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On generalized FenchelMoreau theorem and secondorder characterization for convex vector functions
Fixed Point Theory and Applications volume 2013, Article number: 328 (2013)
Abstract
Based on the concept of conjugate and biconjugate maps introduced in (Tan and Tinh in Acta Math. Viet. 25:315345, 2000) we establish a full generalization of the FenchelMoreau theorem for the vector case. Besides this, by using the Clarke generalized firstorder derivative for locally Lipschitz vector functions, we establish a firstorder characterization for monotone operators. Consequently, a secondorder characterization for convex vector functions is obtained.
MSC: 26B25, 49J52, 49J99, 90C46, 90C29.
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, firstorder as well as secondorder, have been studied intensively. We also know that in convex analysis the theory of Fenchel conjugation plays a central role and the FenchelMoreau 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 noncompletion of the order under consideration. Hence several generalizations are not complete.
The first purpose of the paper is to generalize the FenchelMoreau 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 firstorder derivative for locally Lipschitz vector functions, we establish a firstorder characterization for monotone operators. Consequently, a secondorder 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 FenchelMoreau theorem. The last section deals with a secondorder characterization of convex vector functions.
2 Preliminaries
Let C\subseteq {\mathbb{R}}^{m} be a nonempty set. We recall that 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 yx\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]
Let A\subseteq {\mathbb{R}}^{m} be a nonempty set, and let a\in A. We say that

(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(AC).

(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(AC).
Remark 2.2 When C is pointed and IMin(AC) is nonempty, then IMin(AC) is a singleton and Min(AC)=IMin(AC). The concepts of Max and IMax are defined analogously. It is clear that MinA=Max(A).
Definition 2.3 [16]
Let A\subseteq {\mathbb{R}}^{m} be a nonempty set, and let b\in {\mathbb{R}}^{m}. We say that b is an upper bound of A with respect to C if
The set of upper bounds of A is denoted by Ub(AC).
When Ub(AC)\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(AC).
Definition 2.4 [[16], Definition 2.3]
Let A\subseteq {\mathbb{R}}^{m} be a nonempty set, and let b\in {\mathbb{R}}^{m}. We say that

(i)
b is an ideal supremal point of A with respect to C if b\in IMin(UbAC), 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(AC).\hfill \end{array}The set of ideal supremal points of A is denoted by ISup(AC).

(ii)
b is a supremal point of A with respect to C if b\in Min(UbAC), i.e.,
\{\begin{array}{c}x\u2aafb,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in A,\hfill \\ \mathrm{\forall}y\in Ub(AC),\phantom{\rule{1em}{0ex}}y\u2aafb\Rightarrow b\u2aafy.\hfill \end{array}The set of supremal points of A is denoted by Sup(AC).
Remark 2.5 If ISupA\ne \mathrm{\varnothing}, then ISupA=SupA. In addition, if the ordering cone C is pointed, then ISupA 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)
Now let 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 epif 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],
Subdifferential of 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+(1t)y:t\in [0,1]\}).
3 Generalized FenchelMoreau theorem
Let F be a setvalued 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
The effective domain of F is the set
F is called convex (resp., closed) with respect to C if epiF 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 setvalued map
Definition 3.1 [[16], Definition 3.1]
Assume that domF\ne \mathrm{\varnothing}. The conjugate map of F, denoted by {F}^{\ast}, is a setvalued 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]
Let F be a setvalued 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 setvalued map from {\mathbb{R}}^{n} to {\mathbb{R}}^{m} defined as follows.
Remark 3.3 Let F be a setvalued 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:
we see that {F}^{\ast \ast} is the restriction of {({F}^{\ast})}^{\ast} on {\mathbb{R}}^{n}, 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 setvalued 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 setvalued 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
Pick any {y}_{0}\in riD and {A}_{0}\in \partial f({y}_{0}). By Lemma 3.6, {f}^{\ast}({A}_{0}) is a singleton. For each c\in C, we define a linear map {\beta}_{c}:\mathbb{R}\to {\mathbb{R}}^{m} as follows:
By (1) and by Lemma 2.6(ii),
Then we have
Then there exists {y}_{c}\in {f}^{\ast}({A}_{0}+{\beta}_{c}\xi ) such that
Let z\in {f}^{\ast \ast}({x}_{0}) be arbitrary. From the definition of {f}^{\ast \ast}, one has
By (1), this is impossible since C\ne \{0\} and pointed. Thus, dom{f}^{\ast \ast}\subseteq \overline{D}. The proof is complete. □
Let {x}_{0},x\in {\mathbb{R}}^{n}, {\{{x}_{k}\}}_{k}\subseteq [{x}_{0},x]. Then we write ‘{x}_{k}\uparrow x’ if
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 setvalued structure, under certain conditions, they reduce to singlevalued maps. Such conditions are the convexity and closedness of the functions. Moreover, we have the following theorem.
Theorem 3.9 (Generalized FenchelMoreau 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
Let {\{{\lambda}_{k}\}}_{k}\subseteq (0,1) be an increasing sequence that converges to 1. Put {x}_{k}={\lambda}_{k}x+(1{\lambda}_{k}){x}_{0}. Then {\{{x}_{k}\}}_{k}\subseteq riD\cap [{x}_{0},x] and {x}_{k}\uparrow x. By Lemma 2.7, \partial f({x}_{k})\ne \mathrm{\varnothing}. For each 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,
Take k\to \mathrm{\infty} in (3), by (2) and by Lemma 3.8, we have
which together with Lemma 3.4(ii) implies
Hence, by Lemma 2.6(i), Remark 2.5 and by the definition of biconjugate maps, we have
Finally, we shall show that
Indeed, by the proof above, we have dom{f}^{\ast \ast}\supseteq D. Let {x}_{0}\in dom{f}^{\ast \ast} be arbitrary. By Lemma 3.7, {x}_{0}\in \overline{D}. Let {y}_{0}\in {f}^{\ast \ast}({x}_{0}) and x\in riD. Then (x,f(x)),({x}_{0},{y}_{0})\in epi{f}^{\ast \ast}. For every natural number k\ge 1, put
Obviously, ({x}_{k},{y}_{k})\to ({x}_{0},{y}_{0}) and ({x}_{k},{y}_{k})\in epi{f}^{\ast \ast}, ∀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 FenchelMoreau theorem in convex analysis.
4 Secondorder characterization of convex vector functions
Let X,Y be real finitely dimensional normed spaces. We denote by \mathcal{L}(X,Y) the space of continuous linear maps from 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]
Assume that 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 secondorder derivative of f at {x}_{0} is defined as
where {D}^{2}f({x}_{k}) denotes the secondorder 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.
Now assume that D\subseteq {\mathbb{R}}^{n} is a nonempty, convex and open set. Let F:D\to \mathcal{L}({\mathbb{R}}^{n},{\mathbb{R}}^{m}) be a locally Lipschitz map, x\in D, y\in {\mathbb{R}}^{n}. We denote by I the largest open line segment satisfying x+ty\in D, \mathrm{\forall}t\in I. Define
Set
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
Since φ is linear and ψ is affine, we have
Then, applying a chain rule in [[18], Corollary 2.6.6], one obtains
□
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.
\underline{(\mathrm{ii})\Rightarrow (\mathrm{iii})} Let x\in D, A\in \partial F(x) and u\in {\mathbb{R}}^{n} be arbitrary. By the definition of the Clark generalized derivative, we can represent 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.
\underline{(\mathrm{iii})\Rightarrow (\mathrm{i})} Let x,y\in D be arbitrary. Consider the function
Then Φ is locally Lipschitz on an open line segment I which contains [0,1]. Hence Φ is Lipschitz on any compact line segment [a,b] with
By the mean value theorem, for a vector function [[18], Proposition 2.6.5], there exist {\tau}_{1},\dots ,{\tau}_{k}\in [0,1], {\lambda}_{1},\dots ,{\lambda}_{k}\ge 0, {\lambda}_{1}+\cdots +{\lambda}_{k}=1 such that
Hence
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 secondorder 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
Then
Hence f is convex with respect to C by Corollary 4.7.
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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 (NRF2013R1A1A2A10008908).
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Tinh, P.N., Kim, D.S. On generalized FenchelMoreau theorem and secondorder characterization for convex vector functions. Fixed Point Theory Appl 2013, 328 (2013). https://doi.org/10.1186/168718122013328
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DOI: https://doi.org/10.1186/168718122013328
Keywords
 convex vector function
 biconjugate map
 FenchelMoreau theorem
 monotonicity
 secondorder characterization