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 48, combinatorial optimization 53, or automata theory 85. They also arose in mathematical physics and asymptotic analysis 78, 75. More recently, these structures have appeared in several areas of pure mathematics, in particular in the study of combinatorial aspects of algebraic geometry 69, 96, 88, 74, in algebraic combinatorics 64, and in arithmetics 58. Also, further applications of tropical methods have appeared, including optimal control 79, program invariant computation 43 and timed systems verification 77, 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 96, 82. 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 93, 48, 72, 54.

Tropical methods are linked to the fields of positive systems and of metric geometry 84, 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 76, 3. Such dynamical systems are of fundamental importance in the study of repeated games 81. Monotonicity properties are also essential in the understanding of the fixed points problems which determine program invariants by abstract interpretation 59. 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 94 or in a PDE setting 50. 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 87, 92.

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 61),
and also using metric geometry ideas (abstract boundaries can be used to represent the sets of solutions 73, 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 37, 35, 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) 62. 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” 63. 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) 80, 66.
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. 39 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 48, 56, 40) 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 42, 2, tropical semialgebraic sets 46, 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 36, 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 67, 71.

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 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 46, 45, 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 48. 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 40.

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 89. Applications of tropical methods arise in particular from discrete optimization 54, 55, scheduling problems with and-or constraints 83, or product mix auctions 95.

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 59, 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 91), shadows of sets represented by linear matrix inequalities, disjunctive constraints represented by tropical polyhedra 43, 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. 65, 68 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 8.6.1.

X. Allamigeon, P. Benchimol and S. Gaubert (from the Tropical team) and M. Joswig (from TU Berlin) obtained the SIAM Sigest Award in 2021, for the work Log-barrier interior point methods are not strongly polynomial », SIAM Journal on Applied Algebra and Geometry 2, 1, 2018, p. 140–178, reprinted in extended form in the SIGEST section of SIAM Review, see: What Tropical Geometry Tells Us about the Complexity of Linear Programming, SIAM Review 63, 1, February 2021, p. 123–164, 14. According to SIAM, “The purpose of SIGEST is to make the 10,000+ readers of SIAM Review aware of exceptional papers published in SIAM’s specialized journals.”

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.

Coq-Polyhedra now provides most of the basic operations on polyhedra. They are expressed on a quotient type that avoids reasoning with particular inequality representations. They include : * the construction of elementary polyhedra (half-spaces, hyperplanes, affine spaces, orthants, simplices, etc) * basic operations such as intersection, projection (thanks to the formalization of the Fourier-Motzkin algorithm), image under linear functions, computations of convex hulls, finitely generated cones, etc. * computation of affine hulls of polyhedra, as well as their dimension

Thanks to this, we have made huge progress on the formalization of the combinatorics of polyhedra. The poset of faces, as well as its fundamental properties (lattice, gradedness, atomicity and co-atomicity, etc) are now formalized. The manipulation of the faces is based on an extensive use of canonical structures, that allows to get the most appropriate inequality representations for reasoning. In this way, we arrive at very concise and elegant proofs, closer to the pen-and-paper ones.

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.

Accelerated gradient algorithms in convex optimization were introduced by Nesterov. A fundamental question is whether similar acceleration schemes work for the iteration of nonexpansive mappings. In a joint work with Zheng Qu (Hong Kong University)

13, motivated by the analysis of Markov decision processes and zero-sum repeated games, we study fixed point problems for Shapley operators, i.e., for sup-norm nonexpansive and order preserving mapping. We deal more especially with affine operators, corresponding to zero-player problems – the latter can be used as a building block for one or two player problems, by means of policy iteration. For an affine operator, associated to a Markov chain, the acceleration property can be formalized as follows: one should replace an original scheme with a convergence rate

by a convergence rate

where

is the spectral gap of the Markov chain. We characterize the spectra of Markov chains for which this acceleration is possible. We also characterize the spectra for which a multiple acceleration is possible, leading to a rate of

for

.

In 25 (joint work with Vincent Leclère, ENPC), we study multistage stochastic problems with a linear structure and general cost distribution. We obtain an exact quantization result, showing that a multistage problem with a continuous cost distribution is equivalent to a problem with a discrete distribution, constructed by exploiting results from polyhedral geometry (“nested” analogue of fiber polytopes). We deduce polynomial-time solvability results in fixed dimension, both for exact and approximated versions of the problem.

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 60, 90, 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 is to develop new numerical methods to solve Hamilton-Jacobi-Bellman equations that are less sensitive to curse of dimensionality.

In a first work, 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. A first account of this work has been presentend to the PGMO days in December 2021.

In a recent paper 19, we investigated how the volume of a ball in a Hilbert geometry grows as its radius increases. In particular, we studied the volume entropy

where

We are continuing this work by investigating the volume of balls of finite
radius, rather than the asymptotics. We have found it convenient to turn
our attention to a metric different from the Hilbert metric,
but related to it. The Funk metric, as it is called, lacks the
symmetry property usually assumed for metric spaces.
However, it is somewhat simpler to work with when dealing with volumes,
and it exhibits the same interesting behaviour.
Given a convex body and a radius

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 way that doesn't change the combinatorics. This motivates us to look at the second highest order term. We have developed a formula for this in terms of the position of the vertices of the polytope and the vertices of its dual. Thus we get a new centro-affine invariant for polytopes. We are in the process of studying this invariant, to see where it is maximised and minimised, etc.

This is a joint work with Antonin Guilloux (IMJ).

In a Hilbert geometry, if one knows the shape of a metric ball of radius

The law of reflection in a smooth strictly convex Finsler geometry was established by Gutkin and Tabachnikov who were studying billards in that setting . Establishing it in the non-smooth not-strictly-convex case forced us to give an affine interpretation which also permits one to establish Snell-Descartes laws of refraction. We also get to give an affine interpretation of refraction with negative indicies. We are investigating existence of periodic orbits in that setting.

The existence of certain type of Barycentre in a metric geometry is related to certain types of curvature. For instance in Hyperbolic geometry the fact that the square of the distance is strictly convex gives existence, and is the founding fact in the barycentric methods introduced by Besson-Courtois-Gallot, which allowed them to prove the minimal entropy theorem. We are investigating the existence of such barycentres in the setting of Hilbert geometries, which share some similarity with the hyperbolic geometries, but whose distance is never convex in the sense of Busemman, which is a weaker assumption than what happens in Hyperbolic geometry.

We investigate an analogue of Funk and Hilbert geometries inside tropical convex sets. This is motivated by the construction of barriers adapted to tropical polyhedra, and by nonarchimedean convexity.

A fundamental question in the theory of metric spaces is the long term behaviour of iterates of non-expansive mappings. Particularly interesting is the case where the mapping has no fixed point, because here the iterates have the possibility of escaping to infinity.

In 30, we introduce the notion of
firm non-expansive mapping in an arbitrary (weak) metric space.
We show that that, for these mappings, the minimal displacement,
the linear rate of escape, and the asymptotic step size are all equal.

This generalises a previous result of Reich and Shafrir in the setting of Banach spaces, and also one by Ariza-Ruiz et al. in the setting of “W -hyperbolic spaces”, which are geodesic metric spaces with a certain “negative curvature”-type condition.

The advantage of our definition is that we do not need to assume the existence of geodesics. This is significant since in modern optimisation applications one often deals with discrete spaces or has access to a collection of points of a space whose geometric structure is unknown. Our class of firm non-expansive mappings includes the mappings considered in the setting of Banach spaces or W-hyperbolic spaces.

Our definition of firm non-expansive was inspired by Ćirić's
work on generalisations of Banach's Contraction Theorem,
for which he introduced the notion of generalised contraction.
The relation between this concept and our firm non-expansive mappings
is similar to the relation between strict contractions and non-expansive mappings.

In a joint work with Anders Karlsson (University of Geneva and Uppsala University), we use the explicit formulas for metric functionals on

and on Hilbert spaces to provide a new result for non-expansive mappings in

and to study the well-known invariant subspace problem, respectively.

More precisely, the first result is that for every non-expansive mapping

In a joint work with Ricardo Katz (Conicet, Argentina)
and Pierre-Yves Strub (LIX, Ecole Polytechnique), we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library providing the basic constructions and operations over polyhedra, including projections, convex hulls and images under linear maps. Moreover, we design a special mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. We demonstrate the usability of this approach by establishing some of the most important combinatorial properties of faces, namely that they constitute a family of graded atomistic and coatomistic lattices closed under sublattices. We also prove a theorem due to Balinski on the

In collaboration with P. Y. Strub, X. Allamigeon and Q. Canu are currently working on the enumeration of vertices and the computation of graphs of polytopes within the proof assistant Coq. They have also contributed to the improvement of the implementation of hierarchies of mathematical objects in the MathComp library 21. Another work on the formalization of lattices and ordered structures, with a special emphasis on finite (sub)lattices, is also currently ongoing.

In a joint work with Louis Rowen (Univ. Bar Ilan), we study linear algebra and convexity properties over “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.

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

. Closed ambitropical cones are precisely the fixedpoint sets of Shapley operators. They are characterized by a property of best co-approximation arising from the theory of nonexpansive retracts of normed spaces. Finitely generated ambitropical cones arise when considering Shapley operators of deterministic games with finite action spaces. They are also special cases of hyperconvex spaces. Finitely generated ambitropical cones are special polyhedral complexes whose cells are alcoved poyhedra, and locally, they are in bijection with order preserving retracts of the Boolean cube. This is a joint work with Sara Vannucci. See

23.

In

22, 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 86 (see also 48), and the signed valuation which associates to any series its valuation together with its sign.

We study with these tools the asymptotics of eigenvalues and eigenvectors of symmetric positive definite matrices over the field of Puiseux series. This raises the problem of defining the appropriate notions of positive definite matrices over the symmetrized tropical semiring, eigenvalues and eigenvectors of such matrices, thus roots of polynomials and their multiplicities. This builds on 46 and 49.

The PhD thesis of Antoine Bereau, started in September 2021, deals with systems of polynomial equations over tropical semifields. In the case of linear equations (that is polynomials with degree 1)

2and in some other particular cases

32, the solution can be characterized as the value of a zero-sum deterministic or stochastic game, allowing one to use the associated algorithms. The aim of the thesis is to apply zero-sum games, combinatorial and polyhedral techniques, in order to solve tropical polynomial systems. One shall also use the extended and symmetrized tropical semirings which were already useful in the study of linear systems

86,

34. A first investigation concerns an optimal degree bound for the linearization method based on the tropical analogue of the Macaulay matrix.

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 6, 14 that the answer is negative.

In a joint work 38 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 6.

In a subsequent work, we have 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 provide an explicit linear program that falls in the same class as the Klee–Minty counterexample, i.e., a

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 44 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).

The primary goal of 57 is to better understand the topological properties of various tensor ranks, which have useful practical implications.

In this ongoing project, we would like to study the corresponding problems for the space of tensors of a fixed Barvinok rank.

In 18 (joint work with Lek-Heng Lim and Mateusz Michałek) we show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. More precisely, given a training set

We show that even for

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 40, 41, 51. 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. Besides, we have initiated a new collaboration, with SAMU69, also on dimensioning.

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 15.

The PhD thesis of Maxime Grangereau 70 has been cosupervised by Emmanuel Gobet (CMAP), Stéphane Gaubert, and Wim van Aackooij (EDF Labs), it dealt with the application of stochastic control methods to the optimization of flexibilities in energy management.

A first series of results concern the application of mean-field control methods to the smart grid 27, including the modelling of storage resources and decentralized aspects 28. Another contribution concerns a version of the Newton method in stochastic control 29. A last series of work concern the multistage and stochastic extension of the Optimal Power Flow problem (OPF). We developed semidefinite relaxations, extending the ones which arise in static and deterministic OPF problems. We provided a priori conditions which guarantee the absence of relaxation gap, and also a posteriori methods allowing one to bound this relaxation gap. We applied this approach on examples of grids, with scenario trees representing the random solar power production 16.

The PhD thesis of Quentin Jacquet, is 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 31 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 31.

With Luz Pascal (former internship, now LIS, Marseille) and Iadine Chades (CSIRO, Australia), we have developped an algorithm for Adaptive Management based on a hidden model Markov Decision Process (hmMDP) with a universal set of predefined models in 20. hmMDP is a particular case of Partially observable MDP (POMDP). The main algorithm to solve such a problem is the SARSOP algorithm, which share some similarities with the combination of SDDP and tropical algorithms considered in 33. In a work in progress, we are studying the convergence of SARSOP algorithm using the same techniques as in 33.

See Section 11.1.1.