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# On generalized Fenchel-Moreau theorem and second-order characterization for convex vector functions

- Phan Nhat Tinh
^{1}and - Do Sang Kim
^{2}Email author

**2013**:328

https://doi.org/10.1186/1687-1812-2013-328

© Tinh and Kim; licensee Springer. 2013

**Received:**3 September 2013**Accepted:**2 November 2013**Published:**3 December 2013

## Abstract

Based on the concept of conjugate and biconjugate maps introduced in (Tan and Tinh in Acta Math. Viet. 25:315-345, 2000) we establish a full generalization of the Fenchel-Moreau theorem for the vector case. Besides this, by using the Clarke generalized first-order derivative for locally Lipschitz vector functions, we establish a first-order characterization for monotone operators. Consequently, a second-order characterization for convex vector functions is obtained.

**MSC:** 26B25, 49J52, 49J99, 90C46, 90C29.

## Keywords

- convex vector function
- biconjugate map
- Fenchel-Moreau theorem
- monotonicity
- second-order characterization

## 1 Introduction

Convex functions play an important role in nonlinear analysis, especially in optimization theory since they guarantee several useful properties concerning extremum points. Consequently, characterizations of the class of these functions, first-order as well as second-order, have been studied intensively. We also know that in convex analysis the theory of Fenchel conjugation plays a central role and the Fenchel-Moreau theorem concerning biconjugate functions plays a key role in the duality theory.

In the vector case, there are also many efforts focussing on these topics (see [1–17]). However, the results are still far from the repletion. The main difficulty for ones working on the vector setting is the non-completion of the order under consideration. Hence several generalizations are not complete.

The first purpose of the paper is to generalize the Fenchel-Moreau theorem to the vector case. Based on the concepts of supremum and conjugate and biconjugate maps introduced in [16], we obtain a full generalization of the theorem. Secondly, by using the Clarke generalized first-order derivative for locally Lipschitz vector functions, we establish a first-order characterization for monotone operators. Consequently, a second-order characterization for convex vector functions is obtained.

The paper is organized as follows. In the next section, we present some preliminaries on a cone order in finitely dimensional spaces and on convex vector functions. Section 3 is devoted to a generalization of the Fenchel-Moreau theorem. The last section deals with a second-order characterization of convex vector functions.

## 2 Preliminaries

*C*is said to be a cone if $tx\in C$, $\mathrm{\forall}x\in C$, $t\ge 0$. A cone

*C*is said to be pointed if $C\cap (-C)=\{0\}$. A convex cone $C\subseteq {\mathbb{R}}^{m}$ specifies on ${\mathbb{R}}^{m}$ a partial order defined by

When $intC\ne \mathrm{\varnothing}$, we shall write $x{\ll}_{C}y$ if $y-x\in intC$. From now on we assume that ${\mathbb{R}}^{m}$ is ordered by a convex cone *C*.

**Definition 2.1** [[5], Definition 2.1]

- (i)
*a*is an ideal efficient (or ideal minimum) point of*A*with respect to*C*if$a\u2aafx,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in A.$The set of ideal efficient points of

*A*is denoted by $IMin(A|C)$. - (ii)
*a*is an efficient (or Pareto minimum) point of*A*with respect to*C*if$\mathrm{\forall}x\in A,\phantom{\rule{1em}{0ex}}x\u2aafa\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}a\u2aafx.$The set of efficient points of

*A*is denoted by $Min(A|C)$.

**Remark 2.2** When *C* is pointed and $IMin(A|C)$ is nonempty, then $IMin(A|C)$ is a singleton and $Min(A|C)=IMin(A|C)$. The concepts of Max and IMax are defined analogously. It is clear that $-MinA=Max(-A)$.

**Definition 2.3** [16]

*b*is an upper bound of

*A*with respect to

*C*if

The set of upper bounds of *A* is denoted by $Ub(A|C)$.

When $Ub(A|C)\ne \mathrm{\varnothing}$, we say that *A* is bounded from above. The concept of lower bound is defined analogously. The set of lower bounds of *A* is denoted by $Lb(A|C)$.

**Definition 2.4** [[16], Definition 2.3]

- (i)
*b*is an ideal supremal point of*A*with respect to*C*if $b\in IMin(UbA|C)$,*i.e.*,$\{\begin{array}{c}x\u2aafb,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in A,\hfill \\ b\u2aafy,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in Ub(A|C).\hfill \end{array}$The set of ideal supremal points of

*A*is denoted by $ISup(A|C)$. - (ii)
*b*is a supremal point of*A*with respect to*C*if $b\in Min(UbA|C)$,*i.e.*,$\{\begin{array}{c}x\u2aafb,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in A,\hfill \\ \mathrm{\forall}y\in Ub(A|C),\phantom{\rule{1em}{0ex}}y\u2aafb\Rightarrow b\u2aafy.\hfill \end{array}$The set of supremal points of

*A*is denoted by $Sup(A|C)$.

**Remark 2.5** If $ISupA\ne \mathrm{\varnothing}$, then $ISupA=SupA$. In addition, if the ordering cone *C* is pointed, then ISup*A* is a singleton.

In the sequel, when there is no risk of confusion, we omit the phrase ‘with respect to *C*’ and the symbol ‘${|}_{C}$’ in the definitions above. We list here some properties of supremum which will be needed in the sequel.

**Lemma 2.6**

*Assume that the ordering cone*$C\subseteq {\mathbb{R}}^{m}$

*is closed*,

*convex and pointed*.

- (i)[[16],
*Corollary*2.21]*Let*$A\subseteq {\mathbb{R}}^{m}$*be nonempty*.*If*$UbA\cap \overline{coA}\ne \mathrm{\varnothing}$,*then*$UbA\cap \overline{coA}=ISupA$(

*where*$\overline{coA}$*denotes the closure of the convex hull of**A*). - (ii)[[16],
*Corollary*2.14]*Let*$S\subseteq \mathbb{R}$*be nonempty and bounded from above*.*Then*,*for every*$c\in C$,*we have*$ISup(Sc)=(supS)c$(

*where*$Sc:=\{tc:t\in S\}$). - (iii)[[16],
*Theorem*2.16,*Remark*2.18]*Let*$A\subseteq {\mathbb{R}}^{m}$*be nonempty*.*Then*$SupA\ne \mathrm{\varnothing}$*if and only if**A**is bounded from above*.*In this case*,*we have*$UbA=SupA+C.$ - (iv)[[16],
*Proposition*2.22]*Let*$A,B\subseteq {\mathbb{R}}^{m}$*be nonempty*.*Then*- (a)
*If*$A\subseteq B$,*then*$SupB\subseteq SupA+C$; - (b)$SupA+SupB\subseteq Sup(A+B)+C$.
*If*,*in addition*, $ISupA\cup ISupB\ne \mathrm{\varnothing}$,*then*$SupA+SupB=Sup(A+B).$

- (a)

*f*be a vector function from a nonempty set $D\subseteq {\mathbb{R}}^{n}$ to ${\mathbb{R}}^{m}$, and let $S\subseteq D$, $x\in S$. We say that

*f*is continuous relative to

*S*at

*x*if for every neighborhood

*W*of $f(x)$, there exists a neighborhood

*V*of

*x*such that

*f*is called continuous relative to

*S*if it is continuous relative to

*S*at every $x\in S$. The epigraph of

*f*(with respect to the ordering cone

*C*) is defined as the set

*f*is called closed (with respect to

*C*) if epi

*f*is closed in ${\mathbb{R}}^{n}\times {\mathbb{R}}^{m}$. Now assume that $D\subseteq {\mathbb{R}}^{n}$ is nonempty and convex. We recall that $f:D\to {\mathbb{R}}^{m}$ is said to be convex (with respect to

*C*) if for every $x,y\in D$, $\lambda \in [0,1]$,

*f*at $x\in D$ is defined as the set

Convex vector functions have several nice properties as scalar convex functions (see, [6, 16, 17]). We recall some results which will be used in the sequel.

**Lemma 2.7** [[6], Theorem 4.12]

*Assume that the ordering cone* $C\subseteq {\mathbb{R}}^{m}$ *is closed*, *convex and pointed*. *Let* *f* *be a convex vector function from a nonempty convex set* $D\subseteq {\mathbb{R}}^{n}$ *to* ${\mathbb{R}}^{m}$. *Then* $\partial f(x)\ne \mathrm{\varnothing}$ *for every* $x\in riD$.

From [[17], Theorem 3.6] we immediately have the following lemma.

**Lemma 2.8** *Assume that the ordering cone* $C\subseteq {\mathbb{R}}^{m}$ *is closed*, *convex and pointed with* $intC\ne \mathrm{\varnothing}$. *Let* *f* *be a closed convex vector function from a nonempty convex set* $D\subseteq {\mathbb{R}}^{n}$ *to* ${\mathbb{R}}^{m}$, *and let* $x,y\in D$ *be arbitrary*. *Then* *f* *is continuous relative to* $[x,y]$ (*where* $[x,y]:=\{tx+(1-t)y:t\in [0,1]\}$).

## 3 Generalized Fenchel-Moreau theorem

*F*be a set-valued map from a finitely dimensional normed space

*X*to ${\mathbb{R}}^{m}$. We recall that the epigraph of

*F*with respect to

*C*is defined as the set

*F*is the set

*F*is called convex (resp., closed) with respect to

*C*if epi

*F*is convex (resp., closed) in $X\times {\mathbb{R}}^{m}$. Sometimes a vector function $f:D\subseteq {\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ is identified with the set-valued map

**Definition 3.1** [[16], Definition 3.1]

*F*, denoted by ${F}^{\ast}$, is a set-valued map from $\mathcal{L}(X,{\mathbb{R}}^{m})$ to ${\mathbb{R}}^{m}$ defined as follows.

where $\mathcal{L}(X,{\mathbb{R}}^{m})$ denotes the space of continuous linear maps from *X* to ${\mathbb{R}}^{m}$.

**Definition 3.2** [[16], Definition 3.2]

*F*be a set-valued map from ${\mathbb{R}}^{n}$ to ${\mathbb{R}}^{m}$. Assume that $dom{F}^{\ast}\ne \mathrm{\varnothing}$. The biconjugate map of

*F*, denoted by ${F}^{\ast \ast}$, is a set-valued map from ${\mathbb{R}}^{n}$ to ${\mathbb{R}}^{m}$ defined as follows.

**Remark 3.3**Let

*F*be a set-valued map from ${\mathbb{R}}^{n}$ to ${\mathbb{R}}^{m}$ with $dom{F}^{\ast}\ne \mathrm{\varnothing}$. By identifying $x\in {\mathbb{R}}^{n}$ with the linear map $\overline{x}:\mathcal{L}({\mathbb{R}}^{n},{\mathbb{R}}^{m})\to {\mathbb{R}}^{m}$ defined as follows:

*i.e.*,

In the rest of this section, we assume that the ordering cone $C\subseteq {\mathbb{R}}^{m}$ is closed, convex, pointed and $intC\ne \mathrm{\varnothing}$.

**Lemma 3.4** [[16], Proposition 3.5]

*Let*

*F*

*be a set*-

*valued map from*${\mathbb{R}}^{n}$

*to*${\mathbb{R}}^{m}$

*with*$domF\ne \mathrm{\varnothing}$.

*Then*

- (i)
${F}^{\ast}$

*is closed and convex*. - (ii)
*If*$dom{F}^{\ast}\ne \mathrm{\varnothing}$,*then*$F(x)\subseteq {F}^{\ast \ast}(x)+C$, $\mathrm{\forall}x\in {\mathbb{R}}^{n}$.

**Lemma 3.5** *Let* *F* *be a set*-*valued map from* ${\mathbb{R}}^{n}$ *to* ${\mathbb{R}}^{m}$ *with* $dom{F}^{\ast}\ne \mathrm{\varnothing}$. *Then* ${F}^{\ast \ast}$ *is closed and convex*.

*Proof* It is immediate from Remark 3.3 and Lemma 3.4. □

**Lemma 3.6** [[16], Proposition 3.6]

*Let*

*f*

*be a convex vector function from a nonempty convex set*$D\subseteq {\mathbb{R}}^{n}$

*to*${\mathbb{R}}^{m}$,

*and let*$x\in D$, $A\in \mathcal{L}({\mathbb{R}}^{n},{\mathbb{R}}^{m})$.

*Then*$A\in \partial f(x)$

*if and only if*

**Lemma 3.7**

*Let*

*f*

*be a convex vector function from a nonempty convex set*$D\subseteq {\mathbb{R}}^{n}$

*to*${\mathbb{R}}^{m}$.

*Then*

*Proof*Let $x\in riD$ be arbitrary. By Lemma 2.7, $\partial f(x)\ne \mathrm{\varnothing}$. Then, by Lemma 3.6, $\partial f(x)\subseteq dom{f}^{\ast}$. Consequently, $dom{f}^{\ast}\ne \mathrm{\varnothing}$. Then, by Lemma 3.4, $D\subseteq dom{f}^{\ast \ast}$. Now, suppose on the contrary that $dom{f}^{\ast \ast}\u2288\overline{D}$. Then there is ${x}_{0}\in dom{f}^{\ast \ast}$ such that ${x}_{0}\notin \overline{D}$. Using the strong separation theorem, one can find $\xi \in \mathcal{L}({\mathbb{R}}^{n},\mathbb{R})\setminus \{0\}$ so that

By (1), this is impossible since $C\ne \{0\}$ and pointed. Thus, $dom{f}^{\ast \ast}\subseteq \overline{D}$. The proof is complete. □

**Lemma 3.8** [[16], Lemma 3.16]

*Let*

*f*

*be a convex function from a nonempty convex set*$D\subseteq {\mathbb{R}}^{n}$

*to*${\mathbb{R}}^{m}$,

*and let*$x\in D$.

*If there exists*${x}_{0}\in riD$

*such that*

*then for every sequence*${\{({A}_{k},{x}_{k})\}}_{k}\subseteq \mathcal{L}({\mathbb{R}}^{n},{\mathbb{R}}^{m})\times [{x}_{0},x]$

*such that*${x}_{k}\uparrow x$

*and*${A}_{k}\in \partial f({x}_{k})$,

*we have*

Although biconjugate maps of vector functions have a set-valued structure, under certain conditions, they reduce to single-valued maps. Such conditions are the convexity and closedness of the functions. Moreover, we have the following theorem.

**Theorem 3.9**(

*Generalized Fenchel*-

*Moreau theorem*)

*Let*

*f*

*be a vector function from a nonempty convex set*$D\subseteq {\mathbb{R}}^{n}$

*to*${\mathbb{R}}^{m}$.

*Then*

*f*

*is closed and convex if and only if*

*Proof*$\underline{\Rightarrow}$: Let $x\in D$ be arbitrary. Pick a point ${x}_{0}\in riD$. By Lemma 2.8,

*f*is continuous relative to $[{x}_{0},x]$. Hence

*k*, pick ${A}_{k}\in \partial f({x}_{k})$. By Lemma 3.6, $f({x}_{k})={A}_{k}({x}_{k})-{f}^{\ast}({A}_{k})$. Hence,

*k*, since ${f}^{\ast \ast}$ is convex. By (4), $f({x}_{k})={f}^{\ast \ast}({x}_{k})$ since ${x}_{k}\in D$. Hence, $({x}_{k},{y}_{k})\in epif$ (∀

*k*). This fact together with closedness of

*f*implies

Hence ${x}_{0}\in D$. Thus, $dom{f}^{\ast \ast}=D$ and then $f={f}^{\ast \ast}$.

$\underline{\Leftarrow}$: It is immediate from Lemma 3.5. The theorem is proved. □

When $m=1$ and $C={\mathbb{R}}_{+}$, Theorem 3.9 is the famous Fenchel-Moreau theorem in convex analysis.

## 4 Second-order characterization of convex vector functions

*X*to

*Y*. In $\mathcal{L}(X,Y)$ we equip the norm defined by

Let $D\subseteq X$ be a nonempty open set, ${x}_{0}\in D$, and let $f:D\to Y$ be a vector function.

**Definition 4.1** [18]

*f*is locally Lipschitz. The

*Clarke generalized derivative*of

*f*at ${x}_{0}$ is defined as

where $Df({x}_{k})$ denotes the derivative of *f* at ${x}_{k}$.

The following definition is suggested by [[19], Definition 2.1].

**Definition 4.2**Assume that

*f*is a vector function of class ${C}^{1,1}$. The

*Clarke generalized second-order derivative*of

*f*at ${x}_{0}$ is defined as

where ${D}^{2}f({x}_{k})$ denotes the second-order derivative of *f* at ${x}_{0}$.

In the remainder of this section, we assume that the ordering cone $C\subseteq {\mathbb{R}}^{m}$ is closed and convex.

**Definition 4.3**Let $D\subseteq {\mathbb{R}}^{n}$ be a nonempty set, and let a map $F:D\to \mathcal{L}({\mathbb{R}}^{n},{\mathbb{R}}^{m})$. We say that

*F*is monotone with respect to

*C*if

When $m=1$ and $C={\mathbb{R}}_{+}$, Definition 4.3 collapses to the classical concept of monotonicity.

*I*the largest open line segment satisfying $x+ty\in D$, $\mathrm{\forall}t\in I$. Define

We have the following lemma.

**Lemma 4.4** $\partial \mathrm{\Phi}(t)(\u03f5)\subseteq \u03f5\partial F(x+ty)(y,y)$, $\mathrm{\forall}t\in I$, $\u03f5\in \mathbb{R}$.

*Proof*Observe that $\mathrm{\Phi}=\phi \circ F\circ \psi $, where

*φ*is linear and

*ψ*is affine, we have

□

**Theorem 4.5**

*Let*$D\subseteq {\mathbb{R}}^{n}$

*be a nonempty*,

*convex and open set*,

*and let*$F:D\to \mathcal{L}({\mathbb{R}}^{n},{\mathbb{R}}^{m})$

*be a locally Lipschitz map*.

*Then the following statements are equivalent*:

- (i)
*F**is monotone with respect to**C*. - (ii)
*For every*$x\in D$*at which**F**is differentiable*,$DF(x)(u,u)\in C,\phantom{\rule{1em}{0ex}}\mathrm{\forall}u\in {\mathbb{R}}^{n}.$ - (iii)
*For every*$x\in D$, $A\in \partial F(x)$,$A(u,u)\in C,\phantom{\rule{1em}{0ex}}\mathrm{\forall}u\in {\mathbb{R}}^{n}.$

*Proof*$\underline{(\mathrm{i})\Rightarrow (\mathrm{ii})}$ Let $x\in D$ at which

*F*is differentiable, and let $u\in {\mathbb{R}}^{n}$ be arbitrary. Let ${\{{t}_{k}\}}_{k}$ be a positive sequence converging to 0. Since

*F*is monotone with respect to

*C*, we have

Taking $k\to \mathrm{\infty}$, since *C* is closed, we obtain $DF(x)(u,u)\in C$.

*A*in the form

where ${\lambda}_{i}\ge 0$, ${\sum}_{i=1}^{k}{\lambda}_{i}=1$ and ${A}_{i}={lim}_{j\to \mathrm{\infty}}DF({x}_{ij})$ with ${x}_{ij}\to x$ ($j\to \mathrm{\infty}$), and there exists $DF({x}_{ij})$ for every $i=1,\dots ,k$; $j=1,2,\dots $ . Since $DF({x}_{ij})(u,u)\in C$ and *C* is closed, passing to the limit, we have ${A}_{i}(u,u)\in C$, $\mathrm{\forall}i=1,\dots ,k$. By (5) and by the convexity of *C*, we obtain $A(u,u)\in C$.

*I*which contains $[0,1]$. Hence Φ is Lipschitz on any compact line segment $[a,b]$ with

Thus *F* is monotone. The proof is complete. □

We note that Theorem 4.5 generalizes the corresponding result of Luc and Schaible in [7] in which $m=1$ and $C={\mathbb{R}}_{+}$.

**Theorem 4.6**

*Let*$D\subseteq {\mathbb{R}}^{n}$

*be a nonempty convex and open set*,

*and let*$f:D\to {\mathbb{R}}^{m}$

*be a*${C}^{1,1}$

*vector function*.

*Then*

*f*

*is convex with respect to*

*C*

*if and only if for every*$x\in D$, $A\in {\partial}^{2}f(x)$, $u\in {\mathbb{R}}^{n}$,

*Proof*We have

□

Specially, we have the following.

**Corollary 4.7** [[17], Theorem 4.9]

*Let*$D\subseteq {\mathbb{R}}^{n}$

*be a nonempty convex and open set*,

*and let*$f:D\to {\mathbb{R}}^{m}$

*be a twice continuously differentiable function*.

*Then*

*f*

*is convex with respect to*

*C*

*if and only if*

*Proof* Since continuously differentiable functions are locally Lipschitz, repeating arguments in the proof of the above theorem, we obtain the result. □

We note that when $m=1$, $C={\mathbb{R}}_{+}$, Corollary 4.7 collapses to the classical result on the second-order characterization of convex functions.

**Example 4.8**Let ${\mathbb{R}}^{3}$ be ordered by the cone $C=con(co\{(1,0,1),(0,-1,-1),(0,0,1)\})$. Let $f:{\mathbb{R}}^{2}\to {\mathbb{R}}^{3}$ be defined by $f({x}_{1},{x}_{2}):=(\frac{1}{2}{x}_{1}^{2}+2{x}_{1}-{x}_{2},-\frac{1}{2}{x}_{2}^{2}-{x}_{1}+2{x}_{2},\frac{1}{2}{x}_{1}^{2}+{x}_{1}-\frac{1}{2}{x}_{2}^{2})$. By computing we have

Hence *f* is convex with respect to *C* by Corollary 4.7.

## Declarations

### Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education Science and Technology (NRF-2013R1A1A2A10008908).

## Authors’ Affiliations

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