The tropical analogue of polar cones

Stéphane Gaubert; Ricardo D. Katz

doi:10.1016/j.laa.2009.03.012

The tropical analogue of polar cones

Stéphane Gaubert
, Ricardo D. Katz

Universidad Nacional de Rosario

Research output: Contribution to journal › Article › peer-review

Abstract

We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the inequalities satisfied by the elements of a tropical convex cone. We derive this characterization from a new separation theorem. We also establish variants of these results concerning systems of linear equalities.

Original language	English
Pages (from-to)	608-625
Number of pages	18
Journal	Linear Algebra and Its Applications
Volume	431
Issue number	5-7
DOIs	https://doi.org/10.1016/j.laa.2009.03.012
Publication status	Published - 1 Aug 2009

Keywords

B-convexity
Duality
Extremal convexity
Farkas lemma
Idempotent spaces
Max-plus convexity
Max-plus semiring
Semimodules
Separation theorem
Tropical convexity

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10.1016/j.laa.2009.03.012

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abstract = "We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the inequalities satisfied by the elements of a tropical convex cone. We derive this characterization from a new separation theorem. We also establish variants of these results concerning systems of linear equalities.",

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AU - Katz, Ricardo D.

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AB - We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the inequalities satisfied by the elements of a tropical convex cone. We derive this characterization from a new separation theorem. We also establish variants of these results concerning systems of linear equalities.

KW - B-convexity

KW - Duality

KW - Extremal convexity

KW - Farkas lemma

KW - Idempotent spaces

KW - Max-plus convexity

KW - Max-plus semiring

KW - Semimodules

KW - Separation theorem

KW - Tropical convexity

U2 - 10.1016/j.laa.2009.03.012

DO - 10.1016/j.laa.2009.03.012

M3 - Article

AN - SCOPUS:67349245150

SN - 0024-3795

VL - 431

SP - 608

EP - 625

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 5-7

ER -

The tropical analogue of polar cones

Abstract

Keywords

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