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Idempotent structures in optimization
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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 ISSN 10723374.
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Official URL: http://dx.doi.org/10.1023/A:1009514904187
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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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: |
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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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