GENERALIZED QUASICONVEX SET-VALUED MAPS

. The aim of this paper is to introduce a concept of quasiconvexity for set-valued maps in a general framework, by only considering an abstract conve-xity structure in the domain and an arbitrary binary relation in the codomain. It is shown that this concept can be characterized in terms of usual quasiconvexity of certain real-valued functions. In particular, we focus on cone-quasiconvex set-valued maps with values in a partially ordered vector space.


INTRODUCTION AND PRELIMINARIES
Several generalizations of the classical notions of convexity and quasiconvexity of real-valued functions have been given for vector-valued functions and even for set-valued maps with values in a partially ordered vector space (see e.g.[2]- [7]), the most natural of them being those which preserve the characteristic properties of convex and quasiconvex functions to have a convex epigraph and convex lower level sets, respectively.The aim of this paper is to extend the concept of quasiconvexity for set-valued maps in a very general framework: on one hand, the domain will be a set endowed by a convexity structure induced by a set-valued map Γ, the values of which will replace the linear segments; on the other hand, the order induced by a convex cone in the codomain will be replaced by a general binary relation Ω.The class of so-called (Γ, Ω)-quasiconvex set-valued maps will be introduced in Section 2.Then, by using a technique from [8], in Section 3 we shall characterize this class in terms of quasiconvexity of certain real-valued functions.
Let us firstly recall some notions of Set-Valued Analysis (see e.g.[1]).Given a set-valued map (i.e. a point to set function) Φ : A B between some sets A and B, we denote by the domain and the graph of Φ, respectively.
A set-valued map Φ : A B is said to be an extension of Φ if Graph(Φ) ⊂ Graph(Φ ), which means that Φ(x) ⊂ Φ (x) for all x ∈ A.

The inverse Φ
For any U ⊂ A and V ⊂ B the image of U by Φ and the inverse image of V by Φ are:

If Ψ : B
C is a set-valued map, the composition product Ψ • Φ : A C and the square product Ψ Φ : A C of Ψ and Φ are the set-valued maps defined for all x ∈ A by Ψ(y).

(Γ, Ω)-QUASICONVEX SET-VALUED MAPS
Throughout this paper X and Y will be two nonempty sets.We will introduce a class of generalized quasiconvex set-valued maps defined on X with values in Y .
To this aim, we endow the set X with an abstract convexity structure by means of a set-valued map Γ : X × X X, which assigns to each pair (x 1 , x 2 ) ∈ X × X a subset Γ(x 1 , x 2 ) of X (i.e. a generalized segment).We say that a subset D of X is Γ-convex, if Γ(D × D) ⊂ D. We also consider a set-valued map ∆ : X × X X which assigns to each pair ( On the other hand, we endow the set Y with a binary relation Ω ⊂ Y × Y , which will be regarded as a set-valued map Ω : Y Y , by identifying it with its graph.
Remark 1.If the binary relation Ω satisfies the following additional condition: (2) then the domain Dom(F) is Γ-convex whenever condition (1) holds.
Example 1. Suppose that Y is a partially ordered vector space, with the order Ω induced by a convex cone K ⊂ Y , i.e.K + K ⊂ R + K ⊂ K = ∅ and ( 4) In particular, if X is a vector space and Γ is the usual convex hull, i.e. ( According to Theorem 3 below, condition (6) actually means that F is Kquasiconvex in the sense of Kuroiwa [5].Moreover, if f : D → Y is a vectorvalued function defined on some nonempty convex subset D of X, then f is (Γ, Ω)-quasiconvex in the sense of Definition 2 if and only if it is K-quasiconvex in the sense of Dinh The Luc [6].Finally, for Y = R and K = R + , we recover the classical notion of quasiconvexity of real-valued functions.
The following result shows that (Γ, Ω)-quasiconvexity naturally extends the classical notion of quasiconvexity, since it can be characterized in terms of certain generalized convex level sets (see e.g.[9] for other generalizations based on convex level sets).
Conversely, suppose that for each y ∈ Y the set Proof.In view of Remark 2, the conclusion directly follows from Theorem 3.
To conclude this section, consider the particular case where Y = R is endowed with the usual order relation Ω u , defined by ( 4) with For any set-valued map G : X R with nonempty compact values, we denote by µ G : X → R the lower marginal function of G, defined for all x ∈ X by Actually, for all y ∈ R we have: since G(x) is nonempty compact for every x ∈ X.Thus the desired equivalence holds.
The aim of this section is to characterize (Γ, Ω)-quasiconvex set-valued maps in terms of certain (Γ, Ω u )-quasiconvex set-valued maps and their lower marginal functions.As in [8], our approach is essentially based on the concept of properly characteristic function associated to a binary relation.
is properly characteristic for Ω.
Example 3. Let Y be a topological vector space, partially ordered by a closed convex cone K with nonempty interior, and let Ω be given by ( 4).As shown in [8], for any fixed point e ∈ int K, the function ω : Y × Y → R defined for all (y, z) ∈ Y × Y by ( 7) is properly characteristic for Ω.In this case, for any fixed z ∈ Y , the function ω(•, z) : Y → R represents the "smallest strictly monotonic function at z" in the sense of Dinh The Luc [6].Note that this function is continuous (this property will be used further to obtain an application of Corollary 9).
Given a function ω : Y × Y → R and a set-valued map F : X Y , for each z ∈ Y we denote by ω(F (•), z) : X R the set-valued map defined for all x ∈ X by ω(F (x), z) = {ω(y, z) : y ∈ F (x)}.
Remark 3. If F : X Y is single-valued, i.e. if F (x) = {f (x)} for all x ∈ X, where f : X → Y is a function, then condition (C) becomes trivial.Therefore Theorem 8 extends similar results from [6]- [8], which have been obtained for single-valued functions.
We conclude by presenting a characterization of (Γ, Ω)-quasiconvex set-valued map with values in a topological space in terms of (Γ, Ω u )-quasiconvexity of real-valued functions.
Corollary 9.In addition to the hypotheses of Theorem 8, assume that: Y is a nonempty topological space, F has nonempty compact values, and ω is continuous with respect to the first argument.Then F is (Γ, Ω)-quasiconvex if and only if for each z ∈ Y the lower marginal function of the set-valued map ω(F (•), z) is (Γ, Ω u )-quasiconvex.
Proof.Since F has nonempty compact values and ω is continuous with respect to the first argument, it follows that for every z ∈ Y the set-valued map ω(F (•), z) has nonempty compact values.Hence the conclusion follows directly from Lemma 5 and Theorem 8.
Note that, in view of Example 3 and Remark 4, Corollary 9 may be applied to characterize those (Γ, Ω)-quasiconvex set-valued maps with values in a topological ordered vector space, which have nonempty compact values, each value containing a smallest element.