# On a globalization property

Applicationes Mathematicae (1993)

- Volume: 22, Issue: 1, page 69-73
- ISSN: 1233-7234

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topRolewicz, 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 -

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