The tropical analogue of polar cones

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.

arXiv comment21 pages, 3 figures, example added, figures improved, notation changed

References in corpus (6)

• Duality and separation theorems in idempotent semimodules
• Max-plus (A,B)-invariant spaces and control of timed discrete event systems
• The Minkowski Theorem for Max-plus Convex Sets
• Generators, extremals and bases of max cones
• Rational semimodules over the max-plus semiring and geometric approach of discrete event systems
• Carathéodory, Helly and the others in the max-plus world

Cited by in corpus (10)

• Tropical polyhedra are equivalent to mean payoff games
• Tropical linear-fractional programming and parametric mean payoff games
• Minimal half-spaces and external representation of tropical polyhedra
• Duality between invariant spaces for max-plus linear discrete event systems
• The number of extreme points of tropical polyhedra
• Tropical polar cones, hypergraph transversals, and mean payoff games
• The level set method for the two-sided eigenproblem
• Minimal external representations of tropical polyhedra
• Signed tropical convexity
• Using matrix sparsification to solve tropical linear vector equations