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Vietnam Journal of Mathematics 35:1(2007) 107-119

 Outer γ-Convexity and Inner γ-Convexity of Disturbed Functions

Hoang Xuan Phu

Abstract.  The most kinds of generalized convexities cannot resist perturbations, even linear ones, while real application problems are often affected by disturbances, both linear and nonlinear ones. For instance, we showed earlier that quasiconvexity, explicit quasiconvexity, and pseudoconvexity cannot withstand arbitrarily small linear disturbances to keep their characteristic properties, and convex functions are the only ones which can resist every linear disturbance to preserve property “each local minimizer is a global minimize”, but it fails if perturbation is nonlinear, even with arbitrarily small supremum norm. In this paper, we present some sufficient conditions for the outer γ-convexity and the inner γ-convexity of disturbed functions, for instance, when convex functions are added with arbitrarily wild but accordingly bounded functions. That means, in spite of such nonlinear disturbances, some weakened properties can be saved, namely the properties of outer γ-convex functions and inner γ-convex ones. For instance, each γ-minimizer of an outer γ-convex function f: D -> R defined by $f(x^*)=\inf_{x\in\bar B(x^*,\gamma)\cap D}f(x)$ is a global minimizer, or if an inner γ-convex function f: D -> R defined on some bounded convex subset D of an inner product space attains its supremum, then it does so at least at some strictly γ-extreme point of D, which cannot be represented as midpoint of some segment $[z', z'']\subset D$ with $\|z'-z''\|\geq 2\,\gamma$, etc.

 

2000 Mathematics Subject Classification: 52A01, 52A41, 90C26.

Keywords: Generalized convexity, rough convexity, outer γ-convex function, inner γ-convex function, perturbation of convex function, self-Jung constant, γ-extreme point.

 

 

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