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.

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 57, combinatorial optimization 63, or automata theory 100. They also arose in mathematical physics and asymptotic analysis 91, 86. More recently, these structures have appeared in several areas of pure mathematics, in particular in the study of combinatorial aspects of algebraic geometry 78, 115, 105, 83, in algebraic combinatorics 73, and in arithmetics 67. Also, further applications of tropical methods have appeared, including optimal control 92, program invariant computation 51 and timed systems verification 90, 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 115, 96. 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 111, 57, 81, 64.

Tropical methods are linked to the fields of positive systems and of metric geometry 98, 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 88, 3. Such dynamical systems are of fundamental importance in the study of repeated games 95. Monotonicity properties are also essential in the understanding of the fixed points problems which determine program invariants by abstract interpretation 68. 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 112 or in a PDE setting 59. 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 104, 109.

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 70),
and also using metric geometry ideas (abstract boundaries can be used to represent the sets of solutions 82, 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 41, 38, 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) 71. 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” 72. 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) 93, 75.
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. 44 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.

Several applications (including population dynamics 10 and discrete event systems 57, 66, 45) 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.

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 50,2, tropical semialgebraic sets 54, 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 40, and more generally of systems of tropical polynomial equations. Our research also builds on, and concern, classical convex and discrete geometry methods.

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

when

This entails a relation between classical algorithmic problems
and tropical algorithmic problems, one may first solve the

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 76, 80.

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 43,6, 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 54, 53, with the goal to better understand complexity issues in classical semidefinite and semi-algebraic programming.

One important class of applications of max-plus algebra comes from discrete event dynamical systems 57. 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 45.

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.

We develop, or have developed, several aspects of operations research, including the application of stochastic control to optimal pricing, optimal measurement in networks 106. Applications of tropical methods arise in particular from discrete optimization 64, 65, scheduling problems with and-or constraints 97, or product mix auctions 114.

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 68, 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 108), shadows of sets represented by linear matrix inequalities, disjunctive constraints represented by tropical polyhedra 51, 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. 74, 77 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).

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 recent 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.

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.

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.

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.

In 113, we were interested in the numerical solution of the dynamic programming equation of discrete time stochastic control problems, and we developped and studied several algorithms combining the tropical or the max-plus based numerical method of McEneaney 94, 92, the stochastic max-plus scheme proposed by Zheng Qu 103, and the stochastic dual dynamic programming (SDDP) algorithm of Pereira and Pinto 99. In 35, we also show that in the case of the dynamic programming equation associated to a partially observable Markov Decision Process (POMDP), it is similar to the so called point based algorithms developped in 101, 87, 110, which includes in particular SARSOP algorithm.

In an ongoing work, we are studying the convergence of SARSOP algorithm using the same techniques as in 36.

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 synthetize an optimal feedback control, leading to a solution robust against system perturbations. Several methods have been proposed in the litterature 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”, developped by Sanders, Schultes and coworkers 69, 107, 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.

The aim of the PhD thesis of Shanqing Liu 89 was to develop new numerical methods to solve Hamilton-Jacobi-Bellman equations that are less sensitive to curse of dimensionality.

In 24, we have developped a multilevel fast-marching method, extending to the PDE case the idea of “highway hierarchies”. Given the problem of finding an optimal trajectory between two given points, the method consists in refining the grid only in a neighborhood of the optimal trajectory, which is itself computed using an approximation of the value function on a coarse grid.

In 18, we have considered general finite horizon deterministic optimal control problems, for which we combine the idea of the multilevel grids around optimal trajectories with the max-plus finite element method solving Hamilton-Jacobi equations introduced in 39.

More recently, we combined these ideas with an adaption of SDDP algorithm to semiconcave problems. We also apply our algorithm to numerically solve the N-body problem.

We studied in 19 an algorithm solving stochastic mean-payoff games, combining the ideas of relative value iteration and of Krasnoselskii-Mann damping. We analysed its convergence and derived parameterized complexity bounds for several classes of games satisfying irreducibility conditions. This is a joint work with Ulysse Naepels and Basile Terver, two students at École polytechnique, IP Paris.

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 begin with the case of 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. 56, which generalize the joint spectral radius of sets of nonnegative matrices. See also 37. We established a two-player analogue of “Mañe's Lemma”, characterizing the value of the game.

This is joint work with Dmitry Faifman (Tel Aviv) and Constantin Vernicos (Montpellier).

We are investigating the volume of metric balls in the Funk geometry 32.
This is an asymmetric metric related to the Hilbert metric;
indeed, the Hilbert metric is its symmetrisation. We are interested in it
because it is somewhat easier to work with
than the Hilbert metric when dealing with volumes,
and exhibits similar interesting of behaviour.
Given a convex body in

Similarly, one could conjecture that, under the assumption that the convex body is centrally symmetric, the minimum volume is minimised when the body is a Hanner polytope. Recall that these are the polytopes that can be constructed from copies of the unit interval by taking products and polar duals. This conjecture interpolates between the Mahler conjecture and Kalai’s flag conjecture. We can prove it for unconditional bodies, that is, bodies that are symmetric through reflections in the coordinate hyperplanes.

We have been studying in more detail the volume of balls in the Funk geometry when the convex body is a polytope. Here, as in the case of the Hilbert metric, the volume grows polynomially with order equal to the dimension, and the constant on front of the highest order term depends only on the number of flags of the polytope. Thus, this term does not change when the polytope is perturbed in a way that doesn't change the combinatorics. This motivates us to look at the second highest order term. We have developed the following formula for this in terms of the position of the vertices of the polytope and the vertices of its dual.

Recall that polytopes have the following property:
if monodromy group or connection group
of the polytope complete flip

Here, we are identifying the facet

Thus we get a new centro-affine invariant for polytopes. We have been studying this invariant: we can show that in dimension 2 its only stationary points are the linear images of regular polygons centered at the origin.

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 79, it was established that there is such a metric natural 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?

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 20.

Dealing with polyhedra and their faces raises the problem of formalizing order structures. We study this problem in a joint work 29 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.

In a joint work with Louis Rowen (Univ. Bar Ilan), 14, 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 14, we characterize the semiring systems which arise from hyperrings.

In 25, we are studying linear algebra properties over a generalization of “systems” called “T-pairs”.

Closed tropical convex cones are the most basic examples of modules
over the tropical semifield. They coincide with sub-fixed-point sets of Shapley
operators – dynamic programming operators of zero-sum games. We study a
larger class of cones, which we call “ambitropical” as it includes both tropical
cones and their duals. Ambitropical cones can be defined as lattices in
the order induced by

In 13, we show that the problem consisting in computing a best approximation of a collection of points by a tropical hyperplane is equivalent to solving a mean payoff game, and also, to compute the maximal radius of an inscribed ball in a tropical polytope. We provide an application to a problem of auction theory – measuring the distance to equilibrium. We also study a dual problem — computing the minimal radius of a circumscribed ball to a tropical polytope – and apply it to the rank-one approximation of tropical matrices and tensors.

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 102 (see also 57), and the signed valuation which associates to any series its valuation together with its sign.

In a work 26 which started during the postdoc of Hanieh Tavakolipour (Amirkabir University of Technology) in the team, we study the roots's multiplicities and the factorization of polynomials over the symmetrized tropical semiring. We then deduce a Descartes' rule of sign over ordered valued fields. This builds in particular on 58 (for multiplicities) and on 54.

In a work with Dariush Kiani and Hanieh Tavakolipour (Amirkabir University of Technology), we study with these tools the asymptotics of eigenvalues and eigenvectors of symmetric positive definite matrices over the field of Puiseux series.

The PhD thesis of Antoine Bereau deals with systems of polynomial equations over tropical semifields. In 17, 23, 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.

Solving systems of polynomial equations is an intrinsically hard problem. In applications, almost all the systems that we encounter have certain structure, so it is central to take advantage of this to be able to solve bigger inputs. Among the most common structures, we find sparsity: the polynomial are defined by a few set of monomials. If all the polynomial share the same sparsity pattern we say that the system is “unmixed”; otherwise, we call it “mixed”. Several strategies had been proposed to deal with sparse polynomial systems, being the more efficients ones the ones to deal with unmixed systems. In our recent paper 30 with Pierre-Jean Spaenlehauer (Inria Nancy), we characterise the conditions under which the specialised strategies developed for unmixed systems can be applied to mixed systems. Our analysis relies on the combinatorics of convex polytopes associated to the input polynomials and toric geometry. Furthermore, we show that deciding whenever this can be done is a NP-hard problem.

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 43,6 that the answer is negative.

In a subsequent work 42 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 43,6.

In the work 55, 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

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, valid under certain genericity assumptions, have been subsequently established.

In a joint work 49 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

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 52 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).

With Sergey Sergeev (U. Birmingham), we study in 22 the tropical analogue of the notion of polar of a cone over the symmetrized tropical semiring (see for instance 102, 57). 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.

With Gregorio Malajovich (Univ. Federal Rio de Janeiro (UFRJ), we study the approximation of the solutions of (classical) polynomial systems, using the roots of some associated tropical polynomial systems, or more generally techniques of tropical geometry. In particular, in a work in progress, we generalize the Cauchy and Lagrange bounds to multivariate polynomials.

In a joint work with Khazhgali Kozhasov (Université Côte d'Azur), Alan Muniz (Universidade Estadual de Campinas), and Luca Sodomaco (KTH Stockholm) 34, we study the algebraic complexity of Euclidean distance minimization from a generic tensor to the set of rank-one tensors. More precisely, we study the number of complex critical points of this optimization problem, which is called the Euclidean distance (ED) degree of the Segre-Veronese variety in algebraic geometry. We view this invariant as a function of inner products and show this function achieves its minimal value at Frobenius inner product for the matrix case. In addition, we discuss the above optimization problem for other varieties as well.

With Christian Ikenmeyer (Warwick), we study the complexity classes WR which is defined by symmetric rank and its Zariski closure VWR which is defined by symmetric border rank. In particular, we show these two classes equal, i.e., for a given symmetric tensor

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 45, 46, 60. 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 (APHP), i.e., with SAMU75, 92, 93, 94, in the framework of a project coordinated by Dr. Christophe Leroy from APHP. 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 APHP 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 48.

In 47, 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 are also presented in the Phd thesis 62.

In a followup work, 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).

The PhD thesis of Quentin Jacquet 85 was cosupervised by Stéphane Gaubert, Clémence Alasseur (EDF Labs), and Wim van Ackooij (EDF Labs). It concerns the application of bilevel programming methods to the pricing of electricity contracts. We investigated in 16 a new model of customer's response, based on a quadratic regularization. We showed that this model has qualitative properties and a realism similar to the classical models based on the logit-response, while being amenable to mathematical programming and polyhedral techniques, and so to exact solutions, via a reduction to quadratic complementary problems. An application to a set of instances representative of French electricity contracts was also developed in 16.

In 84, we developed a model representing the response of a population of customers to dynamic price offers. This lead to a mean-field Markov decision model, with ergodic cost. We showed in particular the optimality of pricing policies characterized by periodic discounts.

The preprint 28 develops a mean field game model to model an incentive pricing scheme, in which agents compete to get the best reward.

The conference article 21 develops a method to compute an optimal menu of contracts of a prescribed cardinality, optimizing the income of a provider, taking into account the agent's characteristics.