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CHARACTERIZATION OF QUASI CONVEX SET VALUED MAPS

12 pages
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 ?

  • vector space

  • real

  • inf ?

  • all x1

  • valued maps

  • extended real-valued

  • function ?

  • arbitrary ? ?


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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 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