Stéphane Gaubert
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View article: Signed Tropicalization of Polar Cones
Signed Tropicalization of Polar Cones Open
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 …
View article: Continuity and approximability of competitive spectral radii
Continuity and approximability of competitive spectral radii Open
The competitive spectral radius extends the notion of joint spectral radius to the two-player case: two players alternatively select matrices in prescribed compact sets, resulting in an infinite matrix product; one player wishes to maximiz…
View article: Stationary regimes of piecewise linear dynamical systems with priorities
Stationary regimes of piecewise linear dynamical systems with priorities Open
Dynamical systems governed by priority rules appear in the modeling of emergency organizations and road traffic. These systems can be modeled by piecewise linear time-delay dynamics, specifically using Petri nets with priority rules. A cen…
View article: Computing the Congestion Phases of Dynamical Systems with Priorities and Application to Emergency Departments
Computing the Congestion Phases of Dynamical Systems with Priorities and Application to Emergency Departments Open
Medical emergency departments are complex systems in which patients must be treated according to priority rules based on the severity of their condition. We develop a model of emergency departments using Petri nets with priorities, describ…
View article: Equality of tropical rank and dimension for tropical linear series
Equality of tropical rank and dimension for tropical linear series Open
The tropical rank of a semimodule of rational functions on a metric graph mirrors the concept of rank in linear algebra. Defined in terms of the maximal number of tropically independent elements within the semimodule, this quantity has rem…
View article: Spectral Properties of Positive Definite Matrices over Symmetrized Tropical Algebras and Valued Ordered fields
Spectral Properties of Positive Definite Matrices over Symmetrized Tropical Algebras and Valued Ordered fields Open
We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the eigen…
View article: A Multilevel Fast Marching Method for the Minimum Time Problem
A Multilevel Fast Marching Method for the Minimum Time Problem Open
International audience
View article: The Competive Spectral Radius of Families of Nonexpansive Mappings
The Competive Spectral Radius of Families of Nonexpansive Mappings Open
We consider a new class of repeated zero-sum games in which the payoff is the escape rate of a switched dynamical system, where at every stage, the transition is given by a nonexpansive operator depending on the actions of both players. Th…
View article: Directed Metric Structures arising in Large Language Models
Directed Metric Structures arising in Large Language Models Open
Large Language Models are transformer neural networks which are trained to produce a probability distribution on the possible next words to given texts in a corpus, in such a way that the most likely word predicted is the actual word in th…
View article: Order isomorphisms of sup-stable function spaces: continuous, Lipschitz, c-convex, and beyond
Order isomorphisms of sup-stable function spaces: continuous, Lipschitz, c-convex, and beyond Open
There have been many parallel streams of research studying order isomorphisms of some specific sets $G$ of functions from a set $X$ to $\mathbb{R}\cup\{\pm\infty\}$, such as the sets of convex or Lipschitz functions. We develop in this art…
View article: Exact Quantization of Multistage Stochastic Linear Problems
Exact Quantization of Multistage Stochastic Linear Problems Open
We show that the multistage linear problem (MSLP) with an arbitrary cost distribution is equivalent to a MSLP on a finite scenario tree. We establish this exact quantization result by analyzing the polyhedral structure of MSLPs. In particu…
View article: Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time
Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time Open
International audience
View article: A Quantization Procedure for Nonlinear Pricing with an Application to Electricity Markets
A Quantization Procedure for Nonlinear Pricing with an Application to Electricity Markets Open
We consider a revenue maximization model, in which a company aims at\ndesigning a menu of contracts, given a population of customers. A standard\napproach consists in constructing an incentive-compatible continuum of\ncontracts, i.e., a me…
View article: The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems
The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems Open
Grigoriev and Podolskii (2018) have established a tropical analogue of the effective Nullstellensatz, showing that a system of tropical polynomial equations is solvable if and only if a linearized system obtained from a truncated Macaulay …
View article: Linear algebra over T-pairs
Linear algebra over T-pairs Open
This is part of an ongoing project to find a general algebraic framework for semiring theory. The structure theory of semirings is quite challenging, largely because of the lack of negation, and such basic properties such as unique factori…
View article: The Tropical Nullstellensatz and Positivstellensatz for Sparse Polynomial Systems
The Tropical Nullstellensatz and Positivstellensatz for Sparse Polynomial Systems Open
Grigoriev and Podolskii (2018) have established a tropical analog of the effective Nullstellensatz, showing that a system of tropical polynomial equations is solvable if and only if a linearized system obtained from a truncated Macaulay ma…
View article: Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time
Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time Open
A classical problem in ergodic continuous time control consists of studying the limit behavior of the optimal value of a discounted cost functional with infinite horizon as the discount factor $λ$ tends to zero. In the literature, this pro…
View article: Tropical Linear Regression and Mean Payoff Games: Or, How to Measure the Distance to Equilibria
Tropical Linear Regression and Mean Payoff Games: Or, How to Measure the Distance to Equilibria Open
International audience
View article: Signed tropicalization of polar cones
Signed tropicalization of polar cones Open
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 …
View article: Solving irreducible stochastic mean-payoff games and entropy games by relative Krasnoselskii-Mann iteration
Solving irreducible stochastic mean-payoff games and entropy games by relative Krasnoselskii-Mann iteration Open
We analyse an algorithm solving stochastic mean-payoff games, combining the ideas of relative value iteration and of Krasnoselskii-Mann damping. We derive parameterized complexity bounds for several classes of games satisfying irreducibili…
View article: An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems
An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems Open
We introduce a new numerical method to approximate the solution of a finite horizon deterministic optimal control problem. We exploit two Hamilton-Jacobi-Bellman PDE, arising by considering the dynamics in forward and backward time. This a…
View article: A Quantization Procedure for Nonlinear Pricing with an Application to Electricity Markets
A Quantization Procedure for Nonlinear Pricing with an Application to Electricity Markets Open
We consider a revenue maximization model, in which a company aims at designing a menu of contracts, given a population of customers. A standard approach consists in constructing an incentive-compatible continuum of contracts, i.e., a menu …
View article: A Multi-Level Fast-Marching Method For The Minimum Time Problem
A Multi-Level Fast-Marching Method For The Minimum Time Problem Open
We introduce a new numerical method to approximate the solutions of a class of stationary Hamilton-Jacobi (HJ) partial differential equations arising from minimum time optimal control problems. We rely on nested grid approximations, and lo…
View article: Complexity of Geometric programming in the Turing model and application to nonnegative tensors
Complexity of Geometric programming in the Turing model and application to nonnegative tensors Open
We consider a version of geometric programming problem consisting in minimizing a function given by the maximum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption,…
View article: Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration
Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration Open
We analyse an algorithm solving stochastic mean-payoff games, combining the ideas of relative value iteration and of Krasnoselskii-Mann damping. We derive parameterized complexity bounds for several classes of games satisfying irreducibili…
View article: An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems
An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems Open
We introduce a new numerical method to approximate the solution of a finite horizon deterministic optimal control problem. We exploit two Hamilton-Jacobi-Bellman PDE, arising by considering the dynamics in forward and backward time. This a…