Advertisement
Published in:
2011 | OriginalPaper | Chapter
A complete lattice for vector optimization
Author : Andreas Löhne
Published in: Vector Optimization with Infimum and Supremum
Publisher: Springer Berlin Heidelberg
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. 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.