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Signed tropicalization of polar cones

Marianne Akian, Xavier Allamigeon, Stéphane Gaubert, Sergei Sergeev

Abstract

We study the tropical analogue of the notion of polar of a cone, working over the semiring of tropical numbers with signs. We characterize the cones which arise as polars of sets of tropically nonnegative vectors by an invariance property with respect to a tropical analogue of Fourier-Motzkin elimination. We also relate tropical polars with images by the nonarchimedean valuation of classical polars over real closed nonarchimedean fields and show, in particular, that for semi-algebraic sets over such fields, the operation of taking the polar commutes with the operation of signed valuation (keeping track both of the nonarchimedean valuation and sign). We apply these results to characterize images by the signed valuation of classical cones of matrices, including the cones of positive semidefinite matrices, completely positive matrices, completely positive semidefinite matrices, and their polars, including the cone of co-positive matrices, showing that hierarchies of classical cones collapse under tropicalization. We finally discuss an application of these ideas to optimization with signed tropical numbers.

Signed tropicalization of polar cones

Abstract

We study the tropical analogue of the notion of polar of a cone, working over the semiring of tropical numbers with signs. We characterize the cones which arise as polars of sets of tropically nonnegative vectors by an invariance property with respect to a tropical analogue of Fourier-Motzkin elimination. We also relate tropical polars with images by the nonarchimedean valuation of classical polars over real closed nonarchimedean fields and show, in particular, that for semi-algebraic sets over such fields, the operation of taking the polar commutes with the operation of signed valuation (keeping track both of the nonarchimedean valuation and sign). We apply these results to characterize images by the signed valuation of classical cones of matrices, including the cones of positive semidefinite matrices, completely positive matrices, completely positive semidefinite matrices, and their polars, including the cone of co-positive matrices, showing that hierarchies of classical cones collapse under tropicalization. We finally discuss an application of these ideas to optimization with signed tropical numbers.
Paper Structure (17 sections, 35 theorems, 90 equations, 1 figure, 1 table)

This paper contains 17 sections, 35 theorems, 90 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Main results
  4. Related work
  5. Basic definitions
  6. The symmetrized tropical semiring and its relation with real Puiseux series
  7. Classical matrix cones
  8. Polars and valuations
  9. Tropical polars
  10. Valuation of polars
  11. Tropicalization of matrix cones and their polars
  12. Tropical positive definite matrices and completely positive matrices
  13. Tropical copositive and co-completely positive semidefinite matrices
  14. Optimization in the Signed Tropical World
  15. Optimization problems over the symmetrized tropical semiring
  16. ...and 2 more sections

Key Result

Proposition 2.1

\newlabelp:trans0 Let $a,b,c,d\in\mathbb{T}^2$.

Figures (1)

  • Figure 1: Polynomial $f(x)$ over the symmetrized semiring \newlabelf:poly0

Theorems & Definitions (89)

  • Proposition 2.1
  • Proof 1
  • Definition 2.2
  • Example 2.3
  • Definition 3.1
  • Definition 3.2: GK-09
  • Definition 3.3
  • Theorem 3.4: GK-09
  • Definition 3.5
  • Definition 3.6
  • ...and 79 more