Anzeige
Erschienen in:
2011 | OriginalPaper | Buchkapitel
A complete lattice for vector optimization
verfasst von : Andreas Löhne
Erschienen in: Vector Optimization with Infimum and Supremum
Verlag: Springer Berlin Heidelberg
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Extended real-valued objective functions are characteristic for scalar optimization problems. The space of extended real numbers
$$\bar{\mathbb{R}} := \mathbb{R} \bigcup \{{+ \infty}\} \bigcup \{{-\infty} \}$$
enjoys several properties which are quite important for optimization: (i)
$$ \mathbb{R}\, $$
is a vector space, but
$$\bar{\mathbb{R}}$$
is not. The linear operations can be partially extended to
$$\bar{\mathbb{R}}$$
. (ii) The linear operations on
$$ \mathbb{R}\, $$
are continuous, i.e., the topology is compatible with the linear structure. (iii)
$$\bar{\mathbb{R}}\, \hbox{is totally ordered by the usual ordering}\leq. \hbox{The ordering on}\, \mathbb{R} \, $$
is compatible with the linear operations.(iv)
$$\bar{\mathbb{R}}$$
is a complete lattice, i.e., every subset has an infimum and a supremum.