Kolokoltsov, V. N. (Vasiliĭ Nikitich) (2001) Idempotent structures in optimization. Journal of Mathematical Sciences, Vol.104 (No.1). pp. 847-880. doi:10.1023/A:1009514904187

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Abstract

Consider the set A = R ∪ {+∞} with the binary operations o1 = max
and o2 = + and denote by An the set of vectors v = (v1,...,vn) with entries
in A. Let the generalised sum u o1 v of two vectors denote the vector with
entries uj o1 vj , and the product a o2 v of an element a ∈ A and a vector
v ∈ An denote the vector with the entries a o2 vj . With these operations,
the set An provides the simplest example of an idempotent semimodule.
The study of idempotent semimodules and their morphisms is the subject
of idempotent linear algebra, which has been developing for about
40 years already as a useful tool in a number of problems of discrete optimisation.
Idempotent analysis studies infinite dimensional idempotent
semimodules and is aimed at the applications to the optimisations problems
with general (not necessarily finite) state spaces. We review here
the main facts of idempotent analysis and its major areas of applications
in optimisation theory, namely in multicriteria optimisation, in turnpike
theory and mathematical economics, in the theory of generalised solutions
of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games
and controlled Marcov processes, in financial mathematics.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Mathematical optimization, Idempotents
Journal or Publication Title:	Journal of Mathematical Sciences
Publisher:	Springer
ISSN:	10723374
Official Date:	2001
Dates:	Date Event 2001 Published
Volume:	Vol.104
Number:	No.1
Page Range:	pp. 847-880
DOI:	10.1023/A:1009514904187
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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