy the so-called weak duality (i.e., \(v(\mathcal{P})\geq v(\mathcal{D})\)), but a duality gap may occur (i.e., we may have \(v(\mathcal{P})> v(\mathcal{D})\)). A challenge in convex analysis has been to give sufficient conditions which guarantee the strong duality, that is, \(v(\mathcal{P})= v(\mathcal{D})\) and (\(\mathcal{D}\)) has at least an optimal solution, and guarantee the converse strong duality, which corresponds to the situation in which \(v(\mathcal{P})= v(\mathcal{D})\) and (\(\mathcal{P}\)) has at least an optimal solution (see, for instance, [1, 2, 5]). Several interiority-type conditions and epigraph type conditions have been given in order to establish the strong duality and the converse strong duality for the problem (\(\mathcal{P}\)) and problem (\(\mathcal{D}\)) in the literature (cf. [1, 2, 4–7] and the references therein). Especially, Li et al. established in [2] the strong duality and the converse strong duality in the most general setting, that is, f and g are not necessary lower semicontinuous (lsc) and A is not necessary continuous.
Recently, the mathematical programming problems under uncertainty have received much attention (cf. [8–21] and the references therein). The reason is the study of convex programming problems that are affected by data uncertainty is becoming increasingly important in optimization in many real-word optimization problems (cf. [11]). In particular, Li et al. considered the following uncertain convex programming problem (cf. [16]):
Under some additional assumptions, they established the strong duality between (\(\tilde{P}\)) and its dual problem.
Inspired by the works mentioned above, we continue to study the uncertain convex optimization problem (\(\tilde{P}\)) and the problem
Unlike in [16], we assume in this paper that \(f_{u_{1}}:X\rightarrow\overline{{\mathbb {R}}}\), \(u_{1}\in U_{1}\) and \(g_{u_{2}}:Y\rightarrow\overline{{\mathbb {R}}}\), \(u_{2}\in U_{2}\) are proper convex functions (not necessarily lsc), \(A: X\to Y\) is a linear operator, and \(U_{1}\), \(U_{2}\) are subsets of a locally convex space Z. Following [16], we define the dual problem by
In particular, in the case when \(U_{1}\) and \(U_{2}\) are singletons, problems (\(\tilde{P}\)) and (P) coincide with the problem (\(\mathcal{P}\)).
Let \(v(P)\), \(v(\tilde{P})\) and \(v(D)\) denote the optimal values of problems (P), (\(\tilde{P}\)) and (D), respectively. Obviously, \(v(D)\leq v(P)\leq v(\tilde{P})\), that is, the weak dualities hold between (P) and (D) and between (\(\tilde{P}\)) and (D). Our main aim in the present paper is to give some new regularity conditions which completely characterize the strong dualities and the stable strong dualities between (P) and (D) and between (\(\tilde{P}\)) and (D), and provide sufficient and/or necessary conditions for the total duality, the converse dualities between (P) and (D). Most results obtained in this paper seem new and are proper extensions of the known results in [2, 16]; in particular, our Theorem 3.3 extends and improves the result in [16], Theorem 5.1, and, even in the special case when \(U_{1}\) and \(U_{2}\) are singletons, our Corollary 5.1 improves the corresponding result in [2], Theorem 6.9.
This paper is organized as follows. The next section contains some necessary notations and preliminary results. In Section 3, we give some new regularity conditions which completely characterize the strong dualities and the stable strong dualities between (P) and (D) and between (\(\tilde{P}\)) and (D). Some sufficient and/or necessary conditions for the total duality and the converse duality between (P) and (D) are given in Sections 4 and 5, respectively.
2 Notations and preliminary results
The notations used in the present paper are standard (cf. [3]). In particular, we assume throughout the whole paper that X and Y are real locally convex Hausdorff topological vector spaces, and let \(X^{\ast}\) denote the dual space of X, endowed with the weak∗-topology \(w^{\ast}(X^{\ast},X)\). By \(\langle x^{\ast},x\rangle\), we shall denote the value of the functional \(x^{\ast}\in X^{\ast}\) at \(x\in X\); i.e., \(\langle x^{\ast},x\rangle=x^{\ast}(x)\). Let \(Z\subset X\), the interior, closure, convex hull, and the convex conical hull of Z are denoted by intZ, clZ, coZ, and coneZ, respectively. 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 indicator function \(\delta_{Z}:X\rightarrow \overline{{\mathbb {R}}} \) of the nonempty set Z is defined by
Let \(f:X\rightarrow\overline{{\mathbb {R}}}\) be a proper convex function. The effective domain, the epigraph and the conjugate function of f are defined, respectively, by
and
By definition, the Young-Fenchel inequality holds:
The lsc hull and the lsc convex hull of f, denoted respectively by clf and \(\operatorname{cl}(\operatorname{co} f)\), are defined by
If clf is proper, then the following equality holds (cf. [3]):
If g, h are proper, then
and
In particular, let \(p\in X^{\ast}\). Define a function on X that \(p(x):=\langle p, x\rangle\) for each \(x\in X\). Then, for any \(a\in {\mathbb {R}}\) and any function \(h:X\rightarrow\overline{{\mathbb {R}}}\),
Let \(x\in\operatorname{dom} f\). The subdifferential of f at x is the convex set defined by
Then, by definitions,
and
Moreover, by [3], Theorem 2.4.2(iii), the Young equality holds
For functions \(g, h:X\to\overline{{\mathbb {R}}}\), we define the infimal convolution of g and h as the function \(g\,\square\, h:X\rightarrow {\mathbb {R}}\cup\{\pm\infty\}\) given by
If g and h are lsc and \(\operatorname{dom} g\cap\operatorname{dom} h\neq\emptyset\), then by [3] we have that
and
3 Robust stable Fenchel duality
Let X, Y and Z be real locally convex Hausdorff topological vector spaces, \(U_{1}\) and \(U_{2}\) be subsets of Z. Let \(f_{u_{1}}:X\rightarrow\overline{{\mathbb {R}}}\), \(u_{1}\in U_{1}\), and \(g_{u_{2}}:Y\rightarrow\overline{{\mathbb {R}}}\), \(u_{2}\in U_{2}\) be proper convex functions, \(A:X\rightarrow Y\) be a linear operator such that \(\bigcap_{(u_{1}, u_{2})\in U_{1}\times U_{2}}[A(\operatorname{dom} f_{u_{1}}) \cap(\operatorname{dom} g_{u_{2}})]\neq\emptyset\). For simplicity, we denote
We shall consider the identity map \(\operatorname{id}_{{\mathbb {R}}}\) on ℝ, and the image set \((A^{\ast}\times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} g^{\ast}_{u_{2}})\) of \(\operatorname{epi} g^{\ast}_{u_{2}}\) through the map \(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}}:Y^{\ast}\times {\mathbb {R}}\rightarrow X^{\ast}\times {\mathbb {R}}\), that is,
Given \(p\in X^{\ast}\), consider the following robust optimization problems with a linear perturbation:
and
We define the corresponding Fenchel dual problem by
As usual, we use \(v(P_{p})\), \(v(\tilde{P}_{p})\) and \(v(D_{p})\) to stand for the optimal values of the problems (\(P_{p}\)), (\(\tilde{P}_{p}\)) and (\(D_{p}\)), respectively, that is,
and
By definitions, it is easy to check that the following inequalities hold:
Moreover, by definitions, we can get that
and
In particular, in the case when \(p=0\), problems (\(P_{p}\)), (\(\tilde {P}_{p}\)) and (\(D_{p}\)) are just problems (P), (\(\tilde{P}\)) and (D), which are defined by (1.4), (1.3) and (1.5), respectively. Furthermore, by (3.4) and (3.5), the following equalities hold:
and
This section is devoted to the study of the strong dualities and stable strong dualities between (P) and (D) and between (\(\tilde{P}\)) and (D), which are defined as follows.
Definition 3.1
It is said that
-
(i)
the strong duality holds between (P) and (D) (resp., between (\(\tilde{P}\)) and (D)) if \(v(P)=v(D)\) (resp., \(v(\tilde {P})=v(D)\)) and (D) has an optimal solution;
-
(ii)
the stable strong duality holds between (P) and (D) (resp., between (\(\tilde{P}\)) and (D)) if for each \(p\in X^{\ast}\), the strong duality holds between (\(P_{p}\)) and (\(D_{p}\)) (resp., between (\(\tilde{P}_{p}\)) and (\(D_{p}\))).
Remark 3.1
Clearly, if the strong duality holds between (\(\tilde{P}_{p}\)) and (\(D_{p}\)), by (3.3), then the strong duality holds between (\(P_{p}\)) and (\(D_{p}\)). However, the converse is not true in general.
Definition 3.2
The family \((f_{u_{1}}, g_{u_{2}}; A ; U )\) is said to satisfy
-
(a)
the strong further regularity condition (\(SFRC\)) if
$$\begin{aligned} &\operatorname{cl} \biggl[\operatorname{co}\bigcup _{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}}\circ A)^{\ast}\biggr]\cap \bigl(\{0\}\times {\mathbb {R}}\bigr) \\ &\quad\subseteq\bigcup _{u\in U} \bigl(\operatorname{epi} f_{u_{1}}^{\ast}+ \bigl(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}} \bigr) \bigl( \operatorname{epi} g_{u_{2}}^{\ast}\bigr) \bigr)\cap \bigl(\{ 0\} \times {\mathbb {R}}\bigr); \end{aligned}$$(3.7) -
(b)
the asymptotic further regularity condition (\(AFRC\)) if
$$\begin{aligned} &\operatorname{cl} \biggl[\bigcup_{u\in U} \operatorname{epi}(f_{u_{1}}+g_{u_{2}} \circ A)^{\ast}\biggr]\cap \bigl(\{ 0\}\times {\mathbb {R}}\bigr) \\ &\quad\subseteq\bigcup _{u\in U} \bigl(\operatorname{epi} f_{u_{1}}^{\ast}+ \bigl(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}} \bigr) \bigl( \operatorname{epi} g_{u_{2}}^{\ast}\bigr) \bigr)\cap \bigl(\{0\} \times {\mathbb {R}}\bigr); \end{aligned}$$(3.8) -
(c)
the further regularity condition (\(FRC\)) if
$$\begin{aligned} &\bigcup_{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}} \circ A)^{\ast}\cap \bigl(\{0\}\times {\mathbb {R}}\bigr) \\ &\quad\subseteq\bigcup _{u\in U} \bigl(\operatorname{epi} f_{u_{1}}^{\ast}+ \bigl(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}} \bigr) \bigl( \operatorname{epi} g_{u_{2}}^{\ast}\bigr) \bigr)\cap \bigl(\{0\} \times {\mathbb {R}}\bigr); \end{aligned}$$(3.9) -
(d)
the strong closure condition (\(SCC\)) if
$$ \operatorname{cl} \biggl[\operatorname{co}\bigcup _{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}}\circ A)^{\ast}\biggr]\subseteq\bigcup_{u\in U} \bigl( \operatorname{epi} f_{u_{1}}^{\ast}+ \bigl(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}} \bigr) \bigl(\operatorname{epi} g_{u_{2}}^{\ast}\bigr) \bigr); $$(3.10) -
(e)
the asymptotic closure condition (\(ACC\)) if
$$ \operatorname{cl} \biggl[\bigcup_{u\in U} \operatorname{epi}(f_{u_{1}}+g_{u_{2}} \circ A)^{\ast}\biggr]\subseteq\bigcup_{u\in U} \bigl(\operatorname{epi} f_{u_{1}}^{\ast}+ \bigl(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}} \bigr) \bigl(\operatorname{epi} g_{u_{2}}^{\ast}\bigr) \bigr); $$(3.11) -
(f)
the closure condition (CC) if
$$ \bigcup_{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}} \circ A)^{\ast}\subseteq\bigcup_{u\in U} \bigl( \operatorname{epi} f_{u_{1}}^{\ast}+ \bigl(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}} \bigr) \bigl(\operatorname{epi} g_{u_{2}}^{\ast}\bigr) \bigr). $$(3.12)
Remark 3.2
(i) Note from [2], Lemma 3.2, that
It follows that
Thus, the inclusions in (3.7)-(3.12) can be replaced by equalities. Moreover, by (3.13), we see that
and
(ii) Recall from [2], Definition 3.1, that the triple \((f,g;A)\) satisfies the \((CC)_{A}\) if
and from [2], Definition 4.2, that the triple \((f,g;A)\) satisfies the \((FRC)_{A}\) if
and there exists \(x^{\ast}\in X^{\ast}\) such that \((f^{\ast}\,\square\, A^{\ast}g^{\ast})(0)=f^{\ast}(-x^{\ast})+(A^{\ast}g^{\ast})(x^{\ast})\) and the infimum in the definition of \((A^{\ast}g^{\ast})(x^{\ast})\) is attained, where \(f^{\ast}\,\square\, A^{\ast}g^{\ast}\) denotes the infimal convolution of \(f^{\ast}\) and \(A^{\ast}g^{\ast}\). By [2], Proposition 4.3, the \((FRC)_{A}\) for the triple \((f,g;A)\) is equivalent to
Thus, in the case when \(U_{1}\) and \(U_{2}\) are singletons, (\(SFRC\)), (\(AFRC\)) and (\(FRC\)) are same and turned into \((FRC)_{A}\) for the triple \((f,g;A)\); meanwhile, (\(SCC\)), (\(ACC\)) and (CC) for the triple \((f_{u_{1}},g_{u_{2}};A ;U)\) are same as \((CC)_{A}\) for the triple \((f,g;A)\).
The following proposition describes the relationship between the (\(SCC\)) (resp., the (\(ACC\)), the (CC)) and the (\(SFRC\)) (resp., the (\(AFRC\)), the (\(FRC\))).
Proposition 3.1
The family \((f_{u_{1}}, g_{u_{2}}; A; U)\) satisfies the (\(SCC\)) (resp., the (\(ACC\)), the (CC)) if and only if for each \(p\in X^{\ast}\), \((f_{u_{1}}-p, g_{u_{2}}; A; U)\) satisfies the (\(SFRC\)) (resp., the (\(AFRC\)), the (\(FRC\))).
Proof
Let \(p\in X^{\ast}\) and let \(K_{1}(p)\), \(K_{2}(p)\) be defined by
and
respectively. Then, by (2.2), the following equalities are clear:
Hence, we have that
and
Thus, the conclusion holds by definitions, and the proof is complete. □
Lemma 3.1
Let \(r\in {\mathbb {R}}\). Then the following assertions hold:
-
(i)
\((p, r)\in\bigcup_{u\in U}(\operatorname{epi} f_{u_{1}}^{\ast}+(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} g_{u_{2}}^{\ast}))\) if and only if there exist \(\bar{u}=(\bar{u}_{1}, \bar{u}_{2})\in U\) and \(y^{\ast}\in Y^{\ast}\) satisfying
$$ f_{\bar{u}_{1}}^{\ast}\bigl(p-A^{\ast}y^{\ast}\bigr)+g_{\bar{u}_{2}}^{\ast}\bigl(y^{\ast}\bigr)\leq r. $$ -
(ii)
Suppose that
$$\begin{aligned} \begin{aligned}&f_{u_{1}}:X\rightarrow\overline{{\mathbb {R}}},\quad u_{1} \in U_{1},\qquad g_{u_{2}}:Y\rightarrow\overline{ {\mathbb {R}}},\quad u_{2}\in U_{2}\\ &\textit{are lsc and }A\textit{ is continuous}. \end{aligned} \end{aligned}$$(3.14)Then \((p, r)\in\operatorname{cl} [\operatorname{co} \bigcup_{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}}\circ A)^{\ast}]\) if and only if \(v(\tilde{P}_{p})\geq-r\).
Proof
Without loss of generality, we assume that \(p=0\).
(i) Let \((0, r)\in\bigcup_{u\in U}(\operatorname{epi} f_{u_{1}}^{\ast}+(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} g_{u_{2}}^{\ast}))\). Then there exists \(\bar{u}=(\bar{u}_{1}, \bar{u}_{2})\in U\) such that \((0, r)\in(\operatorname{epi} f_{\bar{u}_{1}}^{\ast}+(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} g_{\bar{u}_{2}}^{\ast}))\). Thus, there exist \((x^{\ast}_{1}, r_{1})\in \operatorname{epi} f^{\ast}_{\bar{u}_{1}}\) and \((x_{2}^{\ast}, r_{2})\in(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} g^{\ast}_{\bar{u}_{2}})\) such that
Moreover, there exists \(\bar{y}^{\ast}\in Y^{\ast}\) such that \(A^{\ast}\bar {y}^{\ast}=x_{2}^{\ast}\) and \((\bar{y}^{\ast}, r_{2})\in\operatorname{epi} g_{\bar {u}_{2}}^{\ast}\). Consequently, \(f_{\bar{u}_{1}}^{\ast}(x_{1}^{\ast})\leq r_{1}\) and \(g_{\bar{u}_{2}}^{\ast}(\bar{y}^{\ast})\leq r_{2}\). This together with (3.15) implies that
Conversely, suppose that there exist \(\bar{u}=(\bar{u}_{1}, \bar{u}_{2})\in U\) and \(y^{\ast}\in Y^{\ast}\) satisfying
Let \(x_{1}^{\ast}:=-A^{\ast}y^{\ast}\), \(x_{2}^{\ast}:=A^{\ast}y^{\ast}\), \(r_{1}:= f_{\bar{u}_{1}}^{\ast}(-A^{\ast}y^{\ast})\) and \(r_{2}:=r-r_{1}\). Then \((x_{1}^{\ast},r_{1})\in\operatorname{epi} f_{\bar{u}_{1}}^{\ast}\), \((\bar{y}^{\ast}, r_{2}) \in\operatorname{epi} g_{\bar{u}_{2}}^{\ast}\) and \((x_{2}^{\ast}, r_{2})\in(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} g^{\ast}_{\bar{u}_{2}})\). Thus,
(ii) Define the function \(h: X\rightarrow\overline{{\mathbb {R}}}\) by
Then, by definition,
This means
Note by [22], Proposition 4.1, that
It follows from (3.17) that
The proof is complete. □
The following theorem provides a characterization for the strong duality between (P) and (D) in terms of the (\(AFRC\)) and the (\(FRC\)).
Theorem 3.1
Consider the following statements:
-
(i)
The family \((f_{u_{1}}, g_{u_{2}}; A ;U)\) satisfies the (\(AFRC\)).
-
(ii)
The strong duality holds between (P) and (D).
-
(iii)
The family \((f_{u_{1}}, g_{u_{2}}; A;U )\) satisfies the (\(FRC\)).
Then (i) ⇒ (ii) ⇒ (iii). Furthermore, (i) ⇔ (ii) ⇔ (iii) if and only if
Proof
(i) ⇒ (ii) Suppose that the family \((f_{u_{1}}, g_{u_{2}}; A ;U)\) satisfies the (\(AFRC\)). If \(v(P)=-\infty\), then, by the weak duality, \(v(D)=-\infty\). Hence, the conclusion follows automatically. Below we assume that \(-r:=v(P)\in {\mathbb {R}}\). Then, by (3.6), \(r=\inf_{u\in U}(f_{u_{1}}+g_{u_{2}}\circ A)^{\ast}(0)\). Thus, for each \(\varepsilon> 0\), there exists \(\bar{u}=(\bar{u}_{1}, \bar{u}_{2})\in U\) such that
This implies that
Letting \(\varepsilon\rightarrow0\). We get that
where the last conclusion holds by the (\(AFRC\)). Then, by Lemma 3.1(i), there exist \(\bar{u}=(\bar{u}_{1}, \bar{u}_{2})\in U\) and \(y^{\ast}\in Y^{\ast}\) such that (3.16) holds. Thus, by definition of \(v(D)\),
Therefore, by the weak duality, we see that \(v(P)=v(D)\) and \(\bar {u}=(\bar{u}_{1},\bar{u}_{2})\in U\) and \(y^{\ast}\in Y^{\ast}\) are the optimal solution of (D).
(ii) ⇒ (iii) Suppose that the strong duality holds between (P) and (D). Let \((0, r)\in\bigcup_{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}}\circ A)^{\ast}\). Then there exists \(\hat{u}=(\hat{u}_{1}, \hat{u}_{2})\in U\) such that \((0, r)\in\operatorname{epi}(f_{\hat{u}_{1}}+g_{\hat{u}_{2}}\circ A)^{\ast}\). Thus,
where the last equality holds by (3.6). Hence, by (ii), we see that there exist \(\bar{u}=(\bar{u}_{1}, \bar{u}_{2})\in U\) and \(y^{\ast}\in Y^{\ast}\) such that (3.16) holds. This together with Lemma 3.1(i) implies that \((0, r)\in\bigcup_{u\in U}(\operatorname{epi} f_{u_{1}}^{\ast}+(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} g_{u_{2}}^{\ast}))\). Therefore, (3.9) is seen to hold and the implication (ii) ⇒ (iii) is proved.
Suppose that (3.18) holds. Then the (\(FRC\)) is equivalent to the (\(AFRC\)). Thus, (i) ⇔ (ii) ⇔ (iii) and the proof is complete. □
By Theorem 3.1 and Proposition 3.1, we get the following theorem straightforwardly.
Theorem 3.2
Consider the following statements.
-
(i)
The family \((f_{u_{1}}, g_{u_{2}}; A ; U)\) satisfies the (\(ACC\)).
-
(ii)
The stable strong duality holds between (P) and (D).
-
(iii)
The family \((f_{u_{1}}, g_{u_{2}}; A ; U )\) satisfies the (CC).
Then (i) ⇒ (ii) ⇒ (iii). Furthermore, (i) ⇔ (ii) ⇔ (iii) if and only if
The following theorem provides a characterization for the strong duality and the stable strong duality to hold.
Theorem 3.3
Suppose that (3.14) holds. Then the following assertions hold.
-
(i)
The strong duality holds between (\(\tilde{P}\)) and (D) if and only if the family \((f_{u_{1}}, g_{u_{2}};A; U)\) satisfies the (\(SFRC\));
-
(ii)
The stable strong duality holds between (\(\tilde{P}\)) and (D) if and only if the family \((f_{u_{1}}, g_{u_{2}}; A )\) satisfies the (\(SCC\)).
Proof
Assertion (ii) is a global version of assertion (i). Hence, by Proposition 3.1, we only need to prove (i). To do this, suppose that the strong duality holds between (\(\tilde{P}\)) and (D). Let \((0, r)\in\operatorname{cl} [\operatorname{co}\bigcup_{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}}\circ A)^{\ast}]\). Then, by Lemma 3.1(ii), we have that \(v(\tilde{P})\geq-r\). Since the strong duality holds between (\(\tilde{P}\)) and (D), it follows that \(v(D)\geq-r\) and there exist \(\bar{u}=(\bar{u}_{1}, \bar{u}_{2})\in U\) and \(y^{\ast}\in Y^{\ast}\) such that (3.16) holds. This together with Lemma 3.1(i) implies that \((0, r)\in\bigcup_{u\in U}(\operatorname{epi} f_{u_{1}}^{\ast}+(A^{\ast}\times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} g_{u_{2}}^{\ast}))\). Therefore, (3.7) holds.
Conversely, suppose that the family \((f_{u_{1}}, g_{u_{2}}; A ;U)\) satisfies the (\(SFRC\)). If \(v(\tilde{P})=-\infty\), then, by the weak duality, \(v(D)=-\infty\). Hence, the conclusion follows automatically. Below we assume that \(-r:=v(\tilde{P})\in {\mathbb {R}}\). Then, by Lemma 3.1(ii),
where the last conclusion holds because of the (\(SFRC\)). This together with Lemma 3.1(i) implies that there exist \(\bar {u}=(\bar{u}_{1}, \bar{u}_{2})\in U\) and \(y^{\ast}\in Y^{\ast}\) such that (3.16) holds. It follows that
Thus, by the weak duality, we see that \(v(\tilde{P})=v(D)\) and \(\bar {u}\in U\), \(y^{\ast}\in Y^{\ast}\) are the optimal solution of (D). Therefore, the strong duality holds between (\(\tilde{P}\)) and (D) and the proof is complete. □
Remark 3.3
Let X, Y, Z be Banach spaces and let \(U_{1}\), \(U_{2}\) be subsets of Z. Li et al. established in [16], Theorem 5.1, the strong duality between (\(\tilde{P}\)) and (D) under the assumptions that (3.14) holds and
In this case, it is easy to see that (3.19) is equivalent to the (\(SFRC\)). Thus, [16], Theorem 5.1, follows from Theorem 3.3 directly.
The corollary follows directly from Theorem 3.3, Theorem 3.1 and Definition 3.2. It provides the friendships between the strong duality of (\(\tilde{P}\)) and (D) and the strong duality of (P) and (D).
Corollary 3.1
Suppose that (3.14) holds. Then the following statements are equivalent.
-
(i)
The strong duality holds between (\(\tilde{P}\)) and (D).
-
(ii)
The strong duality holds between (P) and (D), and \(\bigcup_{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}}\circ A)^{\ast}\cap(\{0\}\times {\mathbb {R}})\) is \(w^{\ast}\)-closed and convex.
Combining Corollary 3.1 and Proposition 3.1, we have the following corollary.
Corollary 3.2
Suppose that (3.14) holds. Then the following statements are equivalent.
-
(i)
The stable strong duality holds between (\(\tilde{P}\)) and (D).
-
(ii)
The stable strong duality holds between (P) and (D), and \(\bigcup_{u\in U}\operatorname{epi}(f_{u_{1}}+g_{u_{2}}\circ A)^{\ast}\) is \(w^{\ast}\)-closed and convex.
4 Total duality
Recall that the problem (\(P_{p}\)) and the corresponding dual problem (\(D_{p}\)) are defined by (3.1) and (3.2), respectively. Let \(p\in X^{\ast}\) and \(u=(u_{1},u_{2})\in U\), we define the subproblem of the problem (\(P_{p}\)) by
and use \(v(P_{p}^{u})\) to denote the optimal value of the problem (\(P_{p}^{u}\)). Let \((u_{1},u_{2})\in U\) and let \(x_{0}\in\bigcap_{u\in U}[(\operatorname{dom} f_{u_{1}})\cap A^{-1}(\operatorname{dom} g_{u_{2}})]\). We say the Moreau-Rockafellar formula holds at \((u_{1}, u_{2}; x_{0})\) if
This section is devoted to the study of characterizing the total dualities. For this purpose, let \(p\in X^{\ast}\) and \(S(P_{p})\) denote the optimal solution set of (\(P_{p}\)), that is,
and for each \(u=(u_{1},u_{2})\in U\), let \(S(P_{p}^{u})\) denote the optimal solution set of (\(P_{p}^{u}\)), that is,
Theorem 4.1
Let \(x_{0}\in\bigcap_{u\in U}[(\operatorname{dom} f_{u_{1}})\cap A^{-1}(\operatorname{dom} g_{u_{2}})]\) and \(\bar{u}= (\bar{u}_{1},\bar{u}_{2})\in U\). If the Moreau-Rockafellar formula holds at \((\bar{u}_{1}, \bar{u}_{2}; x_{0})\), then, for each \(p\in X^{\ast}\) satisfying \((\bar{u}_{1}, \bar{u}_{2}; x_{0})\in S(P_{p})\),
Conversely, if (4.2) holds for each \(p\in X^{\ast}\) satisfying \(x_{0}\in S(P_{p}^{\bar{u}})\), then the Moreau-Rockafellar formula holds at \((\bar{u}_{1}, \bar{u}_{2}; x_{0})\).
Proof
Suppose that the Moreau-Rockafellar formula holds at \((\bar{u}_{1}, \bar {u}_{2}; x_{0})\). Let \(p\in X^{\ast}\) be such that \((\bar{u}_{1}, \bar{u}_{2}; x_{0})\in S(P_{p})\), that is,
Then, by (2.3), we see that \(0\in\partial(f_{\bar{u}_{1}}-p+g_{\bar {u}_{2}}\circ A)(x_{0})\), which is equivalent to \(p\in\partial(f_{\bar {u}_{1}}+g_{\bar{u}_{2}}\circ A)(x_{0})\). This together with (4.1) implies that \(p\in\partial f_{\bar {u}_{1}}(x_{0})+A^{\ast}\partial g_{\bar{u}_{2}}(Ax_{0})\). Therefore, there exist \(p_{1}\in\partial f_{\bar{u}_{1}}(x_{0})\) and \(p_{2}\in\partial g_{\bar{u}_{2}}(Ax_{0})\) such that
Applying the Young equality (2.4), one has
Noting that \(\langle p_{2}, Ax_{0}\rangle=\langle A^{\ast}p_{2}, x_{0}\rangle\) and using (4.4),
Hence,
while, by the definition of \(v(D_{p})\) and noting \(p_{2}\in Y^{\ast}\), one has that
Combining this with the weak duality, we see that
Thus (4.2) holds.
Conversely, let \(p\in\partial(f_{\bar{u}_{1}}+g_{\bar{u}_{2}}\circ A)(x_{0})\). Then \(0\in\partial(f_{\bar{u}_{1}}+g_{\bar{u}_{2}}\circ A-p)(x_{0})\) and by (2.3),
This means \(x_{0}\in S(P_{p}^{\bar{u}})\). Thus, by (4.2), we have that
This implies that there exists \(q\in Y^{\ast}\) such that
Noting the above inequality and using the definition of the conjugate function, we see that
It follows that
Combing this with (2.4), we have that \(q\in\partial g_{\bar {u}_{2}}(x_{0})\) and \(p-A^{\ast}q\in\partial f_{\bar{u}_{1}}(x_{0})\). Consequently,
and the set on the left-hand side of the Moreau-Rockafellar formula is contained in the set on the right-hand side. This completes the proof because the converse inclusion holds automatically. □
Remark 4.1
-
(i)
In the case when \(U_{1}\) and \(U_{2}\) are singletons, the Moreau-Rockafellar formula (4.1) reduces to the Moreau-Rockafellar formula in [23], p.47, which was also introduced in [2] to establish the stable total Fenchel duality for (\(\mathcal{P}\)) and (\(\mathcal{D}\)) (see (1.1) and (1.2)).
-
(ii)
By Theorem 4.1 and Theorem 3.2, it is easy to see that if (\(ACC\)) holds, then the Moreau-Rockafellar formula (4.1) holds at each \((u_{1},u_{2};x_{0})\in U_{1}\times U_{2}\times \bigcap_{u\in U}[(\operatorname{dom} f_{u_{1}})\cap A^{-1}(\operatorname{dom} g_{u_{2}})]\).
5 Converse duality
For each \(p\in X\) and \(q\in Y\), we consider the following perturbed robust optimization problem:
and the corresponding dual problem
Let \(u=(u_{1},u_{2})\in U\). We define the subproblem of (\(CP_{(p, q)}\)) by
As before, we use \(v(CP_{(p, q)})\), \(v(CD_{(p, q)})\) and \(v(CP_{(p,q)}^{u})\) to stand for the optimal values of the problems (\(CP_{(p, q)}\)), (\(CD_{(p, q)}\)) and (\(CP_{(p,q)}^{u}\)), respectively. Clearly, the weak duality holds, that is,
Moreover, by definitions, we have the following equalities:
and
In fact, (5.4) comes from the definitions, (5.3) follows from (5.2), and (5.2) holds because
In particular, in the case when \(p=q=0\), problems (\(CP_{(p, q)}\)) and (\(CD_{(p, q)}\)) are just as the problem (P) and problem (D), respectively. Thus, by (5.3), (5.4) and (5.2), we see that
and
Definition 5.1
It is said that
-
(i)
the converse duality holds between (P) and (D) if \(v(P)=v(D)\) and for each \(u\in U\) satisfying \(v(P^{u})=v(P)\), the problem (\(P^{u}\)) has an optimal solution;
-
(ii)
the stable converse duality holds between (P) and (D) if the converse duality holds between (\(CP_{(p, q)}\)) and (\(CD_{(p, q)}\)) for each \(p\in X\) and \(q\in Y\).
To study the converse duality and the stable converse duality between (P) and (D), we introduce the following regularity conditions.
Definition 5.2
The family \((f_{u_{1}}, g_{u_{2}}; A ;U)\) is said to satisfy
-
(i)
the converse further regularity condition (\(CFRC\)) if
$$\begin{aligned} &\bigcap_{u\in U}\operatorname{epi} \bigl(g_{u_{2}}^{\ast}+f_{u_{1}}^{\ast}\circ \bigl(-A^{\ast}\bigr) \bigr)^{\ast}\cap \bigl(\{0\}\times {\mathbb {R}}\bigr) \\ &\quad\subseteq\bigcap_{u\in U} \bigl[( \operatorname{epi} g_{u_{2}}+ \bigl((-A) \times \operatorname{id}_{{\mathbb {R}}} \bigr) ( \operatorname{epi} f_{u_{1}}) \bigr]\cap \bigl(\{0\}\times {\mathbb {R}}\bigr); \end{aligned}$$(5.7) -
(ii)
the converse closure condition (\(CCC\)) if
$$ \bigcap_{u\in U}\operatorname{epi} \bigl(g_{u_{2}}^{\ast}+f_{u_{1}}^{\ast}\circ \bigl(-A^{\ast}\bigr) \bigr)^{\ast}\subseteq\bigcap _{u\in U} \bigl[(\operatorname{epi} g_{u_{2}}+ \bigl((-A) \times \operatorname{id}_{{\mathbb {R}}} \bigr) (\operatorname{epi} f_{u_{1}}) \bigr]. $$(5.8)
Remark 5.1
Recall from [2], Definition 6.2, that the triple \((f,g;A)\) satisfies the converse \((FRC)_{A}\) if
and there exists \(y\in Y\) such that \(( g \,\square\, (-A)f)(0)=g(-y)+(-A)f(y)\) and the infimum in the definition of \((-A)f(y)\) is attained, where \(g \,\square\,(-A)f\) denotes the infimal convolution of g and \((-A)f\), and from [2], Definition 6.5, that the triple \((f,g;A)\) satisfies the converse \((CC)_{A}\) if
By [2], Proposition 6.3, the converse \((FRC)_{A}\) for the triple \((f,g;A)\) is equivalent to
Thus, in the case when \(U_{1}\) and \(U_{2}\) are singletons, the (\(CFRC\)) and (\(CCC\)) for the family \((f_{u_{1}}, g_{u_{2}}; A ;U)\) are reduced into the converse \((FRC)_{A}\) and the converse \((CC)_{A}\), respectively, for the triple \((f,g;A)\).
Theorem 5.1
-
(i)
The family \((f_{u_{1}}, g_{u_{2}};A; U)\) satisfies the (\(CFRC\)) if and only if the converse duality holds between (P) and (D).
-
(ii)
The family \((f_{u_{1}}, g_{u_{2}};A ; U)\) satisfies the (\(CCC\)) if and only if the stable converse duality holds between (P) and (D).
Proof
The proof of (i) is similar to that of (ii). Hence, we only need to prove (ii). To do it, suppose that the family \((f_{u_{1}}, g_{u_{2}};A;U)\) satisfies the (\(CCC\)). Let \(p\in X\) and \(q\in Y\) and let \(r:=v(CD_{(p, q)})\). Then, by (5.4), \(r=\sup_{u\in U}(g_{u_{2}}^{\ast}+f_{u_{1}}^{\ast}\circ(-A^{\ast}))^{\ast}(Ap+q)\). Thus, for each \(u\in U\),
This implies that \((Ap+q, r)\in\operatorname{epi}(g_{u_{2}}^{\ast}+f_{u_{1}}^{\ast}\circ (-A^{\ast}))^{\ast}\), and by the arbitrariness of \(u\in U\),
thanks to the (\(CCC\)). Let \(u\in U\) be arbitrary. Then
Thus, there exist \((y_{1}, r_{1})\in\operatorname{epi} g_{u_{2}}\) and \((y_{2}, r_{2})\in ((-A) \times \operatorname{id}_{{\mathbb {R}}})(\operatorname{epi} f_{u_{1}}) \) such that
Moreover, there exists \(\hat{x}\in X\) such that \(-A\hat{x}=y_{2}\) and \((\hat{x}, r_{2})\in\operatorname{epi} f_{u_{1}}\). Consequently, \(f_{u_{1}}(\hat {x})\leq r_{2}\) and \(g_{u_{2}}(y_{1})\leq r_{1}\). Therefore,
Note that u is arbitrary, it follows that
This together with (5.3) implies that \(v(CP_{(p,q)})\leq r\). Therefore, by the weak duality, \(v(CP_{(p, q)})=v(CD_{(p, q)})\) and \(\hat{x}+p\) is the optimal solution of (\(CP_{(p, q)}^{u}\)) which satisfies \(v(CP_{(p, q)}^{u})=v(CP_{(p, q)})\).
Conversely, suppose that the stable converse duality holds between (P) and (D). Let \(p\in X\), \(q\in Y\) and let \(\bar{u}=(\bar{u}_{1},\bar {u}_{2})\in\bar{U}\) be such that \(v(CP_{(p, q)})=v(CP_{(p, q)}^{\bar {u}})\). Then there exists \(x_{0}\in X\) such that
Let \((Ap+q, r)\in\bigcap_{u\in U}\operatorname{epi}(g_{u_{2}}^{\ast}+f_{u_{1}}^{\ast}\circ(-A^{\ast}))^{\ast}\). Then, for each \(u\in U\),
This together with (5.4) implies that
and
by (5.9) and the definitions of (\(CP_{(p, q)}\)) and (\(CD_{(p, q)}\)). Thus,
This means, for each \(u=(u_{1},u_{2})\in U\),
Note that \((Ax_{0}+q, g_{u_{2}}(Ax_{0}+q))\in\operatorname{epi} g_{u_{2}}\) holds for each \(u_{2}\in U_{2}\). Then, for each \(u\in U\),
and hence
Therefore, by the arbitrariness of \(p\in X\) and \(q\in Y\), the (\(CCC\)) holds. The proof is complete. □
In the remainder of this section, we assume that \(f_{u_{1}}:X\rightarrow \overline{{\mathbb {R}}}\), \(u_{1}\in U_{1}\) and \(g_{u_{2}}:Y\rightarrow\overline{{\mathbb {R}}}\), \(u_{2}\in U_{2}\) are proper lsc convex functions. We now present a necessary and sufficient condition for the total converse duality to hold. To do this, we first introduce two converse Moreau-Rockafellar formulae.
Definition 5.3
Let \(y_{0}^{\ast}\in\bigcap_{u\in U}[(\operatorname{dom}g_{u_{2}}^{\ast}) \cap(-A)^{\ast}(\operatorname{dom}f_{u_{1}}^{\ast})]\) and \(\bar {u}=(\bar{u}_{1},\bar{u}_{2})\in U\). It is said that
-
(i)
the strong converse Moreau-Rockafellar formula (\(SCMR\)) holds at \((\bar{u}_{1},\bar{u}_{2};y_{0}^{\ast})\) if
$$ \partial \bigl(f_{\bar{u}_{1}}^{\ast}\circ \bigl(-A^{\ast} \bigr)+g_{\bar{u}_{2}}^{\ast} \bigr) \bigl(y_{0}^{\ast} \bigr)\subseteq\bigcap _{u\in U} \bigl[(-A)\partial f_{u_{1}}^{\ast} \bigl(-A^{\ast}y_{0}^{\ast} \bigr)+\partial g_{u_{2}}^{\ast} \bigl(y_{0}^{\ast} \bigr) \bigr]; $$(5.10) -
(ii)
the converse Moreau-Rockafellar formula (\(CMR\)) holds at \((\bar{u}_{1},\bar{u}_{2};y_{0}^{\ast})\) if
$$ \partial \bigl(f_{\bar{u}_{1}}^{\ast}\circ \bigl(-A^{\ast} \bigr)+g_{\bar{u}_{2}}^{\ast} \bigr) \bigl(y_{0}^{\ast} \bigr)\subseteq\bigcup _{u\in U} \bigl[(-A)\partial f_{u_{1}}^{\ast} \bigl(-A^{\ast}y_{0}^{\ast} \bigr)+\partial g_{u_{2}}^{\ast} \bigl(y_{0}^{\ast} \bigr) \bigr]. $$(5.11)
Remark 5.2
In the case when \(U_{1}\) and \(U_{2}\) are singletons, (5.10) and (5.11) are reduced to the converse Moreau-Rockafellar formula at \(y_{0}^{\ast}\in Y^{\ast}\cap(\operatorname{dom} g^{\ast})\cap(-A)^{\ast}(\operatorname{dom} f^{\ast})\) (see [2], Definition 6.8), that is,
Let \(u=(u_{1},u_{2})\in U\), we define the subproblem of the problem (\(CD_{(p,q)}\)) by
and use \(v(CD_{(p,q)}^{\bar{u}})\) to denote the optimal value of the problem (\(CD_{(p,q)}^{\bar{u}}\)). For each \(p\in X\) and \(q\in Y\), let \(S(p, q)\) denote the solution set of the problem (\(CD_{(p,q)}\)), that is,
and let \(S(p, q,u)\) denote the solution set of the problem (\(CD_{(p,q)}^{u}\)), that is,
Theorem 5.2
Let \(y_{0}^{\ast}\in\bigcap_{u\in U} [(\operatorname{dom}g_{u_{2}}^{\ast})\cap(-A)^{\ast}(\operatorname{dom}f_{u_{1}}^{\ast})]\) and \(\bar {u}=(\bar{u}_{1},\bar{u}_{2})\in U\).
-
(i)
If the (\(SCMR\)) holds at \((\bar{u}_{1},\bar{u}_{2};y_{0}^{\ast})\), then for each \(p\in X\) and \(q\in Y\) satisfying \((\bar{u}_{1},\bar {u}_{2};y_{0}^{\ast})\in S(p, q)\), \(v(CD_{(p,q)})=v(CP_{(p,q)})=v(CP_{(p,q)}^{\bar{u}})\), and the problem (\(CP_{(p,q)}^{\bar{u}}\)) has an optimal solution.
-
(ii)
If for each \(p\in X\) and \(q\in Y\) satisfying \(y_{0}^{\ast}\in S(p, q;\bar{u})\), \(v(CD_{(p,q)}^{\bar{u}})=v(CP_{(p,q)}^{\bar{u}})\), and the problem (\(CP_{(p,q)}^{\bar{u}}\)) has an optimal solution, then the (\(CMR\)) holds at \((\bar{u}_{1},\bar{u}_{2};y_{0}^{\ast})\).
Proof
(i) Suppose that the (\(SCMR\)) holds at \((\bar{u}_{1},\bar{u}_{2};y_{0}^{\ast})\). Then (5.10) holds. Let \(p\in X\), \(q\in Y\) be such that \((\bar {u}_{1},\bar{u}_{2};y_{0}^{\ast})\in S(p, q)\). Then
By (2.3), we see that \(0\in \partial(f_{\bar{u}_{1}}^{\ast}\circ(-A^{\ast})+g_{\bar{u}_{2}}^{\ast }-(Ap+q))(y_{0}^{\ast})\), which is equivalent to \(Ap+q\in \partial(f_{\bar{u}_{1}}^{\ast}\circ(-A^{\ast})+g_{\bar{u}_{2}}^{\ast })(y_{0}^{\ast})\). This together with (5.10) implies that
Let \(u=(u_{1},u_{2})\in U\). Then there exist \(p_{1}\in\partial f_{u_{1}}^{\ast}(-A^{\ast}y_{0}^{\ast})\) and \(p_{2}\in\partial g_{u_{2}}^{\ast}(y_{0}^{\ast})\) such that
Applying the Young equality (2.4), one has
Noting that \(\langle p_{1}, A^{\ast}y_{0}^{\ast}\rangle=\langle Ap_{1}, y_{0}^{\ast}\rangle\), it follows that
where the first equality holds because \(f_{u_{1}}\) and \(g_{u_{2}}\) are lsc functions. Then, by the arbitrariness of \(u\in U\), we have that
Moreover, by (5.14) and (5.16),
Thus,
while, by the definition of (\(CP_{(p,q)}\)) and (5.15), we have that
Combing this with (5.17) and (5.18), one can see that
This together with the weak duality implies that \(v(CD_{(p,q)})= v(CP_{(p,q)})=v(CP_{(p,q)}^{\bar{u}})\) and \(p_{1}+p\) is the optimal solution of the problem (\(CP_{(p,q)}^{\bar{u}}\)).
(ii) Suppose that for each \(p\in X\), \(q\in Y\) and \(\bar{u}=(\bar {u}_{1},\bar{u}_{2})\in U\) satisfying \(y_{0}^{\ast}\in S(p, q;\bar{u})\), \(v(CD_{(p,q)}^{\bar{u}}) =v(CP_{(p,q)}^{\bar{u}})\), and the problem (\(CP_{(p,q)}^{\bar{u}}\)) has an optimal solution. Let \(p\in X\) and \(q\in Y\) be such that \(Ap+q\in\partial(f_{\bar{u}_{1}}^{\ast}\circ (-A^{\ast})+g_{\bar{u}_{2}}^{\ast})(y_{0}^{\ast})\). Then
which implies that \(y_{0}^{\ast}\in S(p, q,\bar{u})\). Thus, by assumptions, there exists \(x_{0}\in X\) such that
This together with (5.13) implies that
Noting the above inequality and using the definition of the conjugate function, we see that
It follows from (5.19) that
Then, by the Young equality (2.4), we have that \(x_{0}-p\in \partial f_{\bar{u}_{1}}(-A^{\ast}y_{0}^{\ast})\) and \(Ax_{0}+ q\in\partial g_{\bar{u}_{2}}(y_{0}^{\ast})\). Thus,
and hence \(Ap+q\in\bigcup_{u\in U}[(-A)\partial f_{u_{1}}(-A^{\ast}y_{0}^{\ast})+ \partial g_{u_{2}}(y_{0}^{\ast})]\). Therefore, by the arbitrariness of p and q, the (\(CMR\)) holds at \((\bar{u}_{1},\bar{u}_{2};y_{0}^{\ast})\). The proof is complete. □
By Theorem 5.2, we have the following corollary directly, which extends and improves the corresponding result in [2], Theorem 6.9.
Corollary 5.1
Suppose that f, g are proper lsc convex functions and A is a linear operator. Let \(y_{0}^{\ast}\in(\operatorname{dom} g^{\ast})\cap(-A)^{\ast}(\operatorname{dom} f^{\ast})\). Then (5.12) holds at \(y_{0}^{\ast}\) if and only if, for each \(p\in X\) and \(q\in Y\) satisfying \(y_{0}^{\ast}\in S(p,q)\), there exists \(x_{0}\in X\) such that
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Acknowledgements
The authors are grateful to both anonymous reviewers for valuable suggestions and remarks which helped to improve the quality of the paper. The first and the second authors were supported in part by the National Natural Science Foundation of China (grant 11461027) and the Scientific Research Fund of Hunan Provincial Education Department (grant 13B095), the third author was supported in part by the National Natural Science Foundation of China (grant 71471122).
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This work was carried out in collaboration between the three authors. Authors MW and DH collected the literature and wrote the first draft of the manuscript. Author CZ revised and improved the draft of the manuscript. All authors read and approved the manuscript.
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Wang, M., Fang, D. & Chen, Z. Strong and total Fenchel dualities for robust convex optimization problems. J Inequal Appl 2015, 70 (2015). https://doi.org/10.1186/s13660-015-0592-9
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DOI: https://doi.org/10.1186/s13660-015-0592-9