On a globalization property

Rolewicz, Stefan. "On a globalization property." Applicationes Mathematicae 22.1 (1993): 69-73. <http://eudml.org/doc/219084>.

@article{Rolewicz1993,
abstract = {Let (X,τ) be a topological space. Let Φ be a class of real-valued functions defined on X. A function ϕ ∈ Φ is called a local Φ-subgradient of a function f:X → ℝ at a point $x_0$ if there is a neighbourhood U of $x_0$ such that f(x) - f($x_0$) ≥ ϕ(x) - ϕ($x_0$) for all x ∈ U. A function ϕ ∈ Φ is called a global Φ-subgradient of f at $x_0$ if the inequality holds for all x ∈ X. The following properties of the class Φ are investigated: (a) when the existence of a local Φ-subgradient of a function f at each point implies the existence of a global Φ-subgradient of f at each point (globalization property), (b) when each local Φ-subgradient can be extended to a global Φ-subgradient (strong globalization property).},
author = {Rolewicz, Stefan},
journal = {Applicationes Mathematicae},
keywords = {Φ-subgradients; globalization property; -subgradient; strong globalization},
language = {eng},
number = {1},
pages = {69-73},
title = {On a globalization property},
url = {http://eudml.org/doc/219084},
volume = {22},
year = {1993},
}

TY - JOUR
AU - Rolewicz, Stefan
TI - On a globalization property
JO - Applicationes Mathematicae
PY - 1993
VL - 22
IS - 1
SP - 69
EP - 73
AB - Let (X,τ) be a topological space. Let Φ be a class of real-valued functions defined on X. A function ϕ ∈ Φ is called a local Φ-subgradient of a function f:X → ℝ at a point $x_0$ if there is a neighbourhood U of $x_0$ such that f(x) - f($x_0$) ≥ ϕ(x) - ϕ($x_0$) for all x ∈ U. A function ϕ ∈ Φ is called a global Φ-subgradient of f at $x_0$ if the inequality holds for all x ∈ X. The following properties of the class Φ are investigated: (a) when the existence of a local Φ-subgradient of a function f at each point implies the existence of a global Φ-subgradient of f at each point (globalization property), (b) when each local Φ-subgradient can be extended to a global Φ-subgradient (strong globalization property).
LA - eng
KW - Φ-subgradients; globalization property; -subgradient; strong globalization
UR - http://eudml.org/doc/219084
ER -