- Open Access
On generalized Fenchel-Moreau theorem and second-order characterization for convex vector functions
© Tinh and Kim; licensee Springer. 2013
- Received: 3 September 2013
- Accepted: 2 November 2013
- Published: 3 December 2013
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.
- convex vector function
- biconjugate map
- Fenchel-Moreau theorem
- second-order characterization
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 , 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.
When , we shall write if . From now on we assume that is ordered by a convex cone C.
Definition 2.1 [, Definition 2.1]
- (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 
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 [, Definition 2.3]
- (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.
- (i)[, Corollary 2.21] Let be nonempty. If , then
(where denotes the closure of the convex hull of A).
- (ii)[, Corollary 2.14] Let be nonempty and bounded from above. Then, for every , we have
- (iii)[, 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)[, Proposition 2.22] Let be nonempty. Then
If , then ;
- (b). If, in addition, , then
Lemma 2.7 [, 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 [, 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 ).
Definition 3.1 [, Definition 3.1]
where denotes the space of continuous linear maps from X to .
Definition 3.2 [, Definition 3.2]
In the rest of this section, we assume that the ordering cone is closed, convex, pointed and .
Lemma 3.4 [, Proposition 3.5]
is closed and convex.
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 [, Proposition 3.6]
By (1), this is impossible since and pointed. Thus, . The proof is complete. □
Lemma 3.8 [, Lemma 3.16]
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.
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.
Let be a nonempty open set, , and let be a vector function.
Definition 4.1 
where denotes the derivative of f at .
The following definition is suggested by [, Definition 2.1].
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.
When and , Definition 4.3 collapses to the classical concept of monotonicity.
We have the following lemma.
Lemma 4.4 , , .
F is monotone with respect to C.
- (ii)For every at which F is differentiable,
- (iii)For every , ,
Taking , since C is closed, we obtain .
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 .
Thus F is monotone. The proof is complete. □
We note that Theorem 4.5 generalizes the corresponding result of Luc and Schaible in  in which and .
Specially, we have the following.
Corollary 4.7 [, Theorem 4.9]
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.
Hence f is convex with respect to C by Corollary 4.7.
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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