CHARACTERIZATION OF QUASI CONVEX SET VALUED MAPS

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CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS JOEL BENOIST AND NICOLAE POPOVICI Abstract. The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are K-(quasi)convex with respect to a convex cone K. In particular, we recover some known characterizations of K-(quasi)convex vector-valued functions, given by means of the polar cone of K. 1. Introduction and Preliminaries The classical notions of convexity and quasiconvexity of real-valued functions have been extended to set-valued maps in various ways (see e.g. [2], [4]–[6]). Two of them are of special interest and will be studied here. Recall that a set-valued map F : X ? 2Y , defined on a vector space X with values in a vector space Y endowed with a convex cone K ? Y (i.e. K +K ? R+K ? K 6= ?), is said to be: (a) K-convex, if for all x1, x2 ? X and t ? [0, 1] we have tF (x1) + (1? t)F (x2) ? F (tx1 + (1? t)x2) +K, which means that F has a convex epigraph: epi (F ) = {(x, y) ? X ?

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all x1

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extended real-valued

function ?

arbitrary ? ?

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Vector space

CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS ¨ JOEL BENOIST AND NICOLAE POPOVICI Abstract. The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are K -(quasi)convex with respect to a convex cone K . In particular, we recover some known characterizations of K -(quasi)convex vector-valued functions, given by means of the polar cone of K .

1. Introduction and Preliminaries

The classical notions of convexity and quasiconvexity of real-valued functions have been extended to set-valued maps in various ways (see e.g. [2], [4]–[6]). Two of them are of special interest and will be studied here. Recall that a set-valued map F : X → 2 Y , deﬁned on a vector space X with values in a vector space Y endowed with a convex cone K ⊂ Y (i.e. K + K ⊂ R + K ⊂ K 6 = ∅ ), is said to be: (a) K -convex, if for all x 1 , x 2 ∈ X and t ∈ [0 , 1] we have

tF ( x 1 ) + (1 − t ) F ( x 2 ) ⊂ F ( tx 1 + (1 − t ) x 2 ) + K,

which means that F has a convex epigraph:

epi ( F ) = { ( x, y ) ∈ X × Y : y ∈ F ( x ) + K } ;

Date : March 5, 2002. 1991 Mathematics Subject Classiﬁcation. Primary 26B25; Secondary 47H04, 90C29. Key words and phrases. Convex and quasiconvex set-valued maps, scalarization, polar cones. 1

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