Normal Cones to Sublevel Sets: An Axiomatic Approach
An axiomatic approach of normal operators to sublevel sets is given. Considering the Clarke-Rockafellar subdifferential (resp. quasiconvex functions), the definition given in [4 ] (resp. [5 ]) is recovered. Moreover, the results obtained in [4 ] are exten
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Departementde Mathematiques , Universitede Perpignan, 66860 PerpignanCedex, France E-mail : [email protected] Laboratoirede MathematiquesAppliquees,CNRS ERS 2055 , Universitede Pau et des Pays del'Adour, Avenue del'Universite, 64000 PAU, France. E-mail: [email protected]
Abstract.An axiomaticapproachof normal operatorsto sublevel sets is given . Considering the Clarke-Rockafellarsubdifferential(resp. quasiconvexfunctions), thedefinitiongiven in [4] (resp. [5])is recovered. Moreover, theresultsobtainedin [4] areextendedin this more generals etting.Under mild assumptions,quasiconvex continuousfunctions are classified,establishingan equivalencerelationbetween functionswith thesame normaloperator . Applicationsin pseudo convexity are also discussed. 2000 Mathematics Subject Olassification. Primary 52A01j Secondary49J52, 26E25 Keywords and phrases. Normal cone,quasiconvexity,pseudo convexity .
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Introduction
The notionof a "normal cone to sublevel sets", Le. a multivalued operatorassociatingwith everyfunction f and every point x of its domainthe normalcone to the sublevel set 8 1(%) has first beenintroducedand studied in [5], wheretheauthorsdiscussedcontinuitypropertiesof this operator(or variantsof it) when applied to quasiconvex functions. Subsequently, several authorsusedthisnotion(see[13], [10], [11] e.g.) for dealing with quasiconvex optimizationproblems. In [4], a modification ontheoriginal definition([5]) of the normalo perator has been proposed , consisting inconsideringfor every x the polar cone of the Clarke tangentcone of 8 1(%) at x. This new definition coincides with the previous one whenever thefunctionf is quasiconvex, whereas it has the advantageto allow simplecharacterizat ions of various types of quasiconvexity in termsof correspondingtypesof quasimonotonicityof thenormaloperator. In thiswork, followingthelines of [4],we give an axiomaticformulationfor theconceptof normaloperator,based on anabstractnotionof sub differential , see Section 2.Subsequently , we presentsome applicationsin quasiconvexity (Sections 3 and 5)a nd in pseudoconvexity(Section 4). N. Hadjisavras et al. (eds.), Generalized Convexity and Generalized Monotonicity © Springer-Verlag Berlin Heidelberg 2001
Normal Cones to Sublevel Sets
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Throughoutthis paper,X will be a Banach spacewith dual X', and f a lowersemicontinuous(lsc) function on X with values in IRU {+oo}. For any x E X and any z" E X' we denoteby (x' , x) thevalue ofthefunctional z" at the point x. We also use the standardnotation: Bo(x) for the closed ball centeredat x with radius cS > 0, dom f := {x EX: f(x) :f +oo} for the domain of the function f and Sf( x} := {x' EX : f(x') :S f(x)} (resp. S/(x) = {x' EX : f(x') < f(x)}) for thesub leveland thestrictsublevelsets of f . For x,y E X we set [x,y] = {tx + (1- t)y : 0:S t :S I} and we define thesegmentslx, y], [x, y[ and [z, y[ analogously.
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Abstract subdifferentialand normal operator
Let us first recall from[2) the definitionof an abstractsubdifferential. Definition 1. We call subdifferential oper
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