Characterizations of generalized convex functions in terms of coderivative

In the last decades generalized convexity and generalized monotonicity have been widely and intensively study [ 8 ] since they have vast applications in several fields of sciences such as mathematical optimization, economics, finance etc. The generalized convexity of functions are usually characterized by their generalized derivatives, directional derivatives or subdifferential [7] such as the Clarke generalized gradient and Jacobian [2], the generalized Dini directional derivative [2] and the Clarke-Rockafellar generalized subdifferential [13, 1, 18]. Recently, the concept of coderivative introduced by Mordukhovich has proved to be an efficient tool to treat problems in variational analysis, optimization and control optimization [14]. A natural question arises: can we use Mordukhovich coderivative to investigate characterizations of generalized convexity of


Introduction
In the last decades, generalized convexity and generalized monotonicity have been widely and intensively studied [8] since they have vast applications in several fields of sciences such as mathematical optimization, economics, finance, etc.
Recently, the concept of coderivative introduced by Mordukhovich has proved to be an efficient tool to treat problems in variational analysis, optimization and control optimization [14].A natural question arises: can we use Mordukhovich coderivative to investigate characterizations of generalized convexity of functions, particularly for vector-valued functions?The aim of this paper is to answer this question.Besides convex vector functions, the naturally quasiconvex vector functions introduced by Tanaka [17] are considered.As shown in [17] and [11], this class of functions lies at the center of several kinds of generalized convex vector functions, and it plays an important role in the proof of several basic theorems of vector optimization such as the saddle point theorem, the minimax theorem and the solvalidity theorem.
The paper is organized as follows.In the next section, we introduce some preliminaries.Section 3 studies the characterizations of convex vector functions.The last section is devoted to characterizations of naturally quasiconvex vector functions.

Preliminaries
We denote the convex hull and the interior of a set is the Clarke-Rockafellar directional derivative of  at x in the direction u .Now assume that  is locally Lipschitz at x .Then, the Clarke directional derivative [4] of  at x in the direction u is defined as the following limit .

Let
denotes the Jacobian of f at   .
The link between the Clarke generalized Jacobian of the vector function f and the Clarke directional derivative of the real function f Next, we recall some notions from [14].Let x is the set be a vector function and denote its graph by gph f .
For locally Lipschitz functions, we have the link between the Modurkhovich coderivative and the Clarke generalized Jacobian as follows. 3

Characterizations of convex vector-valued functions
From now on we assume that m R is ordered by a closed and convex cone ) , ( : Observe that when 1 = m and  R K = , Definition 3.1 collapses to the classical concept of monotonicity, i.e.,  is monotone (resp., strictly monotone

 
The following results [9] will be needed.

Lemma 3.1 For any
,  is strictly monotone with respect to K if and only if   is strictly monotone in the classical sense for every . Then, by Lemma 2.4, we have For the 'if' part, suppose in the contrary that  is not monotone.Then, there exist . By using the strong separation theorem, one can find  is not monotone.We get a contradiction.
ii) The proof is quite similar to the one of i).

For any
is monotone with respect to K on E .
Proof.For the 'if' part, we assume that is monotone with respect to K on E .Suppose that f is not convex on E .Then, there exist and the Dini upper direction derivative is positive homogenous , we have Observe that we can find a number By positive homogeneity of Dini upper directional derivatives, (3) and (4) give us Taking into account Lemma 2.4 and the properties of Clarke subdifferential, we have By convexity of the scalar function f The relation ( 5) and ( 6) imply the monotonicity of is monotone with respect to K on E by Proposition 3.2.
is strictly monotone with respect to Proof.It is quite similar the proof of Theorem 3.4.

4
Characterizations of quasiconvex vector-valued functions be as in Section 3.
Definition 4.1 [17] i) The function f is said to be naturally quasiconvex with respect to K on , f is said to be strictly naturally quasiconvex with respect to K on E if for every ) , ( , , , the above definition collapses to the classical concept of quasiconvexity of scalar functions, i.e., f is said to be quasiconvex (resp., strictly quasiconvex) if for every . But this contradicts (7).(7) and (8) imply that f  is not quasiconvex which contradicts assumptions.Hence, f is naturally quasiconvex.ii) Analogously.
, Then, by the positive homogeneity of the Dini upper directional derivative, we have coincides with the Clarke generalized gradient.

For
 -subdifferential of  at

Lemma 3 . 3 (
Diewert's mean value theorem [5]) Let n R D  be a nonempty convex set and R D  :  be a lower semicontinuous function.Then, for every

Remark 4 . 2
Proposition 4.1 i) differ from [[6], Proposition 3.9] since the ordering cone is not required to have a nonempty interior.valued map with nonempty values.Definition 4.2 say that i)  is quasimonotone with respect to

Theorem 4 . 6
on E .Then, by Proposition 4.3, not quasimonotone.We get a contradiction.Assume that int   K and f is locally Lipschitz on E .If  with respect to K on E then, f is strictly naturally quasiconvex with respect to K on E .Proof.It is similar the proof of the 'if' part of Theorem 4.5.We should note that in general the converse statement of Theorem 4.6 is not true.For instant, consider the function monotone on E by definitions.Using Proposition 3.2, we see f is convex with respect to K on E .