TROPICAL

TROPICAL - 2025

2025Activity reportProject-TeamTROPICAL

RNSR: 201621988K‌

Research center Inria Saclay Centre
Team name: Tropical‌ methods: structures, algorithms and interactions
In collaboration with:‌Centre de Mathématiques Appliquées (CMAP)

Creation of the‌ Project-Team: 2018 July 01

Each year, Inria research‌ teams publish an Activity Report presenting their work‌ and results over the reporting period. These reports‌ follow a common structure, with some optional sections‌ depending on the specific team. They typically begin‌ by outlining the overall objectives and research programme,‌ including the main research themes, goals, and methodological‌ approaches. They also describe the application domains targeted‌ by the team, highlighting the scientific or societal‌ contexts in which their work is situated.

The‌ reports then present the highlights of the year,‌ covering major scientific achievements, software developments, or teaching‌ contributions. When relevant, they include sections on software,‌ platforms, and open data, detailing the tools developed‌ and how they are shared. A substantial part‌ is dedicated to new results, where scientific contributions‌ are described in detail, often with subsections specifying‌ participants and associated keywords.

Finally, the Activity Report‌ addresses funding, contracts, partnerships, and collaborations at various‌ levels, from industrial agreements to international cooperations. It‌ also covers dissemination and teaching activities, such as‌ participation in scientific events, outreach, and supervision. The‌ document concludes with a presentation of scientific production,‌ including major publications and those produced during the‌ year.

Keywords

Computer Science and Digital Science

A1.2.4.‌ QoS, performance evaluation
A2.3.3. Real-time systems
A4.5. Formal‌ method for verification, reliability, certification
A6.2.5. Numerical Linear‌ Algebra
A6.2.6. Optimization
A6.4.2. Stochastic control
A6.4.6. Optimal‌ control
A7.2.4. Mechanized Formalization of Mathematics
A8.1. Discrete‌ mathematics, combinatorics
A8.2.1. Operations research
A8.3. Geometry, Topology‌
A8.4. Computer Algebra
A8.9. Performance evaluation
A8.11. Game‌ Theory
A9.6. Decision support

1‌ Team members, visitors, external collaborators

Research Scientists

Stephane‌ Gaubert [Team leader, INRIA, Senior‌ Researcher, HDR]
Marianne Akian [INRIA‌, Senior Researcher, HDR]
Xavier Allamigeon‌ [Inria, Researcher, Corps des Mines,‌ under secondment]
Matías Bender [INRIA,‌ Researcher]
Cormac Walsh [INRIA, Researcher‌]

PhD Students

Amanda Bigel [UNIV MONTPELLIER‌, from Sep 2025]
Amanda Bigel [‌INRIA, until Aug 2025]
Abdellah Bulaich‌ Mehamdi [ECOLE POLY PALAISEAU, CIFRE]‌
Pascal Capetillo [INRIA]
Yiyuan Chen [ECOLE POLY PALAISEAU]‌
Luca Froger [INRIA‌]
Jonathan Hornewall [‌‌ECOLE POLY PALAISEAU]
Loic Marchesini [ECOLE‌ POLY PALAISEAU]
Jad‌ Zeroual [ORANGE LABS‌‌, CIFRE]

Interns and Apprentices

Hanqi Sun‌ [INRIA, Intern‌, from Apr 2025‌‌ until Jul 2025]

Administrative Assistants

Bahar Carabetta‌ [INRIA, from‌ Dec 2025]
Amandine‌‌ Sainsard [INRIA, until Nov 2025]‌

Visiting Scientist

Gleb Koshevoi‌ [Russian Academy of‌‌ Science, from Feb 2025 until Apr 2025‌]

2 Overall objectives‌

The project develops tropical‌‌ methods motivated by applications arising in decision theory‌ (deterministic and stochastic optimal‌ control, game theory, optimization‌‌ and operations research), in the analysis or control‌ of classes of dynamical‌ systems (including timed discrete‌‌ event systems and positive systems), in the verification‌ of programs and systems,‌ and in the development‌‌ of numerical algorithms. Tropical algebra tools are used‌ in interaction with various‌ methods, coming from convex‌‌ analysis, Hamilton–Jacobi partial differential equations, metric geometry, Perron-Frobenius‌ and nonlinear fixed-point theories,‌ combinatorics or algorithmic complexity.‌‌ The emphasis of the project is on mathematical‌ modelling and computational aspects.‌

The subtitle of the‌‌ Tropical project, namely, “structures, algorithms, and interactions”, refers‌ to the spirit of‌ our research, including a‌‌ methodological component, computational aspects, and finally interactions with‌ other scientific fields or‌ real world applications, in‌‌ particular through mathematical modelling.

2.1 Scientific context

Tropical‌ algebra, geometry, and analysis‌ have enjoyed spectacular development‌‌ in recent years. Tropical structures initially arose to‌ solve problems in performance‌ evaluation of discrete event‌‌ systems 68, combinatorial optimization 75, or‌ automata theory 114.‌ They also arose in‌‌ mathematical physics and asymptotic analysis 105, 100‌. More recently, these‌ structures have appeared in‌‌ several areas of pure mathematics, in particular in‌ the study of combinatorial‌ aspects of algebraic geometry‌‌ 91, 129, 119, 99,‌ in algebraic combinatorics 85‌, and in arithmetics‌‌ 79. Also, further applications of tropical methods‌ have appeared, including optimal‌ control 106, program‌‌ invariant computation 62 and timed systems verification 104‌, and zero-sum games‌ 2.

The term‌‌ `tropical' generally refers to algebraic structures in which‌ the laws originate from‌ optimization processes. The prototypical‌‌ tropical structure is the max-plus semifield, consisting of‌ the real numbers, equipped‌ with the maximum, thought‌‌ of as an additive law, and the addition,‌ thought of as a‌ multiplicative law. Tropical objects‌‌ appear as limits of classical objects along certain‌ deformations (“log-limits sets” of‌ Bergman, “Maslov dequantization”, or‌‌ “Viro deformation”). For this reason, the introduction of‌ tropical tools often yields‌ new insights into old‌‌ familiar problems, leading either to counterexamples or to‌ new methods and results;‌ see for instance 129‌‌, 110. In some applications, like optimal‌ control, discrete event systems,‌ or static analysis of‌‌ programs, tropical objects do not appear through a‌ limit procedure, but more‌ directly as a modelling‌‌ or computation/analysis tool; see‌ for instance 124, 68, 95,‌ 76.

Tropical methods are linked to the‌ fields of positive systems and of metric geometry‌ 112, 12. Indeed, tropically linear maps‌ are monotone (a.k.a. order-preserving). They are also nonexpansive‌ in certain natural metrics (sup-norm, Hopf oscillation, Hilbert's‌ projective metric, ...). In this way, tropical dynamical‌ systems appear to be special cases of nonexpansive,‌ positive, or monotone dynamical systems, which are studied‌ as part of linear and non-linear Perron-Frobenius theory‌ 101, 3. Such dynamical systems are‌ of fundamental importance in the study of repeated‌ games 109. Monotonicity properties are also essential‌ in the understanding of the fixed points problems‌ which determine program invariants by abstract interpretation 80‌. The latter problems are actually somehow similar‌ to the ones arising in the study of‌ zero-sum games; see 7. Moreover, positivity or‌ monotonicity methods are useful in population dynamics, either‌ in a discrete space setting 125 or in‌ a PDE setting 70. In such cases,‌ solving tropical problems often leads to solutions or‌ combinatorial insights on classical problems involving positivity conditions‌ (e.g., finding equilibria of dynamical systems with nonnegative‌ coordinates, understanding the qualitative and quantitative behavior of‌ growth rates / Floquet eigenvalues 10, etc).‌ Other applications of Perron-Frobenius theory originate from quantum‌ information and control 118, 123.

3‌ Research program

3.1 Optimal control and zero-sum games‌

The dynamic programming approach allows one to analyze‌ one or two-player dynamic decision problems by means‌ of operators, or partial differential equations (Hamilton–Jacobi or‌ Isaacs PDEs), describing the time evolution of the‌ value function, i.e., of the optimal reward of‌ one player, thought of as a function of‌ the initial state and of the horizon. We‌ work especially with problems having long or infinite‌ horizon, modelled by stopping problems, or ergodic problems‌ in which one optimizes a mean payoff per‌ time unit. The determination of optimal strategies reduces‌ to solving nonlinear fixed point equations, which are‌ obtained either directly from discrete models, or after‌ a discretization of a PDE.

The geometry of‌ solutions of optimal control and game problems Basic‌ questions include, especially for stationary or ergodic problems,‌ the understanding of existence and uniqueness conditions for‌ the solutions of dynamic programming equations, for instance‌ in terms of controllability or ergodicity properties, and‌ more generally the understanding of the structure of‌ the full set of solutions of stationary Hamilton–Jacobi‌ PDEs and of the set of optimal strategies.‌ These issues are already challenging in the one-player‌ deterministic case, which is an application of choice‌ of tropical methods, since the Lax-Oleinik semigroup, i.e.,‌ the evolution semigroup of the Hamilton-Jacobi PDE, is‌ a linear operator in the tropical sense. Recent‌ progress in the deterministic case has been made‌ by combining dynamical systems and PDE techniques (weak‌ KAM theory 82), and also using metric‌ geometry ideas (abstract boundaries can be used to represent the sets of‌ solutions 98, 4‌). The two player‌‌ case is challenging, owing to the lack of‌ compactness of the analogue‌ of the Lax-Oleinik semigroup‌‌ and to a richer geometry. The conditions of‌ solvability of ergodic problems‌ for games (for instance,‌‌ solvability of ergodic Isaacs PDEs), and the representation‌ of solutions are only‌ understood in special cases,‌‌ for instance in the finite state space case,‌ through tropical geometry and‌ non-linear Perron-Frobenius methods 48‌‌, 41, 3.

Algorithmic aspects: from‌ combinatorial algorithms to the‌ attenuation of the curse‌‌ of dimensionality Our general goal is to push‌ the limits of solvable‌ models by means of‌‌ fast algorithms adapted to large scale instances. Such‌ instances arise from discrete‌ problems, in which the‌‌ state space may so large that it is‌ only accessible through local‌ oracles (for instance, in‌‌ some web ranking applications, the number of states‌ may be the number‌ of web pages) 83‌‌. They also arise from the discretization of‌ PDEs, in which the‌ number of states grows‌‌ exponentially with the number of degrees of freedom,‌ according to the “curse‌ of dimensionality”. A first‌‌ line of research is the development of new‌ approximation methods for the‌ value function. So‌‌ far, classical approximations by linear combinations have been‌ used, as well as‌ approximation by suprema of‌‌ linear or quadratic forms, which have been introduced‌ in the setting of‌ dual dynamic programming and‌‌ of the so called “max-plus basis methods” 84‌. We believe that‌ more concise or more‌‌ accurate approximations may be obtained by unifying these‌ methods. Also, some max-plus‌ basis methods have been‌‌ shown to attenuate the curse of dimensionality for‌ very special problems (for‌ instance involving switching) 107‌‌, 87. This suggests that the complexity‌ of control or games‌ problems may be measured‌‌ by more subtle quantities that the mere number‌ of states, for instance,‌ by some forms of‌‌ metric entropy (for example, certain large scale problems‌ have a low complexity‌ owing to the presence‌‌ of decomposition properties, “highway hierarchies”, etc.). A second‌ line of of our‌ research is the development‌‌ of combinatorial algorithms, to solve large scale‌ zero-sum two-player problems with‌ discrete state space. This‌‌ is related to current open problems in algorithmic‌ game theory. In particular,‌ the existence of polynomial-time‌‌ algorithms for games with ergodic payment is an‌ open question. See e.g.‌ 52 for a polynomial‌‌ time average complexity result derived by tropical methods.‌ The two lines of‌ research are related, as‌‌ the understanding of the geometry of solutions allows‌ to develop better approximation‌ or combinatorial algorithms.

3.2‌‌ Non-linear Perron-Frobenius theory, nonexpansive mappings and metric geometry‌

Several applications (including population‌ dynamics 10 and discrete‌‌ event systems 68, 78, 53)‌ lead to studying classes‌ of dynamical systems with‌‌ remarkable properties: preserving a cone, preserving an order,‌ or being nonexpansive in‌ a metric. These can‌‌ be studied by techniques‌ of non-linear Perron-Frobenius theory 3 or metric geometry‌ 11. Basic issues concern the existence and‌ computation of the “escape rate” (which determines the‌ throughput, the growth rate of the population), the‌ characterizations of stationary regimes (non-linear fixed points), or‌ the study of the dynamical properties (convergence to‌ periodic orbits). Nonexpansive mappings also play a key‌ role in the “operator approach” to zero-sum games,‌ since the one-day operators of games are nonexpansive‌ in several metrics, see 8.

3.3 Tropical‌ algebra and convex geometry

The different applications mentioned‌ in the other sections lead us to develop‌ some basic research on tropical algebraic structures and‌ in convex and discrete geometry, looking at objects‌ or problems with a “piecewise-linear ” structure. These‌ include the geometry and algorithmics of tropical convex‌ sets 61,2, tropical semialgebraic sets‌ 65, the study of semi-modules (analogues of‌ vector spaces when the base field is replaced‌ by a semi-field), the study of systems of‌ equations linear in the tropical sense, investigating for‌ instance the analogues of the notions of rank,‌ the analogue of the eigenproblems 47, and‌ more generally of systems of tropical polynomial equations.‌ Our research also builds on, and concern, classical‌ convex and discrete geometry methods.

3.4 Tropical methods‌ applied to optimization, perturbation theory and matrix analysis‌

Tropical algebraic objects appear as a deformation of‌ classical objects thought various asymptotic procedures. A familiar‌ example is the rule of asymptotic calculus,

{e​​​‌}^{- a / ϵ﻿​﻿﻿} + e^{- b​‌﻿﻿ / ϵ} ≍ {e​​﻿﻿}^{- min (a​​​‌, b) /﻿​﻿﻿ ϵ}, e^{-​‌﻿﻿ a / ϵ} \times​​﻿﻿ e^{- b /​​​‌ ϵ} = e^{-﻿​﻿﻿ (a + b​‌﻿﻿) / ϵ},​​﻿﻿

when $ϵ \to‌ 0^{+}$ . Deformations of this kind have‌ been studied in different contexts: large deviations, zero-temperature‌ limits, Maslov's “dequantization method” 105, non-archimedean valuations,‌ log-limit sets and Viro's patchworking method 129,‌ etc.

This entails a relation between classical algorithmic‌ problems and tropical algorithmic problems, one may first‌ solve the $ϵ = 0$ case (non-archimedean problem),‌ which is sometimes easier, and then use the‌ information gotten in this way to solve the‌ $ϵ = 1$ (archimedean) case.

In particular, tropicalization‌ establishes a connection between polynomial systems and piecewise‌ affine systems that are somehow similar to the‌ ones arising in game problems. It allows one‌ to transfer results from the world of combinatorics‌ to “classical” equations solving. We investigate the consequences‌ of this correspondence on complexity and numerical issues.‌ For instance, combinatorial problems can be solved in‌ a robust way. Hence, situations in which the‌ tropicalization is faithful lead to improved algorithms for‌ classical problems. In particular, scalings for the polynomial‌ eigenproblems based on tropical preprocessings have started to‌ be used in matrix analysis 88, 94‌.

Moreover, the tropical approach has been recently‌ applied to construct examples of linear programs in which the central path‌ has an unexpectedly high‌ total curvature 51,‌‌6, and it has also led to‌ positive polynomial-time average case‌ results concerning the complexity‌‌ of mean payoff games. Similarly, we are studying‌ semidefinite programming over non-archimedean‌ fields 65, 64‌‌, with the goal to better understand complexity‌ issues in classical semidefinite‌ and semi-algebraic programming.

4‌‌ Application domains

4.1 Discrete event systems (manufacturing systems,‌ networks, emergency call centers)‌

One important class of‌‌ applications of max-plus algebra comes from discrete event‌ dynamical systems 68.‌ In particular, modelling timed‌‌ systems subject to synchronization and concurrency phenomena leads‌ to studying dynamical systems‌ that are non-smooth, but‌‌ which have remarkable structural properties (nonexpansiveness in certain‌ metrics , monotonicity) or‌ combinatorial properties. Algebraic methods‌‌ allow one to obtain analytical expressions for performance‌ measures (throughput, waiting time,‌ etc). A recent application,‌‌ to emergency call centers, can be found in‌ 53.

4.2 Optimal‌ control and games

Optimal‌‌ control and game theory have numerous well established‌ applications fields: mathematical economy‌ and finance, stock optimization,‌‌ optimization of networks, decision making, etc. In most‌ of these applications, one‌ needs either to derive‌‌ analytical or qualitative properties of solutions, or design‌ exact or approximation algorithms‌ adapted to large scale‌‌ problems.

4.3 Operations Research

We develop, or have‌ developed, several aspects of‌ operations research, including the‌‌ application of stochastic control to optimal pricing, optimal‌ measurement in networks 120‌. Applications of tropical‌‌ methods arise in particular from discrete optimization 76‌, 77, scheduling‌ problems with and-or constraints‌‌ 111, or product mix auctions 127.‌

4.4 Computing program and‌ dynamical systems invariants

A‌‌ number of programs and systems verification questions, in‌ which safety considerations are‌ involved, reduce to computing‌‌ invariant subsets of dynamical systems. This approach appears‌ in various guises in‌ computer science, for instance‌‌ in static analysis of program by abstract interpretation,‌ along the lines of‌ P. and R. Cousot‌‌ 80, but also in control (eg, computing‌ safety regions by solving‌ Isaacs PDEs). These invariant‌‌ sets are often sought in some tractable effective‌ class: ellipsoids, polyhedra, parametric‌ classes of polyhedra with‌‌ a controlled complexity (the so called “templates” introduced‌ by Sankaranarayanan, Sipma and‌ Manna 122), shadows‌‌ of sets represented by linear matrix inequalities, disjunctive‌ constraints represented by tropical‌ polyhedra 62, etc.‌‌ The computation of invariants boils down to solving‌ large scale fixed point‌ problems. The latter are‌‌ of the same nature as the ones encountered‌ in the theory of‌ zero-sum games, and so,‌‌ the techniques developed in the previous research directions‌ (especially methods of monotonicity,‌ nonexpansiveness, discretization of PDEs,‌‌ etc) apply to the present setting, see e.g.‌ 86, 90 for‌ the application of policy‌‌ iteration type algorithms, or for the application for‌ fixed point problems over‌ the space of quadratic‌‌ forms 7. The problem of computation of‌ invariants is indeed a‌ key issue needing the‌‌ methods of several fields:‌ convex and nonconvex programming, semidefinite programming and symbolic‌ computation (to handle semialgebraic invariants), nonlinear fixed point‌ theory, approximation theory, tropical methods (to handle disjunctions),‌ and formal proof (to certify numerical invariants or‌ inequalities).

5 Social and environmental responsibility

5.1 Impact‌ of research results

The team has developed collaborations‌ on the dimensioning of emergency call centers, with‌ Préfecture de Police (Plate Forme d'Appels d'Urgence -‌ PFAU - 17-18-112, operated jointly by Brigade de‌ sapeurs pompiers de Paris and by Direction de‌ la sécurité de proximité de l'agglomération parisienne) and‌ also with the Emergency medical services of Assistance‌ Publique – Hôpitaux de Paris (Centre 15 of‌ SAMU75, 92, 93 and 94). This work is‌ described further in Section 8.7.1. A current‌ extension of this work deals with the modelling‌ of medical emergency services, with the project “URGE”‌ which started at the fall 2022, in the‌ framework of the joint INRIA & AP-HP “Bernoulli”‌ lab.

6 Highlights of the year

6.1 Awards‌

Matías Bender received the “Applications of Computer Algebra‌ (ACA) Early Researcher Award” 2025.

6.2 Other highlights‌

Joint Inria-CWI associate team started in 2025.
ANR‌ collaborative research project (PRC) ZADyG, Zoology of Algorithmic‌ Methods for Dynamic Games (CE48) accepted in 2025,‌ coordinated by Marianne Akian.
Young researcher ANR project‌ PeACE, Polynômes et Applications via Calculs Efficace (CE48)‌ accepted in 2025, coordinated by Matías Bender.
Xavier‌ Allamigeon gave a plenary talk at the conference‌ ROADEF 2025.
The articles 26, 25 give‌ sufficient condition for the existence of stationary regimes‌ of piece-wise linear dynamical systems representing priorities, as‌ well as an algorithm to compute then. This‌ yields a partial answer to an open question‌ raised in 1999, see 116.

7 Latest‌ software developments, platforms, open data

7.1 Latest software‌ developments

7.1.1 Coq-Polyhedra

Name:
Coq-Polyhedra
Keywords:
Coq, Polyhedra,‌ Automated theorem proving, Linear optimization
Scientific Description:

Coq-Polyhedra‌ is a library providing a formalization of convex‌ polyhedra in the Coq proof assistant. While still‌ in active development, it provides an implementation of‌ the simplex method, and already handles the basic‌ properties of polyhedra such as emptiness, boundedness, membership.‌ Several fundamental results in the theory of convex‌ polyhedra, such as Farkas Lemma, duality theorem of‌ linear programming, and Minkowski Theorem, are also formally‌ proved.

The formalization is based on the Mathematical‌ Components library, and makes an extensive use of‌ the boolean reflection methodology.
Functional Description:
Coq-Polyhedra is‌ a library which aims at formalizing convex polyhedra‌ in Coq
URL:
https://github.com/nhojem/Coq-Polyhedra
Publications:
hal-01673390, hal-03151656‌, hal-03915661, hal-01967575, hal-01967576
Contact:
Xavier‌ Allamigeon
Participants:
Xavier Allamigeon, Quentin Canu, Ricardo Katz,‌ Pierre-Yves Strub
Partners:
CIFASIS, Ecole Polytechnique

7.1.2 EmergencyEval‌

Keywords:
Dynamic Analysis, Simulation, Ocaml, Emergency, Firefighters, Police‌
Scientific Description:

This software aims at enabling the‌ definition of a Petri network execution semantic, as‌ well as the instanciation and execution of said‌ network using the aforedefined semantic.

The heart of‌ the project dwells in its kernel which operates the step-by-step execution of‌ the network, obeying rules‌ provided by an oracle.‌‌ This user-defined and separated oracle computes the information‌ necessary to the kernel‌ for building the next‌‌ state using the current state. The base of‌ our software is the‌ framework for the instanciation‌‌ and execution of Petri nets, without making assumptions‌ regarding the semantic.

In‌ the context of the‌‌ study of the dynamics of emergency call centers,‌ a second part of‌ this software is the‌‌ definition and implementation of the semantic of call‌ centers modelized as Petri‌ nets, and more specifically‌‌ timed prioritized Petri nets. A module interoperating with‌ the kernel enables to‌ include all the operational‌‌ specificities of call centers (urgency level, discriminating between‌ operators and callers ...)‌ while guaranteeing the genericity‌‌ of the kernal which embeds the Petri net‌ formalism as such.
Functional‌ Description:

In order to‌‌ enable the quantitative study of the throughput of‌ calls managed by emergency‌ center calls and the‌‌ assesment of various organisationnal configurations considered by the‌ stakeholders (firefighters, police, medical‌ emergency service of the‌‌ 75, 92, 93 and 94 French departments), this‌ software modelizes their behaviours‌ by resorting to extensions‌‌ of the Petri net formalism. Given a call‌ transfer protocol in a‌ call center, which corresponds‌‌ to a topology and an execution semantic of‌ a Petri net, the‌ software generates a set‌‌ of entering calls in accord with the empirically‌ observed statistic ditributions (share‌ of very urgent calls,‌‌ conversation length), then simulates its management by the‌ operators with respect to‌ the defined protocol. Transitional‌‌ regimes phenomenons (peak load, support) which are not‌ yet handled by mathematical‌ analysis could therefore be‌‌ studied. The ouput of the software is a‌ log file which is‌ an execution trace of‌‌ the simulation featuring extensive information in order to‌ enable the analysis of‌ the data for providing‌‌ simulation-based insights for decision makers.

The software relies‌ on a Petri net‌ simulation kernel designed to‌‌ be as modular and adaptable as possible, fit‌ for simulating other Petri-net‌ related phenomenons, even if‌‌ their semantic differ greatly.
Contact:
Xavier Allamigeon
Participants:‌
Xavier Allamigeon, Benjamin Nguyen-Van-Yen‌

8 New results

8.1‌‌ Optimal control and zero-sum games

8.1.1 Tropical numerical‌ methods for stochastic control‌ problems

Participants: Marianne Akian‌‌.

We are interested here in the numerical‌ solution of the dynamic‌ programming equation of discrete‌‌ time deterministic or stochastic control problems.

In several‌ works in collaboration with‌ Jean-Philippe Chancelier (CERMICS) and‌‌ Benoit Tran, and included in the PhD thesis‌ of Benoit Tran 126‌, we developed and‌‌ studied algorithms combining the tropical or the max-plus‌ based numerical method of‌ McEneaney 108, 106‌‌, the stochastic max-plus scheme proposed by Zheng‌ Qu 117, and‌ the stochastic dual dynamic‌‌ programming (SDDP) algorithm of Pereira and Pinto 113‌. In particular in‌ 15, we considered‌‌ a stochastic algorithm for deterministic control problems, and‌ in 39, we‌ considered stochastic control problems.‌‌

In an ongoing work‌ with Luz Pascal (PhD student at Queensland University‌ of Technology, Australia), we are studying the convergence‌ of SARSOP algorithm solving partially observable Markov Decision‌ Process (POMDP), using the same techniques as in‌ 39.

8.1.2 Highway hierarchies for Hamilton-Jacobi-Bellman (HJB)‌ PDEs

Participants: Marianne Akian, Stéphane Gaubert.‌

Hamilton-Jacobi-Bellman equations arise as the dynamic programming equations‌ of deterministic or stochastic optimal control problems. They‌ allow to obtain the global optimum of these‌ problems and to synthesize an optimal feedback control,‌ leading to a solution robust against system perturbations.‌ Several methods have been proposed in the literature‌ to bypass the obstruction of curse of dimensionality‌ of such equations, assuming a certain structure of‌ the problem, and/or using “unstructured discretizations”, that are‌ not based on given grids. Among them, one‌ may cite tropical numerical method, and probabilistic numerical‌ method. On another direction, “highway hierarchies”, developed by‌ Sanders, Schultes and coworkers 81, 121,‌ initially for applications to on-board GPS systems, are‌ a computational method that allows one to accelerate‌ Dijkstra algorithm for discrete time and state shortest‌ path problems.

In several works in collaboration with‌ Shanqing Liu (now in Brown University), and included‌ in his PhD thesis 103, we have‌ developped new numerical methods to solve Hamilton-Jacobi-Bellman equations‌ that are less sensitive to curse of dimensionality.‌ In 43, we have introduced a multilevel‌ fast-marching method, extending to the PDE case the‌ idea of “highway hierarchies”. The complexity of the‌ method is then analysed using a priori error‌ bounds for the discretization of the Hamilton-Jacobi PDE.‌ In 49, we obtain error estimates of‌ order 1 for the semi-Lagrangian scheme of the‌ Hamilton-Jacobi equation of the minimum time problem, under‌ some regularity assumptions on the dynamics and the‌ domain. In a more recent work, we are‌ extending these estimates to state constrained problems in‌ order to deduce the best complexity results for‌ the multilevel fast-marching method.

8.1.3 Escape Rate Games‌

Participants: Marianne Akian, Stéphane Gaubert, Loic‌ Marchesini.

The aim of the PhD thesis‌ of Loic Marchesini is to study a new‌ class of repeated zero-sum games in which the‌ payoff of one player is the escape rate‌ of a dynamical system whose dynamics depends on‌ the actions of both players. We suppose that‌ the dynamics obtained from nonexpansive operators. Considering order‌ preserving finite dimensional linear operators over the positive‌ cone endowed with Hilbert's projective (semi-)metric, we recover‌ the matrix multiplication games introduced by Asarin et‌ al. 67, which generalize the joint spectral‌ radius of sets of nonnegative matrices. See also‌ 40. We show in 44 that escape‌ rate games have a value, which we call‌ the competitive spectral radius as it extends the‌ joint spectral radius as well as the lower‌ spectral rdius, corresponding to “one player” special cases.‌ We also characterize the competitive spectral radius in‌ terms of non-linear eigenproblems, providing a a two-player analogue of “Mañe's Lemma”‌ in ergodic control. In‌ a followup work, joint‌‌ with Ian Morris (Queen Mary, London) 24,‌ we show that the‌ competivive spectral radius of‌‌ a family of operators is continuous and approximable,‌ under conditions which cover‌ in particular families of‌‌ operators sending a cone to its interior.

8.1.4‌ Thresholds for sensitive optimality‌ and Blackwell optimality in‌‌ stochastic games

Participants: Stéphane Gaubert.

In a‌ joint work with Julien‌ Grand-Clément (HEC Paris) and‌‌ Ricardo Katz (CIFASIS, CONICET) 28, we give‌ explicit bounds for the‌ threshods of sensitive optimality‌‌ and Blackwell optimality in turn-based stochastic games. The‌ Blackwell threshold is the‌ value of the discount‌‌ factor above which optimal policies become independent of‌ the discount factor, meaning‌ that the players become‌‌ perfectly farsighted. The sensitive thresholds arise when interpolating‌ between the mean-payoff objective‌ and the vanishing discount‌‌ objective. Our bounds are based on separation results‌ for algebraic numbers.

8.1.5‌ Polyhedral methods in stochastic‌‌ programming

Participants: Stéphane Gaubert, Jonathan Hornewall.‌

The PhD thesis of‌ Jonathan Hornewall, in collaboration‌‌ with CERMICS, ENPC, deals with polyhedral methods in‌ stochastic programmming. The article‌ 97 introduces a decision-focused‌‌ scenario generation method for contextual two-stage stochastic linear‌ programs.

8.2 Non-linear Perron-Frobenius‌ theory, nonexpansive mappings and‌‌ metric geometry

8.2.1 Characterising JB-algebras using order reversing‌ maps

Participants: Cormac Walsh‌.

We have been‌‌ studying non-linear operators on open cones, in particular‌ ones that reverse the‌ order structure associated to‌‌ the cone. A bijective map that reverses the‌ order in both directions‌ is called an order‌‌ antimorphism. These are closely related to the isometries‌ of the Hilbert and‌ Thompson metrics on the‌‌ cone.

An example of an antimorphism on a‌ cone is the inverse‌ map $x \mapsto {x‌‌}^{- 1}$ in the positive cone of a‌ JB-algebra. In fact, it‌ was conjectured by Lemmens‌‌ and van Gaans that these are the only‌ examples. More precisely, they‌ conjectured that the existence‌‌ of an order antimorphism that is homogeneous of‌ degree $- 1$ on‌ the open cone of‌‌ a order unit space implies that the cone‌ is the positive cone‌ of a JB-algebra.

In‌‌ previous work 130, 131, we established‌ an even stronger result‌ in the finite dimensional‌‌ case, where the map was not assumed to‌ be homogeneous. Recently, Roelands‌ and Tiersma have released‌‌ a preprint where they prove the conjecture in‌ infinite dimension. We have‌ been working on extending‌‌ our stronger result, where there is no assumption‌ of homogeneity, to the‌ infinite dimensional case.

8.2.2‌‌ Invariant Finsler metrics on symmetric spaces

Participants: Cormac‌ Walsh.

This is‌ joint work with Bas‌‌ Lemmens (Kent).

We are interested in metrics on‌ symmetric spaces, in particular‌ Finsler metrics that are‌‌ invariant under the symmetries of the space. In‌ a previous paper 93‌, it was established‌‌ that there is such a metric naturally associated‌ to every representation of‌ the symmetric space, which‌‌ depends on the system‌ of weights of the representation. More precisely, one‌ takes the convex hull of the weights and‌ interprets that as the dual ball in the‌ Artin subspace. The lengths of all other vectors‌ can then determined by invariance. We are investigating‌ this correspondence between representations and metrics in more‌ detail. We wish to answer the question, concretely,‌ which known representations correspond to which invariant Finsler‌ metrics?

We have also been looking more closely‌ an especially interesting class of symmetric space: the‌ bounded symmetric domains with their Carathéodory (or equivalently,‌ Kobayashi) metric. Here, the metric restricts to the‌ supremum norm on the Artin subspace. These symmetric‌ spaces are particularly tractable due to their relation‌ with JB*-triples—each one arises as the unit ball‌ of such a triple. The algebraic structure of‌ the triple turns out to be very helpful.‌

We have been studying how these spaces can‌ be embedded in each other isometrically. We show‌ 102 that a bounded symmetric domain cannot be‌ isometrically embedded into one of lower rank. In‌ the case where the rank of the target‌ space is strictly larger, it turns out that‌ there is a lot of flexibility in what‌ the map can look like. When the two‌ spaces have the same rank, however, the situation‌ is much more rigid—we show that the map‌ must respect any product structure of the domain,‌ and furthermore, in the case where the domain‌ is irreducible, that the map is either holomorphic‌ or anti-holomorphic.

These results generalise work of Seo‌ and Kim, who were working under the additional‌ assumption that the maps in question are ${C‌}^{1}$ -smooth. They made this assumption because they‌ were using techniques from differential geometry, which require‌ it. We are able to relax the assumption‌ because, rather than the small scale geometry of‌ the space, we consider the large scale geometry.‌ In particular, a tool we use is the‌ horofunction boundary—for bounded symmetric domains, this was determined‌ in earlier work by Chu–Cueto-Avellaneda—Lemmens.

8.2.3 Isoperimetry in‌ Hilbert geometries

Participants: Amanda Bigel, Stéphane Gaubert‌, Constantin Vernicos, Cormac Walsh.

The‌ PhD work of Amanda Bigel deals with the‌ investigation of isoperimetric inequalities in the geometries of‌ Funk, Thompson, and Hilbert's metric on convex domains.‌ In particular, a characterization of the isoperimetrix with‌ respect to the Thompson metric of the Lorentz‌ cone has been obtained.

8.2.4 Horocyclic products

Participants:‌ Luca Froger, Constantin Vernicos, Cormac Walsh‌.

This is the PhD work of Luca‌ Froger. We are studying horocyclic products of metric‌ spaces, in particular, group actions on these spaces.‌ One question we are interested in answering is‌ under what conditions is there a group making‌ these horospherical products divisible in the sense of‌ Benoist? We are also interested in knowing when‌ a horospherical product is amenable. Here, amenability is‌ to be understood in the purely metric sense,‌ using Følner sets, not the group theoretic notion.

8.2.5 External representation of‌ hyperconvex sets

Participants: Stéphane‌ Gaubert.

MA: j'ai‌‌ enleve “subsets of ${ℝ}^{n}$ ” car cela‌ fait une “erreur” dans‌ radar, il ecrit “Package‌‌ hyperref Warning: Token not allowed in a PDF‌ string (Unicode): removing `math‌ shift' on input line‌‌ 478.” meme avec protect Hyperconvex spaces are another‌ name for injective metric‌ spaces or tight-spans. In‌‌ a current work with Gleb Koshevoy, we are‌ studying hyperconvex subsets of‌ $ℝ^{n}$ , equipped‌‌ with the sup-norm. We provide an external representation‌ of hyperconvex sets, as‌ the intersection of a‌‌ family of non-linear half-spaces associated to norm-like functionals.‌

8.3 Tropical algebra and‌ convex geometry

8.3.1 Formalizing‌‌ convex polyhedra in Coq

Participants: Xavier Allamigeon,‌ Quentin Canu.

In‌ a joint work with‌‌ Pierre-Yves Strub (Meta), we have achieved the formal‌ verification of a counterexample‌ of Santos et al.‌‌ to the so-called Hirsch Conjecture on the diameter‌ of polytopes. In contrast‌ with the pen-and-paper proof,‌‌ our approach is entirely computational: we implement in‌ Coq and prove correct‌ an algorithm that explicitly‌‌ computes, within the proof assistant, vertex-edge graphs of‌ polytopes as well as‌ their diameter. The originality‌‌ of this certificate-based algorithm is to achieve a‌ tradeoff between simplicity and‌ efficiency. Simplicity is crucial‌‌ in obtaining the proof of correctness of the‌ algorithm. This proof splits‌ into the correctness of‌‌ an abstract algorithm stated over proof-oriented data types‌ and the correspondence with‌ a low-level implementation over‌‌ computation-oriented data types. A special effort has been‌ made to reduce the‌ algorithm to a small‌‌ sequence of elementary operations (e.g., matrix multiplications, basic‌ routines on sets and‌ graphs), in order to‌‌ make the derivation of the correctness of the‌ low-level implementation more transparent.‌ Efficiency allows us to‌‌ scale up to polytopes with a challenging combinatorics.‌ For instance, we formally‌ check the two counterexamples‌‌ of Matschke, Santos and Weibel to the Hirsch‌ conjecture, respectively 20- and‌ 23-dimensional polytopes with 36‌‌ 425 and 73 224 vertices involving rational coefficients‌ with up to 40‌ digits in their numerator‌‌ and denominator. We also illustrate the performance of‌ the method by computing‌ the list of vertices‌‌ or the diameter of well-known classes of polytopes,‌ such as (polars of)‌ cyclic polytopes involved in‌‌ McMullen's Upper Bound Theorem. This work has appeared‌ in the proceedings of‌ the conference CPP'23 58‌‌.

Dealing with polyhedra and their faces raises‌ the problem of formalizing‌ order structures. We study‌‌ this problem in a joint work 57 with‌ Cyril Cohen (Inria), Kazuhiko‌ Sakaguchi (Inria) and Pierre-Yves‌‌ Strub (Meta). More precisely, using order structures in‌ a proof assistant naturally‌ raises the problem of‌‌ working with multiple instances of a same structure‌ over a common type‌ of elements. This goes‌‌ against the main design pattern of hierarchies used‌ for instance in Coq's‌ MathComp or Lean's mathlib‌‌ libraries, where types are canonically associated to at‌ most one instance and‌ instances share a common‌‌ overloaded syntax. We introduce‌ new design patterns to leverage these issues, and‌ apply them to the formalization of order structures‌ in the MathComp library. A common idea in‌ these patterns is underloading, i.e., a disambiguation of‌ operators on a common type. In addition, our‌ design patterns include a way to deal with‌ duality in order structures in a convenient way.‌ We hence formalize a large hierarchy which includes‌ partial orders, semilattices, lattices as well as many‌ variants. We finally pay a special attention to‌ order substructures. We introduce a new kind of‌ structure called prelattice. They are abstractions of semilattices,‌ and allow us to deal with finite lattices‌ and their sublattices within a common signature. As‌ an application, we report on significant simplifications of‌ the formalization of the face lattices of polyhedra‌ in the Coq-Polyhedra library.

8.3.2 Linear algebra over‌ systems

Participants: Marianne Akian, Stéphane Gaubert.‌

In a joint work with Louis Rowen (Univ.‌ Bar Ilan), we study the properties of “systems”.‌ The latter provide a general setting encompassing extensions‌ of the tropical semifields and hyperfields. Moreover, they‌ have the advantage to be well adapted to‌ the study of linear or polynomial equations. In‌ particular, in 46, we characterize the semiring‌ systems which arise from hyperrings.

In 45,‌ we are studying linear algebra properties over a‌ generalization of “systems” called “T-pairs”. We are still‌ working on improvements of the results of 45‌.

8.3.3 Roots over the symmetrized tropical semiring‌ and eigenvalues of tropical symmetric matrices

Participants: Marianne‌ Akian, Stéphane Gaubert.

The tropical semifield‌ can be thought of as the image of‌ a field with a non-archimedean valuation. It allows‌ in this way to study the asymptotics of‌ Puiseux series with complex coefficients. When dealing with‌ Puiseux series with real coefficients and with its‌ associated order, it is convenient to use the‌ symmetrized tropical semiring introduced in 115 (see also‌ 68), and the signed valuation which associates‌ to any series its valuation together with its‌ sign.

In a work 16 which started during‌ the postdoc of Hanieh Tavakolipour (Amirkabir University of‌ Technology) in the team, we studied the roots's‌ multiplicities and the factorization of polynomials over the‌ symmetrized tropical semiring. We then deduced a Descartes'‌ rule of sign over ordered valued fields. This‌ builds in particular on 69 (for multiplicities) and‌ on 65.

More recently, in a work‌ with Dariush Kiani and Hanieh Tavakolipour (Amirkabir University‌ of Technology) 42, we studied with the‌ above tools the asymptotics of eigenvalues and eigenvectors‌ of symmetric positive definite matrices over the field‌ of Puiseux series. In a more recent work,‌ we are studying the eigenvalues of general symmetric‌ matrices over the field of Puiseux series and‌ over the tropical semiring.

8.3.4 Tropical Systems of‌ Polynomial Equations

Participants: Marianne Akian, Matías Bender‌, Antoine Bereau, Stéphane Gaubert.

The‌ PhD thesis of Antoine Bereau 71 dealt with systems of polynomial equations‌ over tropical semifields. In‌ 38, 14,‌‌ we established a Nullstellenstatz for sparse tropical polynomial‌ systems. We reduce a‌ polynomial system to a‌‌ linearized system obtained by an appropriate truncation of‌ the Macaulay matrix. Our‌ approach is inspired by‌‌ a construction of Canny-Emiris (1993), refined by Sturmfels‌ (1994). It leads to‌ an improved estimate of‌‌ the truncation degree. We also establish a tropical‌ positivstellensatz, allowing one to‌ decide the containment of‌‌ tropical basic semialgebraic sets. This method leads to‌ the solution of systems‌ of tropical linear equalities‌‌ and inequalities, which reduces to mean payoff games‌ 37, 14.‌

Building on this construction,‌‌ in collaboration with Yue Ren (University of Durham,‌ UK), we are investigating‌ new efficient methods for‌‌ computing resultants and solving structured polynomial systems. These‌ systems include cases where‌ the degeneration into tropical‌‌ forms, such as vertically and horizontally parameterized systems,‌ is well understood 96‌.

8.3.5 Systems of‌‌ sparse polynomial equations and convex polytopes

Participants: Matías‌ Bender.

Polynomial systems‌ are ubiquitous in applications.‌‌ They allow us to model problems involving curved‌ objects without linearizing them.‌ Dealing with polynomials is‌‌ challenging, as worst-case bounds are double exponential. However,‌ the polynomials that we‌ encounter in practice are‌‌ structured, and so these bounds are too pessimistic.‌ Nevertheless, our computational strategies,‌ designed to deal with‌‌ arbitrary systems, cannot be applied successfully to these‌ systems, as they do‌ not exploit the structure‌‌ of the problems that we encounter. For this‌ reason, it is central‌ to develop algorithms that‌‌ can take these structures into account. Among them,‌ one of the most‌ common structures is sparsity,‌‌ that is, the polynomials that we encounter can‌ be written as a‌ linear combination of a‌‌ few monomials. Toric and tropical geometry give us‌ a way of understanding‌ and exploiting sparsity.

As‌‌ part of the PhD project of Charles Checa‌ (now a postdoc at‌ U. Copenhagen, Denmark), together‌‌ with Laurent Busé (Team Aromath, Inria Sophia-Antipolis) and‌ Elias Tsigaridas (Team Ouragan,‌ Inria Paris), we focus‌‌ on multihomogeneous systems, which are sparse polynomials with‌ simple but ubiquitous sparsity‌ structure. By extending the‌‌ seminal work of Bayer and Stillman, in 20‌, we show the‌ first relation between the‌‌ complexity of computing Gröbner bases for these systems‌ and the multigraded Castelnuovo-Mumford‌ regularity introduced by Maclagan‌‌ and Smith. Moreover, we study specialized algorithms to‌ compute projections of the‌ zero set of these‌‌ varieties, when they are finite, establishing a new‌ relation between multiplication maps‌ and higher-dimensional varieties, and‌‌ via them, new complexity bounds to solve these‌ problems 27.

8.4‌ Tropical methods applied to‌‌ optimization, perturbation theory and matrix analysis

8.4.1 Tropicalization‌ of interior point methods‌ and application to complexity‌‌

Participants: Xavier Allamigeon, Stéphane Gaubert.

It‌ is an open question‌ to determine if the‌‌ theory of self-concordant barriers can provide an interior‌ point method with strongly‌ polynomial complexity in linear‌‌ programming. In the special‌ case of the logarithmic barrier, it was shown‌ in 51,6 that the answer is‌ negative.

In a subsequent work 50 with Abdellah‌ Aznag (Columbia University) and Yassine Hamdi (Ecole Polytechnique),‌ we have studied the tropicalization of the central‌ path associated with the entropic barrier studied by‌ Bubeck and Eldan (Proc. Mach. Learn. Research, 2015),‌ i.e., the logarithmic limit of this central path‌ for a parametric family of linear programs defined‌ over the field of Puiseux series. Our main‌ result is that the tropicalization of the entropic‌ central path is a piecewise linear curve which‌ coincides with the tropicalization of the logarithmic central‌ path studied by Allamigeon et al. in 51‌,6.

In the work 66,‌ we have now shown that none of the‌ self-concordant barrier interior point methods is strongly polynomial.‌ This result is obtained by establishing that, on‌ parametric families of convex optimization problems, the log-limit‌ of the central path degenerates to the same‌ piecewise linear curve, independently of the choice of‌ the barrier function. We also provided an improved‌ counter example, with an explicit linear program that‌ falls in the same class as the Klee–Minty‌ counterexample, i.e., a $n$ -dimensional combinatorial cube,‌ in which the number of iterations is ${2‌}^{n}$ .

A key tool in this work‌ consists of metric inequalities, controlling the convergence of‌ the log-images of semialgebraic sets to a polyhedral‌ complex (their tropicalization). Explicit convergence bounds have been‌ subsequently established, see §8.4.4 below. These results‌ were presented in the Phd thesis of Nicolas‌ Vandame 128.

In a joint work 60‌, 17 with Daniel Dadush, Georg Loho, Bento‌ Natura and László Végh, we establish a natural‌ connection between the complexity of interior point methods‌ and that of the simplex method, and deduce‌ combinatorial bounds on the number of iterations. In‌ more details, we introduce a new polynomial-time path-following‌ interior point method where the number of iterations‌ also admits a combinatorial upper bound $O (‌ 2^{n} n^{1 . 5} log n‌)$ for an $n$ -variable linear program in‌ standard form. The number of iterations of our‌ algorithm is at most $O (n^{1‌ . 5} log n)$ times the number‌ of segments of any piecewise linear curve in‌ the wide neighborhood of the central path. In‌ particular, it matches the number of iterations of‌ any path following interior point method up to‌ this polynomial factor. The overall exponential upper bound‌ derives from studying the `max central path', a‌ piecewise-linear curve with the number of pieces bounded‌ by the total length of $2 n$ shadow‌ vertex simplex paths.

8.4.2 Tropical Nash equilibria and‌ complementarity problems over oriented matroids

Participants: Xavier Allamigeon‌, Stéphane Gaubert.

Linear complementarity programming is‌ a generalization of linear programming which encompasses the‌ computation of Nash equilibria for bimatrix games. While‌ the latter problem is PPAD-complete, we show in 63 that the analogue‌ of this problem in‌ tropical algebra can be‌‌ solved in polynomial time. Moreover, we prove that‌ the Lemke–Howson algorithm carries‌ over the tropical setting‌‌ and performs a linear number of pivots in‌ the worst case. A‌ consequence of this result‌‌ is a new class of (classical) bimatrix games‌ for which Nash equilibria‌ computation can be done‌‌ in polynomial time. This is joint work with‌ Frédéric Meunier (Cermics, ENPC).‌ Current work focuses on‌‌ abstracting the notion of Nash equilibra over oriented‌ matroids and showing that‌ the problem of finding‌‌ such an equilibria is still in PPAD (modulo‌ chirotope oracles). This is‌ joint work with Frédéric‌‌ Meunier (ENPC).

8.4.3 Signed Tropicalization of Polars and‌ application to Matrix Cones‌

Participants: Marianne Akian,‌‌ Xavier Allamigeon, Stéphane Gaubert.

With Sergey‌ Sergeev (U. Birmingham), we‌ study in 13 the‌‌ tropical analogue of the notion of polar of‌ a cone over the‌ symmetrized tropical semiring (see‌‌ for instance 115, 68). We characterize‌ in particular the tropical‌ polars of sets of‌‌ nonnegative tropical vectors, and relate them with images‌ by the nonarchimedean valuation‌ of classical polars over‌‌ real closed nonarchimedean fields. We study in particular‌ cones of matrices, and‌ optimization problems.

8.4.4 Log-limits‌‌ of semi-algebraic sets

Participants: Xavier Allamigeon, Stéphane‌ Gaubert.

We study‌ the log-convergence of parametric‌‌ families of semialgebraic sets to their valuation. We‌ obtain explicit metric estimates‌ under genericity conditions for‌‌ basic semialgebraic sets. In the general case, we‌ establish a multivariate extension‌ of the Denef–Pas cell‌‌ decomposition. Using Smale’s $α$ -theory, we give a‌ quantitative version of this‌ decomposition that allows us‌‌ to prove a metric lifting theorem for semialgebraic‌ sets. It yields a‌ constructive bound on the‌‌ one-sided Hausdorff convergence of the parametric family to‌ its valuation. These results‌ were presented in the‌‌ PhD thesis of Nicolas Vandame 128, we‌ are pursuing the work‌ on this topic.SG: kept‌‌ the para as it is ongoing, ok?

8.5‌ Relu and softplus neural‌ nets as zero-sum turn-based‌‌ games

Participants: Stéphane Gaubert.

In a joint‌ work with Yannis Vlassopoulos‌ (Athena research center) 89‌‌, we show that the output of a‌ ReLU neural network can‌ be interpreted as the‌‌ value of a zero-sum, turn-based, stopping game, which‌ we call the ReLU‌ net game. The game‌‌ runs in the direction opposite to that of‌ the network, and the‌ input of the network‌‌ serves as the terminal reward of the game.‌ In fact, evaluating the‌ network is the same‌‌ as running the Shapley-Bellman backward recursion for the‌ value of the game.‌ Using the expression of‌‌ the value of the game as an expected‌ total payoff with respect‌ to the path measure‌‌ induced by the transition probabilities and a pair‌ of optimal policies, we‌ derive a discrete Feynman-Kac-type‌‌ path-integral formula for the network output. This game-theoretic‌ representation can be used‌ to derive bounds on‌‌ the output from bounds‌ on the input, leveraging the monotonicity of Shapley‌ operators, and to verify robustness properties using policies‌ as certificates. Moreover, training the neural network becomes‌ an inverse game problem: given pairs of terminal‌ rewards and corresponding values, one seeks transition probabilities‌ and rewards of a game that reproduces them.‌ Finally, we show that a similar approach applies‌ to neural networks with Softplus activation functions, where‌ the ReLU net game is replaced by its‌ entropic regularization.

8.6 Polynomial optimization and certificates of‌ positivity

Participants: Matías Bender.

As part of‌ the PhD project of Chaoping Zhu (co-advised by‌ Matías Bender and Elias Tsigaridas, IMJ-PRG, Paris), we‌ studied algorithms to compute certificates of non-negativity for‌ polynomials. These certificates are algebraic identities that allow‌ us to prove that a polynomial is non-negative‌ over the reals, and arise, for example, in‌ unconstrained optimization. In the univariate setting, this problem‌ can be solved using sum of squares decomposition.‌ Together with P. di Dio from U. Konstanz,‌ Germany, in 32, we improve the work‌ of Chevillard, Harrison, Joldes, and Lauter to obtain‌ a cubic-complexity algorithm to compute such a sum‌ of squares decomposition. Moreover, we relate the sum‌ of squares decomposition with other certificates, such as‌ the ones studied by Karlin. We also consider‌ the multivariate case. In this setting, as shown‌ by Artin, we can perform our certification by‌ writing the polynomial as a sum of rational‌ functions, but the best known bounds for these‌ polynomials, proved by Lombardi, Perrucci, and Roy, involve‌ a tower of five exponentials. Following previous work‌ by Parrillo, Nie, Demmel, and Sturmfels, an alternative‌ certificate exists by writing the polynomial as a‌ sum of squares modulo its gradient ideal. In‌ 34, we follow this approach and present‌ a single-exponential algorithm that can certify the non-negativity‌ of polynomials that satisfy two assumptions: their minimum‌ is attained, and their gradient variety is zero-dimensional.‌ In a subsequent paper with K. Kozhasov (Université‌ Côte d'Azur) 33, we extend these ideas‌ to arbitrary polynomials by using perturbations to reduce‌ our certification to the previous special case. In‌ this way, we obtain the first single-exponential bounds‌ to solve this problem.

8.7 Applications

8.7.1 Performance‌ evaluation of emergency call centers and emergency services‌

Participants: Xavier Allamigeon, Pascal Capetillo, Stéphane‌ Gaubert, Guillaume Thomas.

Since 2014, we‌ have been collaborating with Préfecture de Police (Régis‌ Reboul and LcL Stéphane Raclot), more specifically with‌ Brigade de Sapeurs de Pompiers de Paris (BSPP)‌ and Direction de Sécurité de Proximité de l'agglomération‌ parisienne (DSPAP), on the performance evaluation of the‌ new organization (PFAU, “Plate forme d'appels d'urgence”) to‌ handle emergency calls to firemen and policemen in‌ the Paris area. We developed analytical models, based‌ on Petri nets with priorities, and fluid limits,‌ see 53, 54, 72. In‌ 2019, with four students of École polytechnique, Céline‌ Moucer, Julia Escribe, Skandère Sahli and Alban Zammit, we performed case studies,‌ showing the improvement brought‌ by the two level‌‌ filtering procedure.

Moreover, in 2019, this work has‌ been extended to encompass‌ the handling of health‌‌ emergency calls, with a new collaboration, involving responsibles‌ from the four services‌ of medical emergency aid‌‌ of Assistance Publique – Hôpitaux de Paris (AP-HP),‌ i.e., with SAMU75, 92,‌ 93, 94, in the‌‌ framework of a project coordinated by Dr. Christophe‌ Leroy from AP-HP. As‌ part of his PhD‌‌ work, Marin Boyet have developed Petri net models‌ capturing the characteristic of‌ the centers (CRRA) handling‌‌ emergency calls the SAMU, in order to make‌ dimensioning recommendations. Following this,‌ we have been strongly‌‌ solicited by AP-HP during the pandemic of Covid-19‌ in order to determine‌ crisis dimensioning of SAMU.‌‌

In parallel, we have further investigated the theoretical‌ properties of timed Petri‌ nets with preselection and‌‌ priority routing. We represent the behavior of these‌ systems by piecewise affine‌ dynamical systems. We use‌‌ tools from the theory of nonexpansive mappings to‌ analyze these systems. We‌ establish an equivalence theorem‌‌ between priority-free fluid timed Petri nets and semi-Markov‌ decision processes, from which‌ we derive the convergence‌‌ to a periodic regime and the polynomial-time computability‌ of the throughput. More‌ generally, we develop an‌‌ approach inspired by tropical geometry, characterizing the congestion‌ phases as the cells‌ of a polyhedral complex.‌‌ These results are illustrated by the application to‌ the performance evaluation of‌ emergency call centers of‌‌ SAMU in the Paris area. These results have‌ been published in 56‌.

In 55,‌‌ we provided explicit formulæ allowing one to compute‌ the time needed by‌ a call center to‌‌ return to a stationary state after a bulk‌ of calls. This is‌ based on a turnpike-type‌‌ theorem for Markov decision processes.

These results were‌ also presented in the‌ Phd thesis 74.‌‌

In the followup work described below (Section 8.7.2‌), in the framework‌ of the URGE project‌‌ (see Section 10.3.3 below) we are extending these‌ models in order to‌ evaluate the dimensioning of‌‌ emergency departments. This is the object of the‌ PhD work of Pascal‌ Capetillo, which started in‌‌ Nov. 2023.

8.7.2 Stationary regimes of piecewise linear‌ dynamical systems with priorities‌

Participants: Xavier Allamigeon,‌‌ Pascal Capetillo, Stéphane Gaubert.

A key‌ question in the study‌ of Petri nets with‌‌ priority is the existence of stationary regimes. (See‌ section above for motivations.)‌ The counter equations of‌‌ these Petri nets, modelling the evolution of events,‌ follow piecewise linear equations‌ that are similar to‌‌ the dynamic programming equations of semi-Markov decision processes,‌ up to a key‌ new feature: the “probabilities'‌‌ can take negative values – which entails that‌ the dynamics is no-longer‌ monotone. Although stationary regimes‌‌ were found (by a case by case analysis)‌ for a number of‌ concrete models, including emergency‌‌ call centers, it has been an open problem‌ to show that these‌ regimes do exist under‌‌ general circumstances. Using techniques‌ of topological degree theory, we show in 59‌ that stationary regimes do exist, independently of the‌ choice of resources, for a broad enough class‌ of systems (including all known examples). This result‌ also extends Kohlberg's theorem (existence of invariant half-lines‌ for nonexpansive piecewise affine maps), relaxing the nonexpansiveness‌ assumption. In the followup work 25, we‌ showed how to enumerate the stationary regimes, in‌ order to compute a phase diagram, showing the‌ different bottleneck regimes, depending on the allocation of‌ resources. SG: updated

8.7.3 Optimal pricing of energy‌ contracts

Participants: Abdellah Bulaich, Stéphane Gaubert.‌

This work on optimal pricing of electricity contracts‌ is carried out within the “Defi EDF Inria”,‌ with a collaboration between Wim van Ackooij, Luce‌ Brotcorne (Inria Lille) and S. Gaubert, leading to‌ the hiring in Cifre PhD of Abdellah Bulaich-Mehamdi.‌ We investigated the problem of design of a‌ menu of contracts of a prescribed cardinality, in‌ particular, we related its formulation in terms of‌ the best approximation of a convex function by‌ a polyhedral function with a prescribed number of‌ facets to the problem of optimal quantization of‌ probability measures, using methods from optimal transport 29‌

8.7.4 Aggregated models of flexibilities in long term‌ investment models

Participants: Stéphane Gaubert.

Within the‌ framework of the “Defi EDF Inria”, we started‌ a joint project with the Polaris team of‌ Inria Rhône-Alpes (Bruno Gaujal and Nicolas Gast) and‌ with Olivier Beaude and Juliette Lesturgie, from EDF‌ Labs, on the development of aggregated models of‌ flexibilities of customers in the electricity field. Examples‌ of flexibilities are given by parks of electric‌ vehicles, or hydroelectric resources. These aggregated models are‌ used to optimize long term investments. This is‌ the object of the PhD thesis of Hélène‌ Arvis, jointly supervised with Polaris. We showed in‌ 31 that flexibilities of electric vehicles can be‌ aggregated exactly using generalized polymatroids, allowing us to‌ use submodular optimization techniques to optimize the production‌ of energy, leveraging flexibilities.

8.7.5 Simulation and optimization‌ of robust strategies for spectrum auctions

Participants: Marianne‌ Akian, Stéphane Gaubert, Jad Zeroual.‌

In order to deploy new 5G mobile networks,‌ telecommunication operators must acquire new frequency bands (called‌ spectrum). The acquisition of these frequencies is done,‌ in each country, through participation in auctions whose‌ rules are specific and dictated by the licences’‌ owners (usually states). The aim of the PhD‌ thesis of Jad Zeroual, supervised by Marianne Akian,‌ Stéphane Gaubert, Aurélien Bechler (Orange Labs) and Mathieu‌ Chardy (Orange Labs), is to develop mathematical models‌ allowing one to facilitate the understanding and participation‌ in licensed spectrum auctions, and in particular to‌ compute optimal acquisition strategies of the desired amount‌ of spectrum.

In a first work 30,‌ we studied a model of auction representative of‌ the 5G auction in France. We determine the‌ optimal strategy of a bidder, assuming that the‌ valuations of competitors are unknown to this bidder and that competitors adopt‌ the straightforward bidding strategy.‌ The model is based‌‌ on a Partially Observable Markov Decision Process (POMDP).‌ We show, under some‌ assumptions on the probability‌‌ distributions of the valuations of the competitors, that‌ this POMDP admits a‌ concise statistics, avoiding the‌‌ solution of a dynamic programming equation in the‌ space of beliefs. The‌ results are also illustrated‌‌ by numerical experiments, comparing the value of the‌ bidder with the value‌ of a perfectly informed‌‌ one. The paper 30 obtained the best paper‌ award at the conference‌ NETGCOOP'2024, an extended version‌‌ is now published in 23.

In a‌ followup work 36,‌ we studied multidimensional simple‌‌ clock options. When the opponents' valuations satisfy the‌ ordinary substitutes condition, we‌ show that it is‌‌ optimal to bid on a fixed lot overtime.‌ In this setting, we‌ consider a continuous-time version‌‌ of the SCA auction in which the prices‌ follow a differential inclusion‌ with a piecewise-constant dynamics.‌‌ We show that there exists a unique solution‌ in the sense of‌ Filippov. This guarantees that‌‌ the continuous-time model coincides with the limit of‌ the discrete-time auction when‌ price increments tend to‌‌ zero. Moreover, we showed that the value function‌ of this limit auction‌ is piecewise linear (though‌‌ possibly discontinuous). We illustrated these results by analyzing‌ a simplified version of‌ the multiband Australian spectrum‌‌ auction of 2017.

In an ongoing work in‌ collaboration with Guido Schäfer‌ (CWI), we are studying‌‌ the consistency and robustness of strategies in Simple‌ Clock Auctions, tooking inspiration‌ from 92, while‌‌ adapting the model to fit spectrum auctions. This‌ work aims to provide‌ guarantees on the robustness‌‌ of strategies such as the Straightforward Bidding and‌ to device robust strategies‌ with the help of‌‌ a black-box prediction tool such as experts’ advice‌ or Machine Learning predictions.‌

9 Bilateral contracts and‌‌ grants with industry

9.1 Bilateral contracts with industry‌

Participants: Marianne Akian,‌ Stéphane Gaubert, Abdellah‌‌ Bulaich Mehamdi, Jad Zeroual.

Simulation and‌ optimization of robust strategies‌ for spectrum auctions.‌‌ Collaboration of Marianne Akian and Stéphane Gaubert with‌ Aurélien Bechler and Mathieu‌ Chardy from Orange Labs,‌‌ for the supervision of the Phd work of‌ Jad Zeroual (CIFRE PhD‌ from January 9, 2024,‌‌ directed by Marianne Akian).
Optimal pricing of energy‌ contracts. Collaboration of‌ Stéphane Gaubert with Luce‌‌ Brotcorne (Inria Lille, INOCS team), Wim Van Ackooij‌ and Quentin Jacquet, from‌ EDF Labs. This is‌‌ part of the “DEFI EDF-inria”. This supports the‌ CIFRE Phd Thesis of‌ Adbellah Bulaich-Mehamdi.
Aggregated models‌‌ of flexibilities in long term investment models.‌ Collaboration of Stéphane Gaubert‌ with Bruno Gaujal and‌‌ Nicolas Gast (Inria Rhône-Alpes, Polaris team) and Olivier‌ Beaude and Juliette Lesturgie,‌ from EDF Labs. This‌‌ is part of the “DEFI EDF-inria”. This supports‌ the CIFRE Phd Thesis‌ of Hélène Arvis.

10‌‌ Partnerships and cooperations

10.1 International initiatives

10.1.1 Inria‌ associate team not involved‌ in an IIL or‌‌ an international program

TropOptim‌

Title:
Tropical Algorithms for Optimization
Duration:
2025 ->‌ 2028
Coordinator:
Daniel Dadush (dadush@cwi.nl)
Partners:
- CWI Amsterdam‌ (Pays-Bas)
Inria contact:
Xavier Allamigeon
Summary:

The objective‌ of the project is to make progress on‌ several problems in optimization and game theory (complexity‌ of linear programming, semidefinite programming, mean payoff games,‌ auction theory and mechanism design) by exploiting their‌ connection with tropical geometry. To this extent, the‌ associate team federates the "Network & Optimization" team‌ from CWI which has an extensive expertise in‌ combinatorial and strongly polynomial algorithms for linear optimization,‌ as well as in mechanism design, and the‌ Tropical team from Inria, which has pioneered the‌ application of tropical geometry to the complexity of‌ convex optimization problems, mean payoff games, bilevel programming,‌ and combinatorial auctions. The associate team will facilitate‌ the joint works between the two teams by‌ funding the organization of joint meetings and visits‌ by permanent members and PhD students.

A kickoff‌ workshop of the team took place at CWI‌ in July 2025, followed by a visit of‌ Jad Zeroual to CWI at the fall, initiating‌ a collaboration with Guido Schäfer on auction theory.‌

10.1.2 Participation in other International Programs

Since 2025,‌ Matias Bender has been part of a two-year‌ Franco-German visiting grant (Procope–Partenariat Hubert Curien, PHC) between‌ Sorbonne University (PI: Elias Tsigaridas) and Goethe University‌ Frankfurt, Germany (PI: Thorsten Theobald).

10.2 International research‌ visitors

10.2.1 Visits of international scientists

Other international‌ visits to the team

Gleb Koshevoy

Status
Chief‌ researcher
Institution of origin:
The Institute for Information‌ Transmission Problems, Russian Academy of Sciences (IITP RAS).‌
Country:
Russia.
Dates:
February–April 2025
Context of the‌ visit:
Collaboration on hyperconvexity and algebraic combinatorics.

10.2.2‌ Visits to international teams

Research stays abroad

Gaubert‌ Stéphane

Visited institution:
University of Chicago
Country:
USA‌
Dates:
July 3 – 7, 2025
Context of‌ the visit:
FACCTS - France Chicago center project‌ on tropical geometry applied to machine learning. Collaboration‌ with Shmuel Friedland (University of Illinois at Chicago)‌ and Lek-Heng Lim (University of Chicago).

10.3 National‌ initiatives

10.3.1 ANR

Participants: Marianne Akian, Xavier‌ Allamigeon, Cormac Walsh, Stéphane Gaubert.‌

ANR project HilbertXField (“Géométries de Hilbert sur tout‌ corps valué”). ANR leader: Antonin Guilloux. Partners: IMJ-PRG/OURAGAN‌ (Sorbonne Université, pole leader Antonin Guilloux), CMAP/TROPICAL (Inria,‌ pole leader: Cormac Walsh), Institut Fourier (Grenoble, pole‌ leader Anne Parreau).

Participants: Marianne Akian, Xavier‌ Allamigeon, Stéphane Gaubert.

ANR project ZADyG‌: Zoology of Algorithmic Methods for Dynamic Games‌ (ANR-25-CE48-7058, 4 years from January 2026). This is‌ a collaborative research project (PRC), the coordinator is‌ Marianne Akian, and the poles/partners are LABRI (Université‌ de Bordeaux) with pole leader Nathanaël Fijalkow, and‌ LACL (Université Paris-Est Créteil) with pole leader Youssouf‌ Oualhadj.

The aim of the project is to‌ develop a unified vision of approaches on complexity‌ and algorithmic issues of stochastic games. The ANR‌ funding includes one PhD thesis (3 years) for‌ the Inria Saclay team, and 18 months of Post-doc for each of‌ the other partners. It‌ also includes the organization‌‌ of joint meetings together with an open conference.‌

Participants: Matías Bender.‌

ANR project PeACE:‌‌ Polynomials and applications via efficient computations (ANR-25-CE48-3760; 226,000‌ euros). This project is‌ funded by ANR as‌‌ part of the program JCJC (young researchers). Its‌ PI is Matias Bender‌ and the project encompass‌‌ collaborations with researchers in Denmark, France, UK, and‌ US, on topics as‌ sparse and structured polynomial‌‌ systems and applications to topological data analysis.

10.3.2‌ Programme Gaspard Monge pour‌ l'optimisation, la recherche opérationnelle‌‌ et leurs interactions avec les sciences des données‌

Participants: Matías Bender.‌

Since 2023, SOAP -‌‌ Sparsity in Optimization via Algebra and Polynomials, with‌ Elias Tsigaridas (IMJ), project‌ funded by FMJH within‌‌ the PGMO programme.

10.3.3 Joint INRIA & AP-HP‌ Bernoulli lab project: “URGE”‌

The project URGE (Analyse‌‌ des parcours patients aux URgences et optimisation des‌ prises en charGE), started‌ at the fall 2022,‌‌ in the framework of the joint Inria &‌ AP-HP Bernoulli lab. The‌ goal of the project‌‌ is to develop modelling, simulation, performance analysis, and‌ visualization tools, in order‌ to help physicians to‌‌ optimize the staffing of emergency services. This collaborative‌ project, of four years,‌ involves the Tropical and‌‌ Aviz teams from INRIA Saclay, the Dyogene team‌ from Inria Paris, and‌ the Fédération Hospitalo-Universitaire (FHU)‌‌ / IMPEC Improving Emergency Care, AP-HP / Sorbonne‌ Université / Inserm. The‌ project is led by‌‌ X. Allamigeon (Tropical) and Y. Yordanov (AP-HP, Saint-Antoine)‌ and involves P. Capetillo,‌ S. Gaubert (Tropical), Ch.‌‌ Fricker (Dyogene), J.D. Fekete (Aviz).

11 Dissemination

11.1‌ Promoting scientific activities

11.1.1‌ Scientific events: organisation

General‌‌ chair, scientific chair

Stéphane Gaubert is the adjunct‌ coordinator of the Gaspard‌ Monge Program for Optimization,‌‌ Operations Research and their interactions with data sciences‌ (PGMO), a corporate sponsorhip‌ program, operated by Fondation‌‌ Mathématique Jacques Hadamard, supported by EDF, see PGMO‌ site.

Member of‌ the organizing committees

Xavier‌‌ Allamigeon co-organizes the colloquium of CMAP at Ecole‌ polytechnique.
Marianne Akian co-organized,‌ with Donato Vasquez-Varas (RICAM)‌‌ an invited session on “Hamilton-Jacobi equations for optimal‌ control and games: new‌ trends in numerical and‌‌ analytical aspects”, at the conference VC 2025,‌ Vienna (July 2025).

Matias‌ Bender was part of‌‌ the organization committee of the conference ISSAC 2025‌ as the publicity chair.‌

Stéphane Gaubert co-organizes the‌‌ “Séminaire Parisien d'Optimisation” at Institut Henri Poincaré.

Stéphane‌ Gaubert co-organized, with Guillaume‌ Vigeral (Paris 1), an‌‌ invited session on “Algebraic and Operator methods for‌ Zero-Sum Games”, at the‌ conference VC 2025,‌‌ Vienna.

11.1.2 Scientific events: selection

Chair of conference‌ program committees

Stéphane Gaubert‌ , co-chair of the‌‌ PGMO Days 2025, EDF Labs, November, 2025.

Stéphane‌ Gaubert co-organized, with Quentin‌ Mérigot (LMO, Orsay), the‌‌ PGMO lectures (advanced lectures for a research audience)‌ of Daniel Dadush (January‌ 2025) and Peter Buergisser‌‌ (April 2025), see the PGMO lectures web site‌.

Member of the‌ conference program committees

Stéphane‌‌ Gaubert was member of‌ the Scientific Committee of the conference VC 2025‌, Vienna.

11.1.3 Journal

Member of the editorial‌ boards

Stéphane Gaubert : member of the editorial‌ board of Journal of Dynamics and Games, Linear‌ and Multilinear Algebra, RAIRO, Springer-SMAI book series.

11.1.4‌ Invited talks

Marianne Akian :
- Talk at the‌ special session on “Real and Positive Tropical Geometry”‌ at the conference SIAM Applied Algebraic Geometry (AG25)‌, Madison, Wisconsin, July 2025, on “Solving sparse‌ tropical polynomial systems by means of parametric mean‌ payoff games” (joint work with Antoine Berau et‌ Stéphane Gaubert).
- Talk at the “Première rencontre nationale‌ du RT Optimisation”, on ”The competitive spectral‌ radius of families of nonexpansive mappings” (joint work‌ with Stéphane Gaubert, Loïc Marchesini, and Ian Morris).‌
Xavier Allamigeon :
- Plenary talk at the conference‌ ROADEF 2025 (ENPC, Champs-sur-Marne), February 2025, on “Simplex-type‌ complexity bounded for interior point methods”.
- Talk at‌ Journées Franco-Chiliennes en Optimisation (INSA, Rouen), July 2025,‌ on “Computing stationary solutions of piecewise linear systems‌ with priorities: application to emergency departments” (joint work‌ with P. Capetillo and S. Gaubert).
Abdellah Bulaich‌ Mehamdi
- Talk at VC 2025, Vienna (invited‌ session on "Hamilton-Jacobi equations for optimal control and‌ games: new trends in numerical and analytical aspects"),‌ title: “Duality between polyhedral approximation of value functions‌ and optimal quantization of measures” (joint work with‌ W. van Ackooij, L. Brotcorne, S. Gaubert, Q.‌ Jacquet).
Pascal Capetillo :
- Talk at the special‌ session on Computational Tropical geometry, at the conférence‌ SIAM Applied Algebraic Geometry (AG25), Madison, Wisconsin,‌ July 2025, on “Stationary Regimes of Piecewise Linear‌ Dynamical Systems with Priorities (joint work with Xavier‌ Allamigeon and Stéphane Gaubert).
Stéphane Gaubert :
- Talk‌ at the special session on Computational Tropical geometry,‌ at the conférence SIAM Applied Algebraic Geometry (AG25)‌, Madison, Wisconsin, July 2025, on “Tropical SVMs‌ via Mean-Payoff Games” (joint work with Xavier Allamigeon,‌ Samuel Boïté and Théo Molfessis).
Loic Marchesini
- Talk‌ at VC 2025, Vienna (invited session on“Algebraic‌ and Operator methods for Zero-Sum Games”, title: “The‌ competitive spectral radius of families of nonexpansive mappings”‌ (joint work with Marianne Akian and Stéphane Gaubert).‌

11.1.5 Additional conferences talks, Seminars

[inline]MA: cette section‌ n'est pas suggeree, on la met toujours mais‌ un peu plus loin, dans une sous-section donc‌ c'est un peu trop, je propose de la‌ mettre ici, il faudrait rajouter tous les exposes‌ de conferences au moins ...., j'ai deja rajoute‌ ceux de Jad et Loic. J'ai aussi rajoute‌ tous les exposes a VC2025: ceux de stephane‌ et moi-meme dans cette section et ceux de‌ Loic et Abdellah dans la section precedente.

Marianne‌ Akian
- Talk at VC 2025, Vienna, title:“Multi-level‌ fast-marching method and semi-Lagrangian scheme for the minimum‌ time problem” (joint work with Shanqing Liu and‌ Stéphane Gaubert).
Pascal Capetillo
- Talk at HSCC 2025‌, Irvine, California, USA, May 2025. Title: “Stationary‌ regimes of piecewise linear dynamical systems with priorities”‌ (joint work with X. Allamigeon and S. Gaubert).
- Talk at QEST+FORMATS 2025‌, Aarhus, Denmark, August‌ 2025. Title: “Computing the‌‌ Congestion Phases of Dynamical Systems with Priorities and‌ Application to Emergency Departments”‌ (joint work with X.‌‌ Allamigeon and S. Gaubert).
Stéphane Gaubert
- Talk VC‌ 2025, Vienna, title:‌ "Extending Markov decision processes:‌‌ negative probabilities and priority dynamics” (joint work with‌ Xavier Allamigeon and Pascal‌ Capetillo).
Loic Marchesini
- Talk‌‌ at SMAI 2025 (Carcans, June 2025), title “The‌ competitive spectral radius of‌ families of nonexpansive mappings”‌‌ (joint work with Marianne Akian and Stéphane Gaubert).‌
- Talk at CDC 2025‌ (Rio de Janeiro, Dec‌‌ 2025) , see 24.
Jad Zeroual
- Talk‌ at the kick off‌ of the Inria associate‌‌ team “Tropical Algorithms for Optimization” with CWI (Amsterdam,‌ July 2025), title: “Optimal‌ bidding in Clock-Plus auctions‌‌ against Straightforward Bidders” (joint work with Marianne Akian,‌ Aurélien Béchler, Matthieu Chardy,‌ Stéphane Gaubert).
- Poster at‌‌ the conference “30 ans du séminaire de théorie‌ des jeux” (IHP, Oct‌ 2025), title: “Oracle strategy‌‌ in Clock-Plus Auctions: Playing with perfect information against‌ Straightforward bidders” (joint work‌ with Marianne Akian, Aurélien‌‌ Béchler, Matthieu Chardy, Stéphane Gaubert).

11.1.6 Research administration‌

Inria research administration

Marianne‌ Akian : Elected member‌‌ of Inria's Scientific Board.
Xavier Allamigeon : Elected‌ member of the Committee‌ of the Applied Mathematics‌‌ Department of Ecole polytechnique
Matìas Bender : Member‌ of the scientific committee‌ of INRIA Saclay.

Other‌‌ research administration

Marianne Akian : “Secrétaire Générale” of‌ SMAI.
Stéphane Gaubert :‌ Member of the Scientific‌‌ council of École Nationale des Ponts et Chaussées.‌

11.2 Teaching - Supervision‌ - Juries - Educational‌‌ and pedagogical outreach

11.2.1 Teaching

[inline]MA: la section‌ n'existait pas et les‌ cours etaient dans la‌‌ section “educational...” mais l'enseignement ce n'est pas de‌ la sensibilisation a l'education‌

Marianne Akian
- Course “Markov‌‌ decision processes: dynamic programming and applications” joint between‌ (3rd year of) ENSTA‌ and M2 “Mathématiques et‌‌ Applications”, U. Paris Saclay, “Optimization”, 30 hours.
Xavier‌ Allamigeon
- Petites classes et‌ encadrement d'enseignements d'approfondissement de‌‌ Recherche Opérationnelle en troisième année à l'École Polytechnique‌ (programme d'approfondissement de Mathématiques‌ Appliquées) (niveau M1).
- Cours‌‌ “Theoretical Aspects of Linear Programming” du M2 “Optimisation”‌ de l'Université Paris Saclay.‌
- Cours “Jeux stochastiques” au‌‌ Master Parisien de Recherche en Informatique.
Matias Bender‌
- Part-time chargé d’enseignement (equivalent‌ to a lecturer, 76‌‌ hours per year) in the computer science department‌ of École Polytechnique, where‌ he teaches algorithmics courses‌‌ for the bachelor program in mathematics and computer‌ science.
Amanda Bigel
- Exercises‌ classes for the first‌‌ year of Bachelor program of Ecole polytechnique in‌ the framework of a‌ “Monitorat”.
Pascal Capetillo
- Exercises‌‌ classes for the first year of Bachelor program‌ of Ecole polytechnique in‌ the framework of a‌‌ “Monitorat”.
Yiyuan Chen
- Exercises classes for the first‌ year of Bachelor program‌ of Ecole polytechnique in‌‌ the framework of a “Monitorat”.
Stéphane Gaubert
- Co-head‌ of the Master “Optimization”‌ of University Paris-Saclay and‌‌ IPP.
- Head for IP Paris of the “Mention”‌ (Master Programme) “Mathématiques et‌ Applications” jointly operated by‌‌ University Paris Saclay and‌ IP Paris.
- Course “Systèmes à Événements Discrets”, option‌ MAREVA, ENSMP.
- Course “Algèbre tropicale pour le contrôle‌ optimal et les jeux” of “Contrôle, Optimisation et‌ Calcul des Variations” (COCV) of M2 “Mathématiques et‌ Applications” of Sorbonne University and École Polytechnique.
- Lecture‌ of Operations Research, third year of École Polytechnique.‌ The lectures notes were published as a book‌ 73.

11.2.2 Supervision

Phd in progress: Chaoping‌ Zhu, codirected by Matías Bender and Elias Tsigaridas‌ (Inria Paris, IMJ-PRG, Sorbonne Université).
PhD in progress:‌ Nicolás Alló Gómez, based in Argentina, codirected by‌ Matías Bender and Teresa Krick (University of Buenos‌ Aires, Argentina).
PhD defended: Quentin Canu, registered at‌ Univ. Paris Saclay (defense in December 2025), thesis‌ supervisor: Georges Gonthier (INRIA), cosupervision: Xavier Allamigeon and‌ Pierre-Yves Strub (LIX).
PhD in progress: Amanda Bigel,‌ registered at IPP (EDMH) since September 2022, main‌ thesis supervisors: Cormac Walsh et Constantin Vernicos, thesis‌ supervisor: Stéphane Gaubert .
PhD in progress: Loïc‌ Marchesini, registered at IPP (EDMH), since September 2023.‌ Thesis supervisor Marianne Akian , co-supervised by Stéphane‌ Gaubert .
PhD in progress: Pascal Capetillo, registered‌ at IPP (EDMH), since November 2023. Thesis supervisor:‌ Stéphane Gaubert , cosupervision: Xavier Allamigeon .
PhD‌ in progress: Jonathan Hornewall, registered at Marne-la-vallée, ENPC‌ (maths and STIC), since November 2023. Thesis supervisor:‌ Vincent Leclere, cosupervision: Stéphane Gaubert .
PhD in‌ progress: Jad Zeroual, registered at IPP (EDMH), since‌ January 2024. Thesis supervisor Marianne Akian , co-supervised‌ by Stéphane Gaubert , Aurélien Bechler (Orange Labs)‌ and Mathieu Chardy (Orange Labs).
PhD in progress:‌ Yiyuan Chen, registered at IPP (EDMH), since September‌ 2024. Thesis supervisor: Stéphane Gaubert , cosupervision: Xavier‌ Allamigeon .
PhD in progress: Abdellah Bulaich, registered‌ at IPP (EDMH), since January 2025. Thesis supervisor:‌ Stéphane Gaubert , cosupervision: Luce Brotcorne (Inria Lille),‌ Wim van Ackooij (EDF), Quentin Jacquet (EDF).
PhD‌ in progress: Hélène Arvis, registered at ED “Mathématiques,‌ Sciences et technologies de l'information, Informatique” in Grenoble,‌ since November 2024. Thesis supervisor: Nicolas Gast (Inria‌ Rhône-Alpes), cosupervision: Stéphane Gaubert , Bruno Gaujal (Inria‌ Rhône-Alpes), Olivier Beaude (EDF), Juliette Lesturgie (EDF).
Matías‌ Bender co-directed two 3-month 2A engineering internships with‌ G. Artana (Engineering Department, U Buenos Aires) and‌ C. Artana (Oceanography Department, Sorbonne U). These internships‌ were part of an exchange program between the‌ U Buenos Aires, Argentina, and SeaTech, Universite ́‌ de Toulon. This was part of an interdisciplinary‌ proof-of-concept project to study eddies using topological data‌ analysis.

11.2.3 Juries

Marianne Akian
- Member of the‌ hiring committee for a junior professor in Optimization‌ at University Paris 1.
- Member of the hiring‌ committee for a junior professor in Optimization or‌ games at University Paris Dauphine PSL.
- Jury Chair‌ of the PhD thesis of Mohamed Gharafi, Ecole‌ polytechnique, Octobre 2025.
- Jury Chair of the PhD‌ thesis of Jules Berry, INSA de Rennes, Octobre‌ 2025.
- Jury chair of the PhD thesis of‌ Orso Forghieri, Ecole polytechnique, décembre 2025.
Stéphane Gaubert‌
- Jury (reviewer) of the PhD thesis of Davide Zorzenon, Fakultät IV -‌ Electrotechnik und Informatik, TU-Berlin,‌ June 2025.
- Jury (reviewer)‌‌ of the PhD thesis of Alaa Ibhrahim, ENS‌ lyon, November 2025.

11.2.4‌ Educational and pedagogical outreach‌‌

[inline]MA: j'ai mis ca ici, sinon on peut‌ mettre dans la section‌ popularization

Marianne Akian took‌‌ part in the program Elles font le numérique‌, an outreach program‌ that highlights the careers‌‌ and achievements of women scientists whose research is‌ in digital sciences, see‌ the article Marianne Akian:‌‌ From “New Maths” to the Search for New‌ Models on the Inria‌ web site.

11.3 Popularization‌‌

11.3.1 Others science outreach relevant activities

Matias Bender‌ was part of the‌ 2025 outreach campaign of‌‌ Inria Saclay, where his research was featured in‌ a short communication poster‌ presented at events such‌‌ as the Science Festival in the southern Paris‌ area.

12 Scientific production‌

12.1 Major publications

1‌‌ articleM.Marianne Akian, S.Stéphane Gaubert‌ and R.Ravindra Bapat‌. Non-archimedean valuations of‌‌ eigenvalues of matrix polynomials.Linear Algebra and‌ its Applications498Also‌ arXiv:1601.00438June 2016,‌‌ 592--627HAL DOI
2 articleM.M. Akian‌, S.S. Gaubert‌ and A.A. Guterman‌‌. Tropical polyhedra are equivalent to mean payoff‌ games.Internat. J.‌ Algebra Comput.221‌‌2012, 1250001, 43URL: http://dx.doi.org/10.1142/S0218196711006674DOI back‌ to text back to‌ text
3 articleM.‌‌Marianne Akian, S.Stephane Gaubert and R.‌Roger Nussbaum. Uniqueness‌ of the fixed point‌‌ of nonexpansive semidifferentiable maps.Transactions of the‌ American Mathematical Society368‌2Also arXiv:1201.1536February‌‌ 2016HAL DOI back to text back to‌ text back to text‌
4 articleM.Marianne‌‌ Akian, S.Stéphane Gaubert and C.Cormac‌ Walsh. The max-plus‌ Martin boundary.Doc.‌‌ Math.142009, 195--240back to text‌
5 articleX.X.‌ Allamigeon, P.P.‌‌ Benchimol, S.S. Gaubert and M.M.‌ Joswig. Combinatorial simplex‌ algorithms can solve mean‌‌ payoff games.SIAM J. Opt.244‌2015, 2096--2117
6‌ articleX.Xavier Allamigeon‌‌, P.Pascal Benchimol, S.Stéphane Gaubert‌ and M.Michael Joswig‌. What Tropical Geometry‌‌ Tells Us about the Complexity of Linear Programming‌.SIAM Review63‌1February 2021,‌‌ 123-164HAL DOI back to text back to‌ text back to text‌
7 inproceedingsX.X.‌‌ Allamigeon, S.S. Gaubert, E.E.‌ Goubault, S.S.‌ Putot and N.N.‌‌ Stott. A scalable algebraic method to infer‌ quadratic invariants of switched‌ systems.Proceedings of‌‌ the International Conference on Embedded Software (EMSOFT)Best‌ paper award. The extended‌ version of this conference‌‌ article appeared in \em ACM Trans. Embed. Comput.‌ Syst., 15(4):69:1--69:20, September 2016‌2015back to text‌‌back to text
8 articleJ.J. Bolte‌, S.S. Gaubert‌ and G.G. Vigeral‌‌. Definable zero-sum stochastic games.Mathematics of‌ Operations Research401‌Also arxiv1301.19672014,‌‌ 171--191URL: http://dx.doi.org/10.1287/moor.2014.0666DOI‌back to text
9 articleS.S. Friedland‌, S.Stéphane Gaubert and L.L. Han‌. Perron--Frobenius theorem for nonnegative multilinear forms and‌ extensions.Linear Algebra and its Applications438‌22013, 738-749HAL DOI
10 article‌S.S. Gaubert and T.Th. Lepoutre.‌ Discrete limit and monotonicity properties of the Floquet‌ eigenvalue in an age structured cell division cycle‌ model.J. Math. Biol.2015, URL:‌ http://dx.doi.org/10.1007/s00285-015-0874-3DOI back to text back to text‌
11 articleS.S. Gaubert and G.G.‌ Vigeral. A maximin characterization of the escape‌ rate of nonexpansive mappings in metrically convex spaces‌.Math. Proc. of Cambridge Phil. Soc.152‌https://arxiv.org/abs/1012.47652012, 341--363URL: http://dx.doi.org/10.1017/S0305004111000673back to‌ text
12 incollectionC.C. Walsh. The‌ horofunction boundary and isometry group of the Hilbert‌ geometry.Handbook of Hilbert Geometry22IRMA‌ Lectures in Mathematics and Theoretical PhysicsEuropean Mathematical‌ Society2014HAL back to text

12.2 Publications‌ of the year

International journals

13 articleM.‌Marianne Akian, X.Xavier Allamigeon, S.‌Stéphane Gaubert and S.Sergei Sergeev. Signed‌ tropicalization of polar cones.Journal of Optimization‌ Theory and Applications20710July 2025HAL‌DOI back to text
14 articleM.Marianne‌ Akian, A.Antoine Béreau and S.Stéphane‌ Gaubert. The Nullstellensatz and Positivstellensatz for Sparse‌ Tropical Polynomial Systems.Foundations of Computational Mathematics‌December 2025HAL DOIback to text back‌ to text
15 articleM.Marianne Akian,‌ J.-P.Jean-Philippe Chancelier and B.Benoît Tran.‌ A stochastic algorithm for deterministic multistage optimization problems‌.Annals of Operations Research345January 2025‌, 1-38HAL DOIback to text
16‌ articleM.Marianne Akian, S.Stephane Gaubert‌ and H.Hanieh Tavakolipour. Factorization of polynomials‌ over the symmetrized tropical semiring and Descartes' rule‌ of sign over ordered valued fields.Journal‌ of Pure and Applied Algebra2299September‌ 2025, 51HALDOI back to text‌
17 articleX.Xavier Allamigeon, D.Daniel‌ Dadush, G.Georg Loho, B.Bento‌ Natura and L.László Végh. Interior Point‌ Methods Are Not Worse than Simplex.SIAM‌ Journal on Computing545March 2025,‌ FOCS22-178-FOCS22-264HAL DOI back to text
18 article‌X.Xavier Allamigeon, S.Stéphane Gaubert,‌ R.Ricardo Katz and M.Mateusz Skomra.‌ Universal complexity bounds based on value iteration for‌ stochastic mean payoff games and entropy games.‌Information and Computation302January 2025, 105236‌HAL DOI
19 articleP.-C.Pierre-Cyril Aubin-Frankowski and‌ S.Stéphane Gaubert. Order isomorphisms of sup-stable‌ function spaces: Continuous, Lipschitz, c-convex, and beyond.‌Communications in Contemporary MathematicsAugust 2025HAL DOI‌
20 articleM. R.Matías R Bender,‌ L.Laurent Busé, C.Carles Checa and‌ E.Elias Tsigaridas. Bigraded Castelnuovo-Mumford regularity and‌ Groebner bases.Journal of Symbolic Computation133March 2026, 26‌HAL DOI back to‌ text
21 articleD.‌‌Dmitry Faifman, C.Constantin Vernicos and C.‌Cormac Walsh. Volume‌ growth of Funk geometry‌‌ and the flags of polytopes.Geometry and‌ Topology2972025‌, 3773-3811HAL DOI‌‌
22 articleQ.Quentin Jacquet, W.Wim‌ van Ackooij, C.‌Clémence Alasseur and S.‌‌Stéphane Gaubert. Ergodic control of a heterogeneous‌ population and application to‌ electricity pricing.IEEE‌‌ Transactions on Automatic Control7072025HAL‌
23 articleJ.Jad‌ Zeroual, M.Marianne‌‌ Akian, A.Aurélien Bechler, M.Matthieu‌ Chardy and S.Stéphane‌ Gaubert. Optimal strategy‌‌ against straightforward bidding in clock auctions.Performance‌ Evaluation169September 2025‌, 102502HAL DOI‌‌back to text

International peer-reviewed conferences

24 inproceedings‌M.Marianne Akian,‌ S.Stéphane Gaubert,‌‌ L.Loïc Marchesini and I.Ian Morris.‌ Continuity and approximability of‌ competitive spectral radii.‌‌CDC 2025 - 64th IEEE Conference on Decision‌ and ControlRio De‌ Janeiro, BrazilDecember 2025‌‌HAL back to textback to text
25‌ inproceedingsX.Xavier Allamigeon‌, P.Pascal Capetillo‌‌ and S.Stéphane Gaubert. Computing the Congestion‌ Phases of Dynamical Systems‌ with Priorities and Application‌‌ to Emergency Departments.FORMATS 2025 - International‌ Conference on Formal Modeling‌ and Analysis of Timed‌‌ SystemsAarhus, DenmarkMay 2025HAL back to‌ text back to text‌
26 inproceedingsX.Xavier‌‌ Allamigeon, P.Pascal Capetillo and S.Stéphane‌ Gaubert. Stationary regimes‌ of piecewise linear dynamical‌‌ systems with priorities.Proceedings of the 28th‌ ACM International Conference on‌ Hybrid Systems: Computation and‌‌ ControlHSCC 2025Irvine, CA, United StatesACM‌May 2025, 1-11‌HAL DOI back to‌‌ text
27 inproceedingsM. R.Matías R Bender‌, L.Laurent Busé‌, C.Carles Checa‌‌ and E.Elias Tsigaridas. Solving bihomogeneous polynomial‌ systems with a zero-dimensional‌ projection.ISSAC '25:‌‌ International Symposium on Symbolic and Algebraic ComputationGuanajuato‌ Mexico, MexicoACM; ACM‌2025, 206-214HAL‌‌DOI back to text
28 inproceedingsS.Stéphane‌ Gaubert, J.Julien‌ Grand-Clément and R. D.‌‌Ricardo D. Katz. Thresholds for sensitive optimality‌ and Blackwell optimality in‌ stochastic games.NeurIPS‌‌ 2025 - 39th Annual Conference on Neural Information‌ Processing SystemsSan Diego,‌ United StatesJune 2025‌‌HAL back to text
29 inproceedingsA. B.‌Abdellah Bulaich Mehamdi,‌ W.Wim van Ackooij‌‌, L.Luce Brotcorne, S.Stéphane Gaubert‌ and Q.Quentin Jacquet‌. Duality between polyhedral‌‌ approximation of value functions and optimal quantization of‌ measures.CDC 2025‌ - 64th IEEE Conference‌‌ on Decision and ControlRio, BrazilSeptember 2025‌HAL back to text‌
30 inproceedingsJ.Jad‌‌ Zeroual, M.Marianne Akian, A.Aurélien‌ Bechler, M.Matthieu‌ Chardy and S.Stéphane‌‌ Gaubert. Optimal Strategy Against Straightforward Bidding in‌ Clock Auctions.NETGCOOP‌ 2024 - Network Games,‌‌ Artificial Intelligence, Control and‌ OptimizationLecture Notes in Computer ScienceNetwork Games,‌ Artificial Intelligence, Control and Optimization 11th International Conference,‌ NETGCOOP 2024LNCS-15185Lille, FranceSpringer Nature Switzerland‌January 2025, 83-93HAL DOI back to‌ text back to text

Reports & preprints

31‌ miscH.Hélène Arvis, O.Olivier Beaude‌, N.Nicolas Gast, S.Stéphane Gaubert‌ and B.Bruno Gaujal. Integrating Aggregated Electric‌ Vehicle Flexibilities in Unit Commitment Models using Submodular‌ Optimization.November 2025HAL back to text‌
32 miscM.Matías Bender, P.Philipp‌ Di Dio and E.Elias Tsigaridas. Positive‌ Univariate Polynomials: SOS certificates, algorithms, bit complexity, and‌ T-systems.September 2025HAL back to text‌
33 miscM. R.Matías R Bender,‌ K.Khazhgali Kozhasov, E.Elias Tsigaridas and‌ C.Chaoping Zhu. Certificates for nonnegativity of‌ multivariate integer polynomials under perturbations.December 2025‌HAL back to text
34 miscM.Matías‌ Bender, E.Elias Tsigaridas and C.Chaoping‌ Zhu. Certificates of nonnegativity of multivariate polynomials‌ with zero-dimensional gradient ideal and infimum attained.‌November 2025HAL back to text
35 misc‌S.Stephane Gaubert and Y.Yiannis Vlassopoulos.‌ Relu and softplus neural nets as zero-sum turn-based‌ games.December 2025HAL
36 miscJ.‌Jad Zeroual, M.Marianne Akian, A.‌Aurélien Bechler, M.Matthieu Chardy and S.‌Stéphane Gaubert. Dynamics of multidimensional Simple Clock‌ Auctions.November 2025HAL back to text‌

12.3 Cited publications

37 inproceedingsM.Marianne Akian‌, A.Antoine Béreau and S.Stéphane Gaubert‌. Eigenvalue Methods for Sparse Tropical Polynomial Systems‌.Lecture Notes in Computer ScienceICMS 2024‌ - International Congress on Mathematical SoftwareDurham, United‌ KingdomSpringer2024HALback to text
38‌ inproceedingsM.Marianne Akian, A.Antoine Béreau‌ and S.Stéphane Gaubert. The Tropical Nullstellensatz‌ and Positivstellensatz for Sparse Polynomial Systems.ISSAC‌ '23: Proceedings of the 2023 International Symposium on‌ Symbolic and Algebraic ComputationISSAC 2023 - International‌ Symposium on Symbolic and Algebraic ComputationTromsø, Norway‌ACM2023HAL DOIback to text
39‌ miscM.Marianne Akian, J.-P.Jean-Philippe Chancelier‌ and B.Benoît Tran. Tropical Dynamic Programming‌ for Lipschitz Multistage Stochastic Programming.December 2020‌HAL back to textback to text
40‌ articleM.Marianne Akian, S.Stéphane Gaubert‌, J.Julien Grand-Clément and J.Jérémie Guillaud‌. The operator approach to entropy games.‌Theory of Computing Systemshttps://arxiv.org/abs/1904.051512019HAL DOI‌back to text
41 articleM.Marianne Akian‌, S.Stéphane Gaubert and A.Antoine Hochart‌. Generic uniqueness of the bias vector of‌ finite stochastic games with perfect information.Journal‌ of Mathematical Analysis and Applications457https://arxiv.org/abs/1610.096512018‌, 1038-1064HAL DOIback to text
42‌ miscM.Marianne Akian, S.Stéphane Gaubert‌, D.Dariush Kiani and H.Hanieh Tavakolipour‌. Spectral Properties of Positive Definite Matrices over Symmetrized Tropical Algebras and‌ Valued Ordered fields.‌December 2024HAL back‌‌ to text
43 articleM.Marianne Akian,‌ S.Stéphane Gaubert and‌ S.Shanqing Liu.‌‌ A Multi-Level Fast-Marching Method For The Minimum Time‌ Problem.SIAM Journal‌ on Control and Optimization‌‌6262024, 2963-2991HAL DOI back‌ to text
44 misc‌M.Marianne Akian,‌‌ S.Stéphane Gaubert and L.Loïc Marchesini.‌ The Competive Spectral Radius‌ of Families of Nonexpansive‌‌ Mappings.October 2024HAL back to text‌
45 miscM.Marianne‌ Akian, S.Stephane‌‌ Gaubert and L.Louis Rowen. Linear algebra‌ over T-pairs.October‌ 2023HAL back to‌‌ text back to text
46 articleM.Marianne‌ Akian, S.Stephane‌ Gaubert and L.Louis‌‌ Rowen. Semiring systems arising from hyperrings.‌Journal of Pure and‌ Applied Algebra2286‌‌June 2024, 107584HAL DOI back to‌ text
47 articleM.‌Marianne Akian, S.‌‌Stéphane Gaubert and M.Meisam Sharify. Log-majorization‌ of the moduli of‌ the eigenvalues of a‌‌ matrix polynomial by tropical roots.Linear Algebra‌ and its ApplicationsAlso‌ arXiv:1304.29672017HAL DOI‌‌back to text
48 articleM.Marianne Akian‌ and S.Stéphane Gaubert‌. Spectral theorem for‌‌ convex monotone homogeneous maps, and ergodic control.‌Nonlinear Anal.522‌2003, 637--679URL:‌‌ http://dx.doi.org/10.1016/S0362-546X(02)00170-0DOI back to text
49 miscM.‌Marianne Akian and S.‌Shanqing Liu. Convergence‌‌ and Error Estimates of A Semi-Lagrangian scheme for‌ the Minimum Time Problem‌.July 2024HAL‌‌back to text
50 miscX.Xavier Allamigeon‌, A.Abdellah Aznag‌, S.Stéphane Gaubert‌‌ and Y.Yassine Hamdi. The tropicalization of‌ the entropic barrier.‌2020HAL back to‌‌ text
51 articleX.Xavier Allamigeon, P.‌Pascal Benchimol, S.‌Stéphane Gaubert and M.‌‌Michael Joswig. Log-barrier interior point methods are‌ not strongly polynomial.‌SIAM Journal on Applied‌‌ Algebra and Geometry21https://arxiv.org/abs/1708.01544 - This‌ paper supersedes arXiv:1405.4161. 31‌ pages, 5 figures, 1‌‌ table2018, 140-178HAL DOI back to‌ text back to text‌back to text
52‌‌ inproceedingsX.Xavier Allamigeon, P.Pascal Benchimol‌ and S.Stéphane Gaubert‌. The tropical shadow-vertex‌‌ algorithm solves mean payoff games in polynomial time‌ on average.ICALP‌ 2014857241st International‌‌ Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014,‌ Proceedings, Part ICopenhagen,‌ FranceSpringer2014,‌‌ 12HAL DOI back to text
53 inproceedings‌X.Xavier Allamigeon,‌ V.Vianney Boeuf and‌‌ S.Stéphane Gaubert. Performance evaluation of an‌ emergency call center: tropical‌ polynomial systems applied to‌‌ timed Petri nets.13th International Conference, Formal‌ Modeling and Analysis of‌ Timed Systems (FORMATS 2015)‌‌9268Formal Modeling and Analysis of Timed Systems‌Madrid, SpainSpringer2015‌HAL DOI back to‌‌ text back to textback to text
54‌ articleX.Xavier Allamigeon‌, V.Vianney Boeuf‌‌ and S.Stephane Gaubert‌. Stationary solutions of discrete and continuous Petri‌ nets with priorities.Performance Evaluation113https://arxiv.org/abs/1612.07661‌2017, 1 - 12HAL DOI back‌ to text
55 inproceedingsX.Xavier Allamigeon,‌ M.Marin Boyet and S.Stephane Gaubert.‌ Computing Transience Bounds of Emergency Call Centers: a‌ Hierarchical Timed Petri Net Approach.PETRI NETS‌ 2022: Application and Theory of Petri Nets and‌ Concurrency13288PETRI NETS 2022: Application and Theory‌ of Petri Nets and Concurrency, Springer Lecture Notes‌ in Computer SciencesBergen, NorwaySpringer2022,‌ 90-112HAL DOI back to text
56 article‌X.Xavier Allamigeon, M.Marin Boyet and‌ S.Stéphane Gaubert. Piecewise Affine Dynamical Models‌ of Timed Petri Nets -- Application to Emergency‌ Call Centers.Fundamenta Informaticae1833-42021‌, 169-201HAL back to text
57 misc‌X.Xavier Allamigeon, Q.Quentin Canu,‌ C.Cyril Cohen, K.Kazuhiko Sakaguchi and‌ P.-Y.Pierre-Yves Strub. Design patterns of hierarchies‌ for order structures.February 2023HAL back‌ to text
58 inproceedingsX.Xavier Allamigeon,‌ Q.Quentin Canu and P.-Y.Pierre-Yves Strub.‌ A Formal Disproof of Hirsch Conjecture.Proceedings‌ of the 12th ACM SIGPLAN International Conference on‌ Certified Programs and ProofsCPP 2023 - 12th‌ ACM SIGPLAN International Conference on Certified Programs and‌ ProofsBoston, United StatesACMJanuary 2023,‌ 17-29HAL DOI back to text
59 inproceedings‌X.Xavier Allamigeon, P.Pascal Capetillo and‌ S.Stéphane Gaubert. Stationary regimes of piecewise‌ linear dynamical systems with priorities.Proceedings of‌ the 28th ACM International Conference on Hybrid Systems:‌ Computation and ControlHSCC ’25ACMMay 2025‌, 1–11URL: http://dx.doi.org/10.1145/3716863.3718053DOI back to text‌
60 inproceedingsX.Xavier Allamigeon, D.Daniel‌ Dadush, G.Georg Loho, B.Bento‌ Natura and L.Laszlo Vegh. Interior point‌ methods are not worse than Simplex.2022‌ IEEE 63rd Annual Symposium on Foundations of Computer‌ Science (FOCS)Denver, United StatesIEEEOctober 2022‌, 267-277HAL DOIback to text
61‌ articleX.Xavier Allamigeon, S.Stéphane Gaubert‌ and E.Eric Goubault. Computing the Vertices‌ of Tropical Polyhedra using Directed Hypergraphs.Discrete‌ and Computational Geometry4922013, 247-279‌URL: http://hal.inria.fr/hal-00782862DOI back to text
62 incollection‌X.X. Allamigeon, S.S. Gaubert and‌ É.É. Goubault. Inferring Min and Max‌ Invariants Using Max-plus Polyhedra.Proceedings of the‌ 15th International Static Analysis Symposium (SAS'08)5079LNCS‌Valencia, SpainSpringer2008, 189--204URL: http://dx.doi.org/10.1007/978-3-540-69166-2_13‌back to text back to text
63 article‌X.Xavier Allamigeon, S.Stéphane Gaubert and‌ F.Frédéric Meunier. Tropical Complementarity Problems and‌ Nash Equilibria.SIAM Journal on Discrete Mathematics‌373August 2023, 1645--1665URL: https://doi.org/10.1137/21m1446861‌DOI back to text
64 articleX.Xavier‌ Allamigeon, S.Stephane Gaubert and M.Mateusz‌ Skomra. Solving generic nonarchimedean semidefinite programs using stochastic game algorithms.‌Journal of Symbolic Computation‌85An abridged version‌‌ of this article appeared in the proceedings of‌ ISSAC 20162018,‌ 25-54HAL DOI back‌‌ to text
65 articleX.Xavier Allamigeon,‌ S.Stéphane Gaubert and‌ M.Mateusz Skomra.‌‌ Tropical spectrahedra.Discrete and Computational Geometry63‌February 2020, 507–548‌HAL DOI back to‌‌ text back to textback to text
66‌ inproceedingsX.Xavier Allamigeon‌, S.Stéphane Gaubert‌‌ and N.Nicolas Vandame. No self-concordant barrier‌ interior point method is‌ strongly polynomial.STOC‌‌ '22: 54th Annual ACM SIGACT Symposium on Theory‌ of ComputingRome Italy,‌ FranceACMJune 2022‌‌, 515-528HAL DOIback to text
67‌ inproceedingsE.E. Asarin‌, J.J. Cervelle‌‌, A.A. Degorre, C.C. Dima‌, F.F. Horn‌ and V.V. Kozyakin‌‌. Entropy Games and Matrix Multiplication Games.‌Proceedings of the 33rd‌ International Symposium on Theoretical‌‌ Aspects of Computer Science (STACS)47LIPIcs. Leibniz‌ Int. Proc. Inform.Wadern‌Schloss Dagstuhl--Leibniz-Zentrum für Informatik‌‌2016, 11:1--11:14back to text
68 book‌F.F. Baccelli,‌ G.G. Cohen,‌‌ G.-J.G.-J. Olsder and J.-P.J.-P. Quadrat.‌ Synchronization and linearity: an‌ algebra for discrete event‌‌ systems.Wiley1992back to text back‌ to text back to‌ text back to text‌‌back to text back to text
69 article‌M.Matthew Baker and‌ O.Oliver Lorscheid.‌‌ Descartes' rule of signs, Newton polygons, and polynomials‌ over hyperfields.J.‌ Algebra5692021,‌‌ 416--441URL: https://doi.org/10.1016/j.jalgebra.2020.10.024DOIback to text
70‌ articleG.G. Barles‌, S.S. Mirrahimi‌‌ and B.B. Perthame. Concentration in Lotka-Volterra‌ parabolic or integral equations:‌ a general convergence result‌‌.Methods Appl. Anal.1632009,‌ 321--340URL: http://dx.doi.org/10.4310/MAA.2009.v16.n3.a4DOI‌back to text
71‌‌ phdthesisA.Antoine Bereau. Tropical Polynomial Systems‌ and Game Theory.‌Ecole polytechniqueNovember 2024‌‌back to text
72 articleV.Vianney Boeuf‌ and P.Philippe Robert‌. A Stochastic Analysis‌‌ of a Network with Two Levels of Service‌.Queueing Systems92‌3-4https://arxiv.org/abs/1708.095902019,‌‌ 30HAL DOI back to text
73 book‌F.Frédéric Bonnans and‌ S.Stéphane Gaubert.‌‌ Recherche opérationnelle. Aspects mathématiques et applications.Ellipse‌2016, 391HAL‌back to text
74‌‌ phdthesisM.M. Boyet. Piecewise affine dynamical‌ systems applied to the‌ performance evaluation of emergency‌‌ call centers.Ecole polytechnique2022, URL:‌ http://www.cmapx.polytechnique.fr/~marin.boyet/These_MB.pdfback to text‌
75 articleP.P.‌‌ Butkoviċ. Max-algebra: the linear algebra of combinatorics?‌Linear Algebra and its‌ applications3672003,‌‌ 313--335back to text
76 bookP.Peter‌ Butkoviċ. Max-linear systems:‌ theory and algorithms.‌‌Springer Monographs in MathematicsSpringer-Verlag London, Ltd., London‌2010, xviii+272URL:‌ http://dx.doi.org/10.1007/978-1-84996-299-5DOI back to‌‌ text back to text
77 inproceedingsJ.J.‌ Cochet-Terrasson, G.G.‌ Cohen, S.S.‌‌ Gaubert, M. M.‌M. Mc Gettrick and J.-P.J.-P. Quadrat.‌ Numerical computation of spectral elements in max-plus algebra‌.Proc. of the IFAC Conference on System‌ Structure and ControlNantes1998back to text‌
78 articleG.Guy Cohen, S.Stéphane‌ Gaubert and J.-P.Jean-Pierre Quadrat. Max-plus algebra‌ and system theory: where we are and where‌ to go now.Annual Reviews in Control‌231999, 207--219back to text
79‌ articleA.Alain Connes and C.Caterina Consani‌. Geometry of the arithmetic site.Adv.‌ Math.2912016, 274--329back to text‌
80 articleP.P. Cousot and R.R.‌ Cousot. Abstract Interpretation: A unified lattice model‌ for static analysis of programs by construction of‌ approximations of fixed points.Principles of Programming‌ Languages 41977, 238--252back to text‌back to text
81 incollectionD.Daniel Delling‌, P.Peter Sanders, D.Dominik Schultes‌ and D.Dorothea Wagner. Highway hierarchies star‌.The shortest path problem74DIMACS Ser.‌ Discrete Math. Theoret. Comput. Sci.Amer. Math. Soc.,‌ Providence, RI2009, 141--174back to text‌
82 articleA.Albert Fathi and A.Antonio‌ Siconolfi. Existence of $C^{1}$ critical subsolutions‌ of the Hamilton-Jacobi equation.Invent. Math.155‌22004, 363--388URL: http://dx.doi.org/10.1007/s00222-003-0323-6DOI back‌ to text
83 articleO.Olivier Fercoq,‌ M.Marianne Akian, M.Mustapha Bouhtou and‌ S.Stephane Gaubert. Ergodic control and polyhedral‌ approaches to PageRank optimization.IEEE Trans. Automat.‌ Control5812013, 134--148URL: http://dx.doi.org/10.1109/TAC.2012.2226103‌DOI back to text
84 articleW.W.H.‌ Fleming and W.W.M. McEneaney. A max-plus‌ based algorithm for an HJB equation of non-linear‌ filtering.SIAM J. Control and Opt.2000‌, 683--710back to text
85 articleS.‌S. Fomin and A.A. Zelevinsky. Cluster‌ algebras. I. Foundations.J. Amer. Math. Soc.‌1522002, 497--529URL: http://arxiv.org/abs/math.RT/0104151back‌ to text
86 inproceedingsS.S. Gaubert,‌ E.E. Goubault, A.A. Taly and‌ S.S. Zennou. Static Analysis by Policy‌ Iteration in Relational Domains.Proceedings of the‌ Proc. of the 16th European Symposium on Programming‌ (ESOP'07)4421LNCSBraga (Portugal)Springer2007,‌ 237--252URL: http://dx.doi.org/10.1007/978-3-540-71316-6_17DOIback to text
87‌ inproceedingsS.S. Gaubert, W.W.M. McEneaney‌ and Z.Z. Qu. Curse of dimensionality‌ reduction in max-plus based approximation methods: theoretical estimates‌ and improved pruning algorithms.Proceedings of the‌ 50th IEEE Conference on Decision and Control and‌ European Control Conference (CDC-ECC 11)Orlando, FL, USA‌2011, 1054-1061URL: http://arxiv.org/abs/1109.5241back to text‌
88 incollectionS.S. Gaubert and M.M.‌ Sharify. Tropical scaling of polynomial matrices.‌Positive systems389Lecture Notes in Control and‌ Inform. Sci.BerlinSpringer2009, 291--303URL:‌ http://dx.doi.org/10.1007/978-3-642-02894-6_28back to text
89 miscS.Stephane‌ Gaubert and Y.Yiannis Vlassopoulos. Relu and softplus neural nets as‌ zero-sum turn-based games.‌arXiv:2512.205822025back to‌‌ text
90 articleT. M.T. M. Gawlitza‌, H.H. Seidl‌, A.A. Adjé‌‌, S.S. Gaubert and E.E. Goubault‌. Abstract interpretation meets‌ convex optimization.J.‌‌ Symbolic Comput.4712Special issue on Invariant‌ generation and reasoning about‌ loops2012, 1416--1446‌‌URL: http://dx.doi.org/10.1016/j.jsc.2011.12.048DOI back to text
91 book‌I. M.I. M.‌ Gelfand, M. M.‌‌M. M. Kapranov and A. V.A. V.‌ Zelevinsky. Discriminants, resultants‌ and multidimensional determinants.‌‌Modern Birkhäuser ClassicsReprint of the 1994 edition‌Birkhäuser Boston, Inc., Boston,‌ MA2008, x+523‌‌back to text
92 inbookV.Vasilis Gkatzelis‌, D.Daniel Schoepflin‌ and X.Xizhi Tan‌‌. Clock Auctions Augmented with Unreliable Advice.‌Proceedings of the 2025‌ Annual ACM-SIAM Symposium on‌‌ Discrete Algorithms (SODA)2629-2655URL: https://epubs.siam.org/doi/abs/10.1137/1.9781611978322.86DOI back‌ to text
93 unpublished‌T.Thomas Haettel,‌‌ A.-S.Anna-Sofie Schilling, A.Anna Wienhard and‌ C.Cormac Walsh.‌ Horofunction Compactifications of Symmetric‌‌ Spaces.2018, https://arxiv.org/abs/1705.05026 - working paper‌ or preprintHAL back‌ to text
94 article‌‌S.Sven Hammarling, C. J.Christopher J.‌ Munro and F.Françoise‌ Tisseur. An algorithm‌‌ for the complete solution of quadratic eigenvalue problems‌.ACM Trans. Math.‌ Software3932013‌‌, Art. 18, 19URL: http://dx.doi.org/10.1145/2450153.2450156DOI back‌ to text
95 book‌B.Bernd Heidergott,‌‌ G. J.Geert Jan Olsder and J.Jacob‌ van der Woude.‌ Max Plus at Work:‌‌ Modeling and Analysis of Synchronized Systems: A Course‌ on Max-Plus Algebra and‌ Its Applications.Princeton‌‌2005back to text
96 articleP. A.‌Paul Alexander Helminck and‌ Y.Yue Ren.‌‌ Generic root counts and flatness in tropical geometry‌.arXiv preprint arXiv:2206.07838‌2022back to text‌‌
97 inproceedingsJ.Jonathan Hornewall, S.Solène‌ Delannoy‐Pavy, V.Vincent‌ Leclère and T.Tito‌‌ Homem‐De‐Mello. Decision Focused Scenario Generation for Contextual‌ Two‐Stage Stochastic Linear Programming‌.NeurIPS 2025 Workshop‌‌ on ML×OR: Mathematical Foundations and Operational Integration of‌ Machine Learning for Uncertainty‐Aware‌ Decision‐Making2025, URL:‌‌ https://openreview.net/forum?id=j8BVbta5lgback to text
98 articleH.Hitoshi‌ Ishii and H.Hiroyoshi‌ Mitake. Representation formulas‌‌ for solutions of Hamilton-Jacobi equations with convex Hamiltonians‌.Indiana Univ. Math.‌ J.5652007‌‌, 2159--2183URL: http://dx.doi.org/10.1512/iumj.2007.56.3048DOI back to text‌
99 bookI.I.‌ Itenberg, G.G.‌‌ Mikhalkin and E.E. Shustin. Tropical algebraic‌ geometry.35Oberwolfach‌ SeminarsBaselBirkhäuser Verlag‌‌2007, viii+103back to text
100 book‌V.V.N. Kolokoltsov and‌ V.V.P. Maslov.‌‌ Idempotent analysis and applications.Kluwer Acad. Publisher‌1997back to text‌
101 bookB.B.‌‌ Lemmens and R.R. Nussbaum. Nonlinear Perron-Frobenius‌ theory.189Cambridge‌ Tracts in MathematicsCambridge‌‌ University Press, Cambridge2012, xii+323URL: http://dx.doi.org/10.1017/CBO9781139026079‌DOI back to text‌
102 miscB.Bas‌‌ Lemmens and C.Cormac‌ Walsh. Distance-preserving maps between bounded symmetric domains‌.2024HAL back to text
103 phdthesis‌S.Shanqing Liu. Highway Hierarchies for HJB‌ Equations.Ecole polytechniqueDecember 2023back to‌ text
104 articleQ.Q. Lu, M.‌M. Madsen, M.M. Milata, S.‌S. Ravn, U.U. Fahrenberg and K.‌ G.K. G. Larsen. Reachability Analysis for‌ Timed Automata using Max-Plus Algebra.J. Logic‌ Alg. Prog.8132012, 298-313back‌ to text
105 bookV.V.P. Maslov.‌ Méthodes Operatorielles.MoscouEdition Mir1987back‌ to text back to text
106 articleW.‌ M.W. M. McEneaney. A curse-of-dimensionality-free numerical‌ method for solution of certain HJB PDEs.‌SIAM J. Control Optim.4642007,‌ 1239--1276URL: http://dx.doi.org/10.1137/040610830DOIback to text back‌ to text
107 inproceedingsW.W.M. McEneaney,‌ A.A. Deshpande and S.S. Gaubert.‌ Curse-of-Complexity Attenuation in the Curse-of-Dimensionality-Free Method for HJB‌ PDEs.Proc. of the 2008 American Control‌ ConferenceSeattle, Washington, USA2008back to text‌
108 bookW. M.William M. McEneaney.‌ Max-plus methods for nonlinear control and estimation.‌Systems & Control: Foundations & ApplicationsBoston, MA‌Birkhäuser Boston Inc.2006, xiv+241back to‌ text
109 bookJ.-F.J.-F. Mertens, S.‌S. Sorin and S.S. Zamir. Repeated‌ Games.Cambridge2015back to text
110‌ articleG.G. Mikhalkin. Enumerative tropical algebraic‌ geometry in $^{2}$ .J. Amer. Math. Soc.‌1822005, 313--377URL: http://dx.doi.org/10.1090/S0894-0347-05-00477-7DOI‌back to text
111 articleR. H.Rolf‌ H. Möhring, M.Martin Skutella and F.‌Frederik Stork. Scheduling with AND/OR precedence constraints‌.SIAM J. Comput.3322004,‌ 393--415URL: http://dx.doi.org/10.1137/S009753970037727XDOIback to text
112‌ bookA.A. Papadopoulos. Metric spaces, convexity‌ and non-positive curvature.6IRMA Lectures in‌ Mathematics and Theoretical PhysicsEuropean Mathematical Society (EMS),‌ Zürich2014, xii+309URL: http://dx.doi.org/10.4171/132DOI back‌ to text
113 articleM. V.M. V.‌ F. Pereira and L. M.L. M. V.‌ G. Pinto. Multi-stage stochastic optimization applied to‌ energy planning.Math. Programming522, Ser.‌ B1991, 359--375URL: http://dx.doi.org/10.1007/BF01582895DOI back‌ to text
114 incollectionJ.-E.J.-E. Pin.‌ Tropical Semirings.Idempotency11Publications of the‌ Isaac Newton InstituteCambridge University Press1998,‌ 50–69back to text
115 inproceedingsM.M.‌ Plus. Linear systems in (max,+) algebra.‌Proceedings of the 29th IEEE Conference on Decision‌ and ControlIEEE1990, 151--156back to‌ text back to text
116 incollectionM.Max‌ Plus. Max-plus-times linear systems.Open Problems‌ in Mathematical Systems and Control TheoryMax Plus‌ is a collective name for M. Akian, G.‌ Cohen, S. Gaubert, J.P. Quadrat and M. Viot‌Springer1999, 167--170back to text
117‌ phdthesisZ.Zheng Qu. Théorie de Perron-Frobenius non linéaire et méthodes‌ numériques max-plus pour la‌ résolution d'équations d'Hamilton-Jacobi.‌‌Ecole Polytechnique X2013, URL: http://hal.inria.fr/pastel-00927122back‌ to text
118 article‌D.D. Reeb,‌‌ M. J.M. J. Kastoryano and M. M.‌M. M. Wolf.‌ Hilbert's projective metric in‌‌ quantum information theory.J. Math. Phys.52‌82011, 082201,‌ 33URL: http://dx.doi.org/10.1063/1.3615729DOI‌‌back to text
119 incollectionJ.J. Richter-Gebert‌, B.B. Sturmfels‌ and T.T. Theobald‌‌. First steps in tropical geometry.Idempotent‌ mathematics and mathematical physics‌377Contemp. Math.Providence,‌‌ RIAmer. Math. Soc.2005, 289--317back‌ to text
120 incollection‌G.G. Sagnol,‌‌ S.S. Gaubert and M.M. Bouhtou.‌ Optimal monitoring on large‌ networks by Successive c-Optimal‌‌ Designs.Proceedings of the 22nd international teletraffic‌ congress (ITC22), Amsterdam, The‌ Netherlands, Septemberhttp://dx.doi.org/10.1109/ITC.2010.5608717IEEE‌‌2010back to text
121 articleP.Peter‌ Sanders and D.Dominik‌ Schultes. Engineering highway‌‌ hierarchies.ACM J. Exp. Algorithmics172012‌, Article 1.6, 40‌URL: https://doi.org/10.1145/2133803.2330080DOI back‌‌ to text
122 inproceedingsS.S. Sankaranarayanan,‌ H.H. Sipma and‌ Z.Z. Manna.‌‌ Scalable Analysis of Linear Systems using Mathematical Programming‌.VMCAI3385LNCS‌2005back to text‌‌
123 inproceedingsR.R. Sepulchre, A.A.‌ Sarlette and P.P.‌ Rouchon. Consensus in‌‌ noncommutative spaces.Proceedings of the 49th IEEE‌ Conference on Decision and‌ ControlAtlanta, USA2010‌‌, 6596-6601URL: http://dx.doi.org/10.1109/CDC.2010.5717072back to text
124‌ inproceedingsI.I. Simon‌. Limited subsets of‌‌ a free monoid.Proc. 19th Annual Symposium‌ on Foundations of Computer‌ SciencePiscataway, NJ1978‌‌, 143--150back to text
125 bookH.‌ L.H. L. Smith‌. Monotone dynamical systems‌‌.41Mathematical Surveys and MonographsAn introduction‌ to the theory of‌ competitive and cooperative systems‌‌American Mathematical Society, Providence, RI1995, x+174‌back to text
126‌ phdthesisD.-N.Duy-Nghi Tran‌‌. Programmation dynamique tropicale en optimisation stochastique multi-étapes‌.Université Paris-Est2020‌HAL back to text‌‌
127 articleN. M.Ngoc Mai Tran and‌ J.Josephine Yu.‌ Product-Mix Auctions and Tropical‌‌ Geometry.Math. O.R.444arXiv:1505.057372019‌, 1396--1411back to‌ text
128 phdthesisN.‌‌Nicolas Vandame. Convergence Estimates in Tropical Geometry‌ and Application in Optimization‌.Ecole polytechnique2024‌‌back to text back to text
129 incollection‌O.O. Viro.‌ Dequantization of real algebraic‌‌ geometry on logarithmic paper.European Congress of‌ Mathematics, Vol. I (Barcelona,‌ 2000)201Progr. Math.‌‌BaselBirkhäuser2001, 135--146back to text‌back to text back‌ to text
130 article‌‌C.Cormac Walsh. Gauge-reversing maps on cones,‌ and Hilbert and Thompson‌ isometries.Geometry and‌‌ Topology221https://arxiv.org/abs/1312.7871 - Preprint arXiv:1312.78712018‌, 55-104HAL back‌ to text
131 article‌‌C.Cormac Walsh. Order antimorphisms of finite-dimensional‌ cones.Selecta Mathematica‌ (New Series)264‌‌2020, paper number‌ 53HAL back to text

TROPICAL - 2025

TROPICAL - 2025

2025Activity report​​﻿﻿Project-TeamTROPICAL

Keywords

Computer Science​​﻿﻿ and Digital Science

Other Research Topics and​‌﻿﻿ Application Domains

1​​​‌ Team members, visitors, external﻿​﻿﻿ collaborators

Research Scientists

PhD Students

Interns﻿​​﻿ and Apprentices

Administrative Assistants

Visiting Scientist

2 Overall objectives﻿﻿﻿‌

2.1 Scientific context

3​​​‌ Research program

3.1 Optimal﻿​﻿﻿ control and zero-sum games​‌﻿﻿

3.2﻿‌​‌ Non-linear Perron-Frobenius theory, nonexpansive﻿​​﻿ mappings and metric geometry​​​‌

3.3 Tropical​​​‌ algebra and convex geometry﻿​﻿﻿

3.4 Tropical methods​​​‌ applied to optimization, perturbation﻿​﻿﻿ theory and matrix analysis​‌﻿﻿

4﻿‌​‌ Application domains

4.1 Discrete﻿​​﻿ event systems (manufacturing systems,​​​‌ networks, emergency call centers)﻿﻿﻿‌

4.2 Optimal﻿﻿﻿‌ control and games

4.3 Operations Research﻿​​﻿

4.4 Computing program and﻿﻿﻿‌ dynamical systems invariants

5 Social and﻿​﻿﻿ environmental responsibility

5.1 Impact​‌﻿﻿ of research results

6 Highlights of​​﻿﻿ the year

6.1 Awards​​​‌

6.2 Other highlights​​​‌

7 Latest​‌﻿﻿ software developments, platforms, open​​﻿﻿ data

7.1 Latest software​​​‌ developments

7.1.1 Coq-Polyhedra

7.1.2 EmergencyEval​​​‌

8 New results

8.1﻿‌​‌ Optimal control and zero-sum﻿​​﻿ games

8.1.1 Tropical numerical​​​‌ methods for stochastic control﻿﻿﻿‌ problems

8.1.2 Highway﻿​﻿﻿ hierarchies for Hamilton-Jacobi-Bellman (HJB)​‌﻿﻿ PDEs

8.1.3 Escape Rate Games​​​‌

8.1.4​​​‌ Thresholds for sensitive optimality﻿﻿﻿‌ and Blackwell optimality in﻿‌​‌ stochastic games

8.1.5﻿﻿﻿‌ Polyhedral methods in stochastic﻿‌​‌ programming

8.2 Non-linear Perron-Frobenius﻿﻿﻿‌ theory, nonexpansive mappings and﻿‌​‌ metric geometry

8.2.1 Characterising﻿​​﻿ JB-algebras using order reversing​​​‌ maps

8.2.2﻿‌​‌ Invariant Finsler metrics on﻿​​﻿ symmetric spaces

8.2.3 Isoperimetry in​‌﻿﻿ Hilbert geometries

8.2.4 Horocyclic products

8.2.5 External representation of​​​‌ hyperconvex sets

8.3 Tropical algebra and﻿﻿﻿‌ convex geometry

8.3.1 Formalizing﻿‌​‌ convex polyhedra in Coq﻿​​﻿

8.3.2 Linear algebra over​‌﻿﻿ systems

8.3.3 Roots over​​﻿﻿ the symmetrized tropical semiring​​​‌ and eigenvalues of tropical﻿​﻿﻿ symmetric matrices

8.3.4 Tropical Systems of​‌﻿﻿ Polynomial Equations

8.3.5 Systems of﻿‌​‌ sparse polynomial equations and﻿​​﻿ convex polytopes

8.4﻿﻿﻿‌ Tropical methods applied to﻿‌​‌ optimization, perturbation theory and﻿​​﻿ matrix analysis

8.4.1 Tropicalization​​​‌ of interior point methods﻿﻿﻿‌ and application to complexity﻿‌​‌

8.4.2﻿​﻿﻿ Tropical Nash equilibria and​‌﻿﻿ complementarity problems over oriented​​﻿﻿ matroids

8.4.3 Signed﻿​​﻿ Tropicalization of Polars and​​​‌ application to Matrix Cones﻿﻿﻿‌

8.4.4 Log-limits﻿‌​‌ of semi-algebraic sets

8.5​​​‌ Relu and softplus neural﻿﻿﻿‌ nets as zero-sum turn-based﻿‌​‌ games

8.6 Polynomial​​﻿﻿ optimization and certificates of​​​‌ positivity

8.7 Applications

8.7.1 Performance​‌﻿﻿ evaluation of emergency call​​﻿﻿ centers and emergency services​​​‌

8.7.2 Stationary﻿​​﻿ regimes of piecewise linear​​​‌ dynamical systems with priorities﻿﻿﻿‌

8.7.3﻿​﻿﻿ Optimal pricing of energy​‌﻿﻿ contracts

8.7.4 Aggregated models of﻿​﻿﻿ flexibilities in long term​‌﻿﻿ investment models

8.7.5 Simulation and optimization​‌﻿﻿ of robust strategies for​​﻿﻿ spectrum auctions

9 Bilateral contracts and﻿‌​‌ grants with industry

9.1﻿​​﻿ Bilateral contracts with industry​​​‌

10﻿‌​‌ Partnerships and cooperations

10.1﻿​​﻿ International initiatives

10.1.1 Inria​​​‌ associate team not involved﻿﻿﻿‌ in an IIL or﻿‌​‌ an international program

TropOptim​​​‌

10.1.2 Participation in other​​﻿﻿ International Programs

10.2 International research​​​‌ visitors

10.2.1 Visits of﻿​﻿﻿ international scientists

Other international​‌﻿﻿ visits to the team​​﻿﻿

Gleb Koshevoy

10.2.2​​​‌ Visits to international teams﻿​﻿﻿

Research stays abroad

Gaubert​‌﻿﻿ Stéphane

10.3 National​‌﻿﻿ initiatives

2025Activity reportProject-TeamTROPICAL

Computer Science and Digital Science

Other Research Topics and‌ Application Domains

1‌ Team members, visitors, external collaborators

Interns and Apprentices

2 Overall objectives‌

3‌ Research program

3.1 Optimal control and zero-sum games‌

3.2‌‌ Non-linear Perron-Frobenius theory, nonexpansive mappings and metric geometry‌

3.3 Tropical‌ algebra and convex geometry

3.4 Tropical methods‌ applied to optimization, perturbation theory and matrix analysis‌

4‌‌ Application domains

4.1 Discrete event systems (manufacturing systems,‌ networks, emergency call centers)‌

4.2 Optimal‌ control and games

4.3 Operations Research

4.4 Computing program and‌ dynamical systems invariants

5 Social and environmental responsibility

5.1 Impact‌ of research results

6 Highlights of the year

6.1 Awards‌

6.2 Other highlights‌

7 Latest‌ software developments, platforms, open data

7.1 Latest software‌ developments

7.1.2 EmergencyEval‌

8.1‌‌ Optimal control and zero-sum games

8.1.1 Tropical numerical‌ methods for stochastic control‌ problems

8.1.2 Highway hierarchies for Hamilton-Jacobi-Bellman (HJB)‌ PDEs

8.1.3 Escape Rate Games‌

8.1.4‌ Thresholds for sensitive optimality‌ and Blackwell optimality in‌‌ stochastic games

8.1.5‌ Polyhedral methods in stochastic‌‌ programming

8.2 Non-linear Perron-Frobenius‌ theory, nonexpansive mappings and‌‌ metric geometry

8.2.1 Characterising JB-algebras using order reversing‌ maps

8.2.2‌‌ Invariant Finsler metrics on symmetric spaces

8.2.3 Isoperimetry in‌ Hilbert geometries

8.2.5 External representation of‌ hyperconvex sets

8.3 Tropical algebra and‌ convex geometry

8.3.1 Formalizing‌‌ convex polyhedra in Coq

8.3.2 Linear algebra over‌ systems

8.3.3 Roots over the symmetrized tropical semiring‌ and eigenvalues of tropical symmetric matrices

8.3.4 Tropical Systems of‌ Polynomial Equations

8.3.5 Systems of‌‌ sparse polynomial equations and convex polytopes

8.4‌ Tropical methods applied to‌‌ optimization, perturbation theory and matrix analysis

8.4.1 Tropicalization‌ of interior point methods‌ and application to complexity‌‌

8.4.2 Tropical Nash equilibria and‌ complementarity problems over oriented matroids

8.4.3 Signed Tropicalization of Polars and‌ application to Matrix Cones‌

8.4.4 Log-limits‌‌ of semi-algebraic sets

8.5‌ Relu and softplus neural‌ nets as zero-sum turn-based‌‌ games

8.6 Polynomial optimization and certificates of‌ positivity

8.7.1 Performance‌ evaluation of emergency call centers and emergency services‌

8.7.2 Stationary regimes of piecewise linear‌ dynamical systems with priorities‌

8.7.3 Optimal pricing of energy‌ contracts

8.7.4 Aggregated models of flexibilities in long term‌ investment models

8.7.5 Simulation and optimization‌ of robust strategies for spectrum auctions

9 Bilateral contracts and‌‌ grants with industry

9.1 Bilateral contracts with industry‌

10‌‌ Partnerships and cooperations

10.1 International initiatives

10.1.1 Inria‌ associate team not involved‌ in an IIL or‌‌ an international program

TropOptim‌

10.1.2 Participation in other International Programs

10.2 International research‌ visitors

10.2.1 Visits of international scientists

Other international‌ visits to the team

10.2.2‌ Visits to international teams

Gaubert‌ Stéphane

10.3 National‌ initiatives

10.3.2‌ Programme Gaspard Monge pour‌ l'optimisation, la recherche opérationnelle‌‌ et leurs interactions avec les sciences des données‌

10.3.3 Joint INRIA & AP-HP‌ Bernoulli lab project: “URGE”‌

11.1‌ Promoting scientific activities

11.1.1‌ Scientific events: organisation

General‌‌ chair, scientific chair

Member of‌ the organizing committees

11.1.2 Scientific events: selection

Chair of conference‌ program committees

Member of the‌ conference program committees

11.1.3 Journal

Member of the editorial‌ boards

11.1.4‌ Invited talks

11.1.5 Additional conferences talks, Seminars

11.1.6 Research administration‌