Duality and interval analysis over idempotent semirings

In this paper semirings with an idempotent addition are considered. These algebraic structures are endowed with a partial order. This allows to consider residuated maps to solve systems of inequalities A⊗X⪯B (see [3]). The purpose of this paper is to consider a dual product, denoted ⊙, and the dual...

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Bibliographic Details
Published in:Linear algebra and its applications 2012-11, Vol.437 (10), p.2436-2454
Main Authors: Brunsch, Thomas, Hardouin, Laurent, Maia, Carlos Andrey, Raisch, Jörg
Format: Article
Language:English
Subjects:
Algebra
Combinatorics. Ordered structures
Error analysis
Exact sciences and technology
Idempotent semiring
Interval analysis
Linear and multilinear algebra, matrix theory
Mathematics
Max-plus algebra
Numerical analysis
Numerical analysis. Scientific computation
Order, lattices, ordered algebraic structures
Residuation theory
Sciences and techniques of general use
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Summary:In this paper semirings with an idempotent addition are considered. These algebraic structures are endowed with a partial order. This allows to consider residuated maps to solve systems of inequalities A⊗X⪯B (see [3]). The purpose of this paper is to consider a dual product, denoted ⊙, and the dual residuation of matrices, in order to solve the following inequality A⊗X⪯X⪯B⊙X. Sufficient conditions ensuring the existence of a non-linear projector in the solution set are proposed. The results are extended to semirings of intervals such as they were introduced in [25].
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2012.06.025