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Substitution for minimizing/maximizing a tropical linear (fractional) programming

L. Truffet

Abstract

Tropical polyhedra seem to play a central role in static analysis of softwares. These tropical geometrical objects play also a central role in parity games especially mean payoff games and energy games. And determining if an initial state of such game leads to win the game is known to be equivalent to solve a tropical linear optimization problem. This paper mainly focus on the tropical linear minimization problem using a special substitution method on the tropical cone obtained by homogenization of the initial tropical polyhedron. But due to a particular case which can occur in the minimization process based on substitution we have to switch on a maximization problem. Nevertheless, forward-backward substitution is known to be strongly polynomial. The special substitution developed in this paper inherits the strong polynomiality of the classical substitution for linear systems. This special substitution must not be confused with the exponential execution time of the tropical Fourier-Motzkin elimination. Tropical fractional minimization problem with linear objective functions is also solved by tropicalizing the Charnes-Cooper's transformation of a fractional linear program into a linear program developed in the usual linear algebra. Let us also remark that no particular assumption is made on the polyhedron of interest. Finally, the substitution method is illustrated on some examples borrowed from the litterature.

Substitution for minimizing/maximizing a tropical linear (fractional) programming

Abstract

Tropical polyhedra seem to play a central role in static analysis of softwares. These tropical geometrical objects play also a central role in parity games especially mean payoff games and energy games. And determining if an initial state of such game leads to win the game is known to be equivalent to solve a tropical linear optimization problem. This paper mainly focus on the tropical linear minimization problem using a special substitution method on the tropical cone obtained by homogenization of the initial tropical polyhedron. But due to a particular case which can occur in the minimization process based on substitution we have to switch on a maximization problem. Nevertheless, forward-backward substitution is known to be strongly polynomial. The special substitution developed in this paper inherits the strong polynomiality of the classical substitution for linear systems. This special substitution must not be confused with the exponential execution time of the tropical Fourier-Motzkin elimination. Tropical fractional minimization problem with linear objective functions is also solved by tropicalizing the Charnes-Cooper's transformation of a fractional linear program into a linear program developed in the usual linear algebra. Let us also remark that no particular assumption is made on the polyhedron of interest. Finally, the substitution method is illustrated on some examples borrowed from the litterature.

Paper Structure

This paper contains 25 sections, 9 theorems, 222 equations.

Table of Contents

  1. Main notations and definitions
  2. Introduction
  3. Problem description
  4. Motivations
  5. Organization of the paper
  6. The switch to a maximizing problem can occur
  7. Preliminary results
  8. Elementary results from interval arithmetic in dioids
  9. The minimization problem $\textsf{mpr}$
  10. Study of the cone $\mathcal{C}(i):=\{z \geq c^{\intercal} \otimes x \oplus c_{h} \otimes h, \; a^{+}_{i,.} \otimes w \geq a^{-}_{i,.} \otimes w\}$
  11. How to choose the next variable for substitution in mpr
  12. Classification of the variables and the linear functions of mpr
  13. Choosing a variable for a new substitution after $k$ substitutions in mpr
  14. The new characteristic elements of the mpr after substitution
  15. Linear fractional programming
  16. ...and 10 more sections

Key Result

Proposition 3.1

Let $[u_{l}, +\infty]$, $l=1, \ldots,k$, $k \geq 2$ be $k$ intervals of $\mathcal{I}$ with $u_{1} \leq \ldots \leq u_{k}$. Then, we have the following series of intervals inclusion:

Theorems & Definitions (14)

  • Remark 2.1
  • Proposition 3.1
  • Proposition 3.2: Valid inequality
  • Theorem 4.1: Saturation of an inequality
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.2: Minimality and reachability at $\boldsymbol{-\infty}$ of $h \mapsto c_{h} \otimes h$
  • Theorem 4.3
  • Remark 5.1
  • Proposition A.1
  • ...and 4 more