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.
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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
Let be a nonempty set. We recall that C is said to be a cone if , , . A cone C is said to be pointed if . A convex cone specifies on a partial order defined by
When , we shall write if . From now on we assume that is ordered by a convex cone C.
Definition 2.1 [[5], Definition 2.1]
Let be a nonempty set, and let . We say that
-
(i)
a is an ideal efficient (or ideal minimum) point of A with respect to C if
The set of ideal efficient points of A is denoted by .
-
(ii)
a is an efficient (or Pareto minimum) point of A with respect to C if
The set of efficient points of A is denoted by .
Remark 2.2 When C is pointed and is nonempty, then is a singleton and . The concepts of Max and IMax are defined analogously. It is clear that .
Definition 2.3 [16]
Let be a nonempty set, and let . 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 .
When , 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 .
Definition 2.4 [[16], Definition 2.3]
Let be a nonempty set, and let . We say that
-
(i)
b is an ideal supremal point of A with respect to C if , i.e.,
The set of ideal supremal points of A is denoted by .
-
(ii)
b is a supremal point of A with respect to C if , i.e.,
The set of supremal points of A is denoted by .
Remark 2.5 If , then . 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 ‘’ 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 is closed, convex and pointed.
-
(i)
[[16], Corollary 2.21] Let be nonempty. If , then
(where denotes the closure of the convex hull of A).
-
(ii)
[[16], Corollary 2.14] Let be nonempty and bounded from above. Then, for every , we have
(where ).
-
(iii)
[[16], Theorem 2.16, Remark 2.18] Let be nonempty. Then if and only if A is bounded from above. In this case, we have
-
(iv)
[[16], Proposition 2.22] Let be nonempty. Then
-
(a)
If , then ;
-
(b)
. If, in addition, , then
-
(a)
Now let f be a vector function from a nonempty set to , and let , . We say that f is continuous relative to S at x if for every neighborhood W of , 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 . 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 . Now assume that is nonempty and convex. We recall that is said to be convex (with respect to C) if for every , ,
Subdifferential of f at 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 is closed, convex and pointed. Let f be a convex vector function from a nonempty convex set to . Then for every .
From [[17], Theorem 3.6] we immediately have the following lemma.
Lemma 2.8 Assume that the ordering cone is closed, convex and pointed with . Let f be a closed convex vector function from a nonempty convex set to , and let be arbitrary. Then f is continuous relative to (where ).
3 Generalized Fenchel-Moreau theorem
Let F be a set-valued map from a finitely dimensional normed space X to . 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 . Sometimes a vector function is identified with the set-valued map
Definition 3.1 [[16], Definition 3.1]
Assume that . The conjugate map of F, denoted by , is a set-valued map from to defined as follows.
where denotes the space of continuous linear maps from X to .
Definition 3.2 [[16], Definition 3.2]
Let F be a set-valued map from to . Assume that . The biconjugate map of F, denoted by , is a set-valued map from to defined as follows.
Remark 3.3 Let F be a set-valued map from to with . By identifying with the linear map defined as follows:
we see that is the restriction of on , i.e.,
In the rest of this section, we assume that the ordering cone is closed, convex, pointed and .
Lemma 3.4 [[16], Proposition 3.5]
Let F be a set-valued map from to with . Then
-
(i)
is closed and convex.
-
(ii)
If , then , .
Lemma 3.5 Let F be a set-valued map from to with . Then 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 to , and let , . Then if and only if
Lemma 3.7 Let f be a convex vector function from a nonempty convex set to . Then
Proof Let be arbitrary. By Lemma 2.7, . Then, by Lemma 3.6, . Consequently, . Then, by Lemma 3.4, . Now, suppose on the contrary that . Then there is such that . Using the strong separation theorem, one can find so that
Pick any and . By Lemma 3.6, is a singleton. For each , we define a linear map as follows:
By (1) and by Lemma 2.6(ii),
Then we have
Then there exists such that
Let be arbitrary. From the definition of , one has
By (1), this is impossible since and pointed. Thus, . The proof is complete. □
Let , . Then we write ‘’ if
Lemma 3.8 [[16], Lemma 3.16]
Let f be a convex function from a nonempty convex set to , and let . If there exists such that
then for every sequence such that and , 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 to . Then f is closed and convex if and only if
Proof : Let be arbitrary. Pick a point . By Lemma 2.8, f is continuous relative to . Hence
Let be an increasing sequence that converges to 1. Put . Then and . By Lemma 2.7, . For each k, pick . By Lemma 3.6, . Hence,
Take 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 . Let be arbitrary. By Lemma 3.7, . Let and . Then . For every natural number , put
Obviously, and , ∀k, since is convex. By (4), since . Hence, (∀k). This fact together with closedness of f implies
Hence . Thus, and then .
: It is immediate from Lemma 3.5. The theorem is proved. □
When and , Theorem 3.9 is the famous Fenchel-Moreau theorem in convex analysis.
4 Second-order characterization of convex vector functions
Let be real finitely dimensional normed spaces. We denote by the space of continuous linear maps from X to Y. In we equip the norm defined by
Let be a nonempty open set, , and let be a vector function.
Definition 4.1 [18]
Assume that f is locally Lipschitz. The Clarke generalized derivative of f at is defined as
where denotes the derivative of f at .
The following definition is suggested by [[19], Definition 2.1].
Definition 4.2 Assume that f is a vector function of class . The Clarke generalized second-order derivative of f at is defined as
where denotes the second-order derivative of f at .
In the remainder of this section, we assume that the ordering cone is closed and convex.
Definition 4.3 Let be a nonempty set, and let a map . We say that F is monotone with respect to C if
When and , Definition 4.3 collapses to the classical concept of monotonicity.
Now assume that is a nonempty, convex and open set. Let be a locally Lipschitz map, , . We denote by I the largest open line segment satisfying , . Define
Set
We have the following lemma.
Lemma 4.4 , , .
Proof Observe that , 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 be a nonempty, convex and open set, and let be a locally Lipschitz map. Then the following statements are equivalent:
-
(i)
F is monotone with respect to C.
-
(ii)
For every at which F is differentiable,
-
(iii)
For every , ,
Proof Let at which F is differentiable, and let be arbitrary. Let be a positive sequence converging to 0. Since F is monotone with respect to C, we have
Taking , since C is closed, we obtain .
Let , and be arbitrary. By the definition of the Clark generalized derivative, we can represent A in the form
where , and with (), and there exists for every ; . Since and C is closed, passing to the limit, we have , . By (5) and by the convexity of C, we obtain .
Let be arbitrary. Consider the function
Then Φ is locally Lipschitz on an open line segment I which contains . Hence Φ is Lipschitz on any compact line segment with
By the mean value theorem, for a vector function [[18], Proposition 2.6.5], there exist , , 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 and .
Theorem 4.6 Let be a nonempty convex and open set, and let be a vector function. Then f is convex with respect to C if and only if for every , , ,
Proof We have
□
Specially, we have the following.
Corollary 4.7 [[17], Theorem 4.9]
Let be a nonempty convex and open set, and let 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 , , Corollary 4.7 collapses to the classical result on the second-order characterization of convex functions.
Example 4.8 Let be ordered by the cone . Let be defined by . 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 (NRF-2013R1A1A2A10008908).
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Tinh, P.N., Kim, D.S. On generalized Fenchel-Moreau theorem and second-order characterization for convex vector functions. Fixed Point Theory Appl 2013, 328 (2013). https://doi.org/10.1186/1687-1812-2013-328
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DOI: https://doi.org/10.1186/1687-1812-2013-328