Tropical linear programming and parametric mean payoff games

17 pages•Published: June 22, 2012

Stephane Gaubert, Ricardo Katz and Sergei Sergeev

Abstract

Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, the tropical analogues of classical linear programming results need to be developed. This motivation leads us to study a general tropical linear programming problem. We construct an associated parametric mean payoff game problem, and show that the optimality of a given point, or the unboundedness of the problem, can be certified by exhibiting a strategy for one of the players having certain infinitesimal properties (involving the value of the game and its derivative) that we characterize combinatorially. In other words, strategies play in tropical linear programming the role of Lagrange multipliers in classical linear programming. We use this idea to design a Newton-like algorithm to solve tropical linear programming problems, by reduction to a sequence of auxiliary mean payoff game problems.

Keyphrases: discrete event systems, disjunctive domains, mean payoff games, policy iteration, static analysis, tropical algebra

In: Andrei Voronkov, Laura Kovacs and Nikolaj Bjorner (editors). WING 2010. Workshop on Invariant Generation 2010, vol 1, pages 94-110.

Links:	https://easychair.org/publications/paper/Nq4R
	https://doi.org/10.29007/jcrz

BibTeX entry

@inproceedings{WING2010:Tropical_linear_programming_parametric,
  author    = {Stephane Gaubert and Ricardo Katz and Sergei Sergeev},
  title     = {Tropical linear programming and parametric mean payoff games},
  booktitle = {WING 2010. Workshop on Invariant Generation 2010},
  editor    = {Andrei Voronkov and Laura Kovacs and Nikolaj Bjorner},
  series    = {EPiC Series in Computing},
  volume    = {1},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {/publications/paper/Nq4R},
  doi       = {10.29007/jcrz},
  pages     = {94-110},
  year      = {2012}}

Download PDF Open PDF in browser