NDLI: Supremizers of inner γ-convex functions

Content Provider	Springer Nature Link
Author	Phu, Hoang Xuan
Copyright Year	2007
Abstract	A real-valued function f defined on a convex subset D of some normed linear space is said to be inner γ-convex w.r.t. some fixed roughness degree γ > 0 if there is a $$\nu \in]0, 1]$$ such that $${\rm sup}_{\lambda\in[2,1+1/\nu]} \left(f((1 - \lambda)x_0 + \lambda x_1) - (1 - \lambda) f (x_0)-\right. \left.\lambda f(x_1)\right) \geq 0$$ holds for all $$x_0, x_1 \in D$$ satisfying \|\|x 0 − x 1\|\| = νγ and $$-(1/\nu)x_0+(1+1/\nu)x_1\in D$$ . This kind of roughly generalized convex functions is introduced in order to get some properties similar to those of convex functions relative to their supremum. In this paper, numerous properties of their supremizers are given, i.e., of such $$x^* \in D$$ satisfying lim $${\rm sup}_{x \to x^*}f(x) = {\rm sup}_{x \in D} f(x)$$ . For instance, if an upper bounded and inner γ-convex function, which is defined on a convex and bounded subset D of some inner product space, has supremizers, then there exists a supremizer lying on the boundary of D relative to aff D or at a γ-extreme point of D, and if D is open relative to aff D or if dim D ≤ 2 then there is certainly a supremizer at a γ-extreme point of D. Another example is: if D is an affine set and $$f : D \to {\mathbb{R}}$$ is inner γ-convex and bounded above, then $${\rm sup}_{x'\in \bar B(x,\gamma/2)\cap D}f(x')= \sup_{x'\in D}f(x')$$ for all $$x \in D$$ , and if 2 ≤ dim D < ∞ then each $$x \in D$$ is a supremizer of f.
Ending Page	222
Page Count	16
Starting Page	207
File Format	PDF
ISSN	14322994
e-ISSN	14325217
Journal	Mathematical Methods of Operations Research
Issue Number	2
Volume Number	67
Language	English
Publisher	Springer-Verlag
Publisher Date	2007-10-24
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Supremizer Axiomatic and generalized convexity γ-Extreme point Calculus of Variations and Optimal Control; Optimization Business/Management Science Operations Research/Decision Theory Convex functions and convex programs Nonconvex programming, global optimization Inner γ-convex function Generalized convexity Rough convexity
Content Type	Text
Resource Type	Article
Subject	Management Science and Operations Research Mathematics Software

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