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, Vasileios Charisopoulos, 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

8.1.1 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 107, 105, the stochastic max-plus scheme proposed by Zheng Qu  116, and the stochastic dual dynamic programming (SDDP) algorithm of Pereira and Pinto 112. In particular in 13, we considered a stochastic algorithm for deterministic control problems, and in 40, we considered stochastic control problems.

In 39, we also show that in the case of the dynamic programming equation associated to a partially observable Markov Decision Process (POMDP), the algorithm studied in 40 is similar to the so called point based algorithms developed in 114, 99, 123, which includes in particular SARSOP algorithm. In an ongoing work with Luz Pascal (PhD student at Queensland University of Technology, Australia), we are studying the convergence of SARSOP algorithm using the same techniques as in 40.

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 80, 120, 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 Liu101 (now in Brown University) was to develop new numerical methods to solve Hamilton-Jacobi-Bellman equations that are less sensitive to curse of dimensionality.

In 14, we have developed 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. The complexity of the method is analysed using a priori error bounds for the discretization of the Hamilton-Jacobi PDE. In 29, 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.

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. 64, which generalize the joint spectral radius of sets of nonnegative matrices. See also 41. We show in 28 that escape rate games have a value, which is characterized in terms of non-linear eigenproblems, providing a a two-player analogue of “Mañe's Lemma” in ergodic control.

8.1.4 Vanishing discount limit in optimal control

Participants: Stéphane Gaubert.

In a joint work with Piermarco Cannarsa, Cristian Mendico (Roma Tor Vergata), and Marc Quincampoix (Brest) 19, we characterize the vanishing discount limit of the value function of an optimal control problem, in the absence of controllability conditions. We show that the limit in the uniform topology, if it exists, is the maximal subsolution of a system of Hamilton-Jacobi equations. We also provide an analogue of this result for problems in discrete time.

8.2.1 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  90, 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 36 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.

8.2.2 Isoperimety 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 and Hilbert's metric on convex domains.

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 56.

Dealing with polyhedra and their faces raises the problem of formalizing order structures. We study this problem in a joint work 55 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 15, we characterize the semiring systems which arise from hyperrings.

In 43, we are studying linear algebra properties over a generalization of “systems” called “T-pairs”. We are still working on improvements of the results of 43.

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 65), and the signed valuation which associates to any series its valuation together with its sign.

In a work 46 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 66 (for multiplicities) and on 62.

More recently, in a work with Dariush Kiani and Hanieh Tavakolipour (Amirkabir University of Technology) 27, we studied with the above tools the asymptotics of eigenvalues and eigenvectors of symmetric positive definite matrices over the field of Puiseux series.

8.3.4 Tropical Systems of Polynomial Equations

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

The PhD thesis of Antoine Bereau 70 deals with systems of polynomial equations over tropical semifields. In 38, 37, 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 23.

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 93.

8.3.5 Systems of sparse polynomial equations and convex polytopes

Participants: Matías Bender.

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 18 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, i.e., Newton polytopes, and toric geometry. Furthermore, we show that deciding whenever this can be done is a NP-hard problem.

In 31, in collaboration with Laurent Busé (Inria d'Université Côte d'Azur), Carles Checa (University of Copenhagen, Denmark), and Elias Tsigaridas (Inria Paris), we studied mixed sparse systems whose Newton polytopes are Cartesian products of simplices. We showed that the complexity of computing Groebner bases for these systems can be characterized in terms of the multigraded Castelnuovo-Mumford regularity, providing a partial generalization of the milestone work of Bayer and Stillman on homogeneous polynomials 69. In 32, we use our characterization to introduce a novel method for computing the solutions to projections of these systems onto a subset of variables. This linear algebra-based approach offers new techniques for computing Gröbner bases and obtaining numerical approximations for such projections in single exponential time, but polynomial time relative to the worst-case size of the output.

8.4.1 Tropicalization of interior point methods and application to complexity

Participants: Xavier Allamigeon, Stéphane Gaubert, Nicolas Vandame.

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 49,6 that the answer is negative.

In a subsequent work 48 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 49,6.

In the work 63, 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 are presented in the Phd thesis of Nicolas Vandame, defended in December 2024 128.

In a joint work 57 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 $O(2^n{n}^{1.5}logn)$ for an $n$-variable linear program in standard form. The number of iterations of our algorithm is at most $O(n^{1.5}logn)$ 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 $2n$ shadow vertex simplex paths.

8.4.2 Tropical Nash equilibria and complementarity problems

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 60 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).

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 26 the tropical analogue of the notion of polar of a cone over the symmetrized tropical semiring (see for instance 115, 65). 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, Nicolas Vandame.

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 $\alpha$-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 are presented in the PhD thesis of Nicolas Vandame, defended in December 2024 128.

8.7.1 Performance evaluation of emergency call centers and emergency services

Participants: Xavier Allamigeon, Pascal Capetillo, Stéphane Gaubert, Benjamin Nguyen-Van-Yen, 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 51, 52, 71. 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 54.

In 53, 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 73.

In a current followup work, in the framework of the URGE project (see Section 10.2.3 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 30 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.

8.7.3 Optimal pricing of energy contracts

Participants: Abdellah Bulaich, Stéphane Gaubert.

This work builds on the PhD thesis of Quentin Jacquet 97, which 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 22 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 22.

In 21, 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 47 develops a mean field game model to model an incentive pricing scheme, in which agents compete to get the best reward.

The conference article 96 developed 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.

This work was subsequently pursued within the “Defi EDF Inria”, with a collaboration between Wim van Ackooij, Luce Brotcorne (Inria Lille) and S. Gaubert, leading to the hiring in internship of Abdellah Bulaich-Mehamdi at the spring 2024, pursuing in CIFRE PhD in 2025. We further 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. A first account of these results was presented at the conference “PGMO Days 2024” at EDF Labs, Saclay.

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 starting PhD thesis of Hélène Arvis, jointly supervised with Polaris (main supervisor: Nicolas Gast). An internship preparatory to the PhD took place during the last six months of 2024, Hélène Arvis is now starting her CIFRE PhD.

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 25, 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 25 obtained the best paper award at the conference NETGCOOP'2024.

10.1.1 Visits of international scientists

10.2.1 ANR

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

• Project ANR 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).

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

Participants: Matías Bender.

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

10.2.3 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 S. Gaubert, B. Nguyen (Tropical), Ch. Fricker (Dyogene), J.D. Fekete (Aviz).

10.3.1 Performance evaluation of emergency call centers

Participants: Xavier Allamigeon, Stéphane Gaubert, Benjamin Nguyen-Van-Yen, Guillaume Thomas.

Collaboration with Préfecture de Police on the performance evaluation of the call center “PFAU” (“Plate forme d'appels d'urgence”), to handle emergency calls to firemen and policemen in the Paris area. See Section 8.7.1 for more information.

11.1.1 Scientific events: organisation

11.1.2 Scientific events: selection

11.1.3 Journal

11.1.4 Invited talks

• Marianne Akian :
  • Talk in the minisymposium “Tropical Methods for Numerical Linear Algebra” at SIAM Conference on Applied Linear Algebra (LA24), May 13–17 , 2024, Paris, on “Spectral elements of positive definite matrices over symmetrized tropical algebra” (work with Stéphane Gaubert, Dariush Kiani, and Hanieh Tavakolipour).
  • Talk at the Dagstuhl seminar on Stochastic Games, on “Solving irreducible stochastic mean-payoff games and entropy games by relative Krasnoselskii-Mann iteration” ( Jun 02 – Jun 07, 2024).
  • Talk in the special session “Positive operators and their dynamics” at the 35th International Workshop on Operator Theory and its Applications (IWOTA 2024), University of Kent, Canterbury, UK, on “Escape rate games and competitive spectral radi” (work with Stéphane Gaubert and Loïc Marchesini.
  • Talk in the minisymposium “Solution of Hamilton-Jacobi Equations” at the 26th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2024) 19-23 August 2024, Cambridge, UK, on “Escape Rate Games” (work with Stéphane Gaubert and Loïc Marchesini.
• Matías Bender :
  • Talk at “Back to the roots” seminar of the Department of Electrical Engineering, KU Leuven, Belgium. March 2024.
  • Invitation to the BIRS program Representation Theory and Topological Data Analysis at Banff, Canada. April 7 - 12, 2024.
  • Invitation to the BIRS program Positive Solutions of Polynomial Systems Arising from Real-life Applications at Granada, Spain. May 19 - 24, 2024.
  • Talk at minisymposium “Numerical Linear Algebra Algorithms to Solve (Multivariate) Polynomial Systems” at the 2024 SIAM Conference on Applied Linear Algebra. Paris. May 2024.
  • Talk at the “International Workshop on Recent Advances in MAthematics (RAMA 2024)”, Hangzhou, China. November 2024.
• Antoine Béreau :
  • Talk in the special session “Polyhedral geometry and combinatorics” at International Congress on Mathematical Software 2024 (ICMS), Durham U., UK, 22-25 July, 2024, on “Eigenvalue Methods for Sparse Tropical Polynomial Systems” (work with Marianne Akian and Stéphane Gaubert).
• Stéphane Gaubert :
  • Plenary talk at the 35th International Workshop on Operator Theory and its Applications (IWOTA 2024), University of Kent, Canterbury, UK, on the topic of “Tropical geometry, operators and games”. The video is available here.
  • Plenary talk at the “CaLISTA Workshop Geometry-Informed Machine Learning”, on “Tropical convexity and games: regression, separation, and beyond”, École des Mines, Paris, Sep 2-5, 2024.
  • Talk at the Dagstuhl seminar on Stochastic Games, on “Tropical geometry, linear programming and games” ( Jun 02 – Jun 07, 2024).
  • Talk at the Dagstuhl seminar on Network calculus, on “Tropical geometry and piecewise-linear dynamical systems applied to the dimensioning of emergency call centers” ( April 02 – 04, 2024).
  • Talk in the minisymposium “Recent Trends in Tensors and Quantum Information” at SIAM Conference on Applied Linear Algebra (LA24), May 13–17 , 2024, Paris, on “Polynomial Computability of Spectral Radius of Nonnegative Weakly Irreducible Tensors”.

11.1.5 Research administration

11.2.1 Teaching

• 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.
• Matías Bender
  • Exercises classes for the first and second years of Bachelor program of École Polytechnique in the framework of a “Chargé d'ensegnement”.
• Antoine Béreau
  • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”.
• 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 (since Sep. 2024).
  • 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  72.
• Jonathan Hornewall
  • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”.

11.2.2 Supervision

• PhD in progress: Quentin Canu, registered at Univ. Paris Saclay since October 2020, thesis supervisor: Georges Gonthier (INRIA), cosupervision: Xavier Allamigeon and Pierre-Yves Strub (LIX).
• PhD of Antoine Bereau, registered at IPP (EDMH), started in September 2021, defended in November 2024. Thesis supervisor: Stéphane Gaubert , cosupervision: Marianne Akian .
• PhD of Nicolas Vandame, registered at IPP (EDMH), started in September 2021, defended in December 2024, supervisor: Stéphane Gaubert , cosupervision: Xavier Allamigeon .
• 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 .

11.2.3 Juries

• Marianne Akian
  • Chair of the Inria junior researchers hiring committee at Inria Bordeaux.
  • Jury of the PhD thesis of Christelle Kozaily, Université de Rennes, Oct. 29, 2024.
• Stéphane Gaubert
  • Jury of the PhD thesis of Antoine Béreau, Ecole polytechnique, November 2024.
  • Jury of the PhD thesis of Nicolas Vandame, Ecole polytechnique, December 2024.
  • Jury (president) of the PhD thesis of Lucas Gierczak, Ecole polytechnique, December 2024.
  • Jury of the PhD thesis of Maël Bompais, Université Paris-Saclay, December 2024.
  • Jury (reviewer) of the PhD thesis of Germano Schafaschek, Fakultät IV - Electrotechnik und Informatik, TU-Berlin, December 2024.

International journals

• 13 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, 1-38HALDOIback to text
• 14 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, 2963-2991HALDOIback to text
• 15 articleM.Marianne Akian, S.Stephane Gaubert and L.Louis Rowen. Semiring systems arising from hyperrings.Journal of Pure and Applied Algebra2286June 2024, 107584HALDOIback to text
• 16 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, 105236HALDOI
• 17 articleP.-C.Pierre-Cyril Aubin-Frankowski and S.Stéphane Gaubert. Tropical reproducing kernels and optimization.Integral Equations and Operator Theory96192024HAL
• 18 articleM. R.Matías R Bender and P.-J.Pierre-Jean Spaenlehauer. Dimension results for extremal-generic polynomial systems over complete toric varieties.Journal of Algebra6462024, 156-182HALDOIback to text
• 19 articleP.Piermarco Cannarsa, S.Stéphane Gaubert, C.Cristian Mendico and M.Marc Quincampoix. Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time.Mathematical Control and Related Fields1442024, 1275-1305HALDOIback to text
• 20 articleM.Maël Forcier, S.Stéphane Gaubert and V.Vincent Leclère. Exact quantization of multistage stochastic linear problems.SIAM Journal on Optimization341February 2024, 533-562HALDOI
• 21 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 Control2025. In press. HALback to text
• 22 articleQ.Quentin Jacquet, W.Wim van Ackooij, C.Clémence Alasseur and S.Stéphane Gaubert. Quadratic regularization of bilevel pricing problems and application to electricity retail markets.European Journal of Operational Research3133March 2024, 841-857HALDOIback to textback to text

International peer-reviewed conferences

• 23 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
• 24 inproceedingsM.Marianne Akian, S.Stéphane Gaubert and L.Loïc Marchesini. Escape Rate Games.26th International Symposium on Mathematical Theory of Networks and Systems (Extended Abstract)Cambridge (Angleterre), United KingdomAugust 2024HAL
• 25 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 Optimization15185Lecture Notes in Computer ScienceLille, FranceSpringer Nature SwitzerlandJanuary 2025, 83-93HALDOIback to textback to text

Reports & preprints

• 26 miscM.Marianne Akian, X.Xavier Allamigeon, S.Stéphane Gaubert and S.Sergei Sergeev. Signed tropicalization of polar cones.February 2024HALback to text
• 27 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 2024HALback to text
• 28 miscM.Marianne Akian, S.Stéphane Gaubert and L.Loïc Marchesini. The Competive Spectral Radius of Families of Nonexpansive Mappings.October 2024HALback to text
• 29 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
• 30 miscX.Xavier Allamigeon, P.Pascal Capetillo and S.Stéphane Gaubert. Stationary regimes of piecewise linear dynamical systems with priorities.November 2024HALback to text
• 31 miscM. R.Matías R Bender, L.Laurent Busé, C.Carles Checa and E.Elias Tsigaridas. Multigraded Castelnuovo-Mumford regularity and Groebner bases.July 2024HALback to text
• 32 miscM. R.Matías R Bender, L.Laurent Busé, C.Carles Checa and E.Elias Tsigaridas. Solving bihomogeneous polynomial systems with a zero-dimensional projection.January 2025HALback to textback to text
• 33 miscM. R.Matías R Bender, E.Elias Tsigaridas and C.Chaoping Zhu. Rational SOS certificates of any polynomial over its zero-dimensional gradient ideal.January 2025HALback to text
• 34 miscS.Stéphane Gaubert and Y.Yiannis Vlassopoulos. Directed Metric Structures arising in Large Language Models.May 2024HALback to text
• 35 miscQ.Quentin Jacquet, A.Agnes Bialecki, L. E.Laurent El Ghaoui, S.Stéphane Gaubert and R.Riadh Zorgati. Entropic Lower Bound of Cardinality for Sparse Optimization.April 2024HAL
• 36 miscB.Bas Lemmens and C.Cormac Walsh. Distance-preserving maps between bounded symmetric domains.2024HALback to text