A categorical approach to the maximum theorem
Category Theory
2018-07-03 v2 Optimization and Control
Abstract
Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories. This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.
Cite
@article{arxiv.1704.00209,
title = {A categorical approach to the maximum theorem},
author = {Seerp Roald Koudenburg},
journal= {arXiv preprint arXiv:1704.00209},
year = {2018}
}
Comments
45 pages. Minor changes in v2: this is the final preprint for publication in JPAA