Uniformly quasiconvexlike functions

Abstract

Let f be a function on a convex subset X of a normed real linear space with values from a connected quasiorder.f is quasiconvexiike if for any x,y ∈ X such that f(x), ≦ f(y) there is an α ∈ (0,1), such that f(αx+(1-alpha;)y),≦f(y).f is α-quasiconvex if there is an α ∈ (0,1), such that for any x,y ∈ X such that f(x) ≦ f(y) we have f(αx+(1-α)y) ≦ f(y).This paper studies some properties of a class of functions which have recently been introduced by H. Hartwig (this journal),: uniformly quasiconvexlike functions f is uniformly δ ∈(o,½), such that for any x,y∈ X there is a λ ∈[δ,1-δ] such that [f(x), ≦ f(y),]⇒[f(λx=(1-λ),y), ≦ f(y),].One result discussed in this paper is that f is uniformly quasiconvexlike if and only if its lower level set (with respect to f(y),), on any section [x,y] (such that f(x), ∈ f(y),),is dense in [x,y].The paper also contains some results on local and global minimizers of uniformly strictly quasiconvexiike functions and a very general version of Kabamabdiah's theorem: If f is lower semicontinuous and uniformly strictly quasiconvexiike.then it is explicitly quasiconvex (both quasiconvex and strictly quasiconvex),.(It was recently shown (Zeitschr.Ang.Math.Mech.60 (1980),), that strictly quasiconvexiike functions do not have this property, whereas strictly a-quasiconvex functions have it.Uniformly strict quasiconvexlikeness lies in between.).

Keywords:

• Nonlinear programming
• generalized convexity
• optimality conditions

AMS 1980 Subject Classifications:

• Primary: 90 C 30
• Secondary: 26 A 51
• Secondary: 26 B 25
• Secondary: 54 C 08
• Secondary: 54 C 30