2023Activity reportProject-TeamTROPICAL

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, Luz Pascal.

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.

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

Participants: Marianne Akian, Stéphane Gaubert, Shanqing Liu, Yang Qi.

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.

8.1.3 Relative Krasnoselskii-Mann iteration for irreducible stochastic mean-payoff games and entropy games

Participants: Marianne Akian, Stéphane Gaubert.

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.

8.1.4 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 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.

8.2.1 Volume in Funk geometries

Participants: Cormac Walsh.

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 ${ℝ}^n$ and a radius $r$, there is a unique point $x$ such that the Funk ball of radius $r$ centered at $x$ minimises the volume over all Funk balls of the same radius. It is natural to conjecture that this minimum volume, which depends on the convex body, it maximised when the body is a Euclidean ball. If this is true, one could recover Blaschke–Santaló inequality by letting the radius tend to zero, and the centro-affine isoperimetric inequality by letting the radius tend to infinity.

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 $F$ and $F^{\text{'}}$ are two faces of dimension $i-1$ and $i+1$, respectively, and $F\subset F^{\text{'}}$, then there are exactly two faces $G$ of dimension $i$ such that $F\subset G\subset F^{\text{'}}$. So, for each $i\in \{0,\cdots ,n-1\}$ and flag $f$, there is a unique flag, which we denote ${\theta }_{i}f$, that differs from $f$ only by having a different face of dimension $i$. The group $\langle {\theta }_0,\cdots ,{\theta }_{n-1}\rangle$ generated by these maps is known as the monodromy group or connection group of the polytope $P$. To express the second term in the asymptotics of the volume, we will need a particular element of this group, the complete flip$\theta :={\theta }_{n-1}\circ \cdots \circ {\theta }_0$. With this notation, the second term is then $\alpha r^{n-1}$, where

Here, we are identifying the facet ${\left(\theta f\right)}_{n-1}$ with the associated vertex of the dual polytope.

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.

8.2.2 Invariant Finsler metrics on symmetric spaces

Participants: Cormac Walsh.

This is joint work with Bas Lemmens (Kent).

We are interested in metrics on symmetric spaces, in particular Finsler metrics that are invariant under the symmetries of the space. In a previous paper  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?

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 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.

8.3.2 Linear algebra over systems

Participants: Marianne Akian, Stéphane Gaubert.

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”.

8.3.3 Ambitropical convexity and Shapley retracts

Participants: Marianne Akian, Stéphane Gaubert.

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 ${ℝ}^n$. 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. Moreover, 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. We also showed that a cone is ambitropical if and only if it is hyperconvex. This is a joint work with Sara Vannucci (Praha). See 27.

8.3.4 Tropical linear regression and applications

Participants: Marianne Akian, Stéphane Gaubert, Yang Qi, Omar Saadi.

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.

8.3.5 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  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.

8.3.6 Tropical Systems of Polynomial Equations

Participants: Marianne Akian, Antoine Bereau, Stéphane Gaubert.

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.

8.3.7 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 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.

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  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 $n$-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, 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 $O(2^n{n}^{1.5}logn)$ 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  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).

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 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.

8.4.4 Bounds on roots of (multivariate) polynomial systems

Participants: Marianne Akian, Stéphane Gaubert.

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.

8.5.1 Algebraic complexity of low-rank approximations

Participants: Yang Qi.

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.

8.7.1 Performance evaluation of emergency call centers and emergency services

Participants: Xavier Allamigeon, Pascal Capetillo, Stéphane Gaubert, Benjamin Nguyen-Van-Yen.

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).

8.7.2 Optimal pricing of energy contracts

Participants: Stéphane Gaubert, Quentin Jacquet.

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.

10.1.1 Participation in other International Programs

10.2.1 Visits of international scientists

10.2.2 Visits to international teams

10.3.1 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).

10.3.2 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.

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

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

11.1.1 Scientific events: organisation

11.1.2 Scientific events: selection

11.1.3 Journal

11.1.4 Invited talks

• Marianne Akian :
  • Minisymposium “Manifestations of the Mixed Volume” at SIAM Conference on Applied Algebraic Geometry (AG23), July 2023, Eindhoven, The Netherlands. Talk : “Cauchy and Lagrange Bounds for Multivariate Polynomials” (work with Stéphane Gaubert and Gregorio Malajovich).
  • Minisymposium “Numerical methods for Hamilton-Jacobi equations and their applications” at 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), August 2023, Tokyo, Japan. Talk: “Tropical and multi-level numerical methods for solving optimal control problems” (work with Stéphane Gaubert and Shanqing Liu).
  • Invited talk at Mathematics & Decision Conference, 11-15 December 2023, UM6P Ben Guerir, Morocco. Title: “Repeated zero-sum games: from operators to algorithms and tropical geometry”.
• Xavier Allamigeon
  • Journée des lauréats des prix de l'Académie des Sciences, section “Mécanique et Informatique”, Paris, France, mai 2023. Titre : A Formal Disproof of the Hirsch Conjecture.
  • Exposé invité à la conférence “Highlights of Algorithms”, Prague, République Tchèque, juin 2023. Titre : No self-concordant barrier interior point method is strongly polynomial.
  • Minisymposium “Computational Tropical Geometry” at SIAM Conference on Applied Algebraic Geometry (AG23), July 2023, Eindhoven, The Netherlands. Talk: “Tropical Nash Equilibria and Oriented Matroids” (work with Stéphane Gaubert and Frédéric Meunier).
  • Exposé invité au 27ème séminaire “Polyèdres et Optimisation Combinatoire”, Paris, France, décembre 2023. Titre : No self-concordant barrier interior point method is strongly polynomial.
• Matìas Bender
  • Minisymposium “Manifestations of the Mixed Volume” at SIAM Conference on Applied Algebraic Geometry (AG23), July 2023, Eindhoven, The Netherlands. Talk: “Dimension Results for Polynomial Systems Over Complete Toric Varieties”.
  • Minisymposium “Recent Advances on Polynomial System Solving” at 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), August 2023, Tokyo, Japan. Talk: “Dimension Results for Polynomial Systems Over Complete Toric Varieties”.
  • Weekly seminar of the FCFS at Universidad Autónoma de Chiapas.
• Antoine Béreau :
  • Minisymposium “Manifestations of the Mixed Volume” at SIAM Conference on Applied Algebraic Geometry (AG23), July 2023, Eindhoven, The Netherlands. Talk: “The Tropical Nullstellensatz and Positivstellensatz for Sparse Polynomial Systems” (work with Marianne Akian and Stéphane Gaubert).
• Stéphane Gaubert :
  • 25th Conference of the International Linear Algebra Society, Madrid, June 2023 (Invited Minisymposium). Talk: “Turing-model complexity estimates in geometric programming and application to the computation of the spectral radius of nonnegative tensors.”.
  • Jon-Shmuel Halfway to Twelfty celebration, ENPC, July 2023. Talk: “Ambitropical convexity, injective hulls of metric spaces, and mean-payoff games.”
  • Minisymposium “Computational Tropical Geometry” at SIAM Conference on Applied Algebraic Geometry (AG23), July 2023, Eindhoven, The Netherlands. Talk: “Ambitropical Convexity, Injective Hulls and Mean-Payoff Games” (work with Marianne Akian and Sara Vannucci).
  • Minisymposium “Advances in Optimization” at 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), August 2023, Tokyo, Japan. Talk: “Tropical convexity: application to linear programming and mean-payoff games”.
  • Combinatorics and Arithmetic for Physics, IHES, Nov. 2023. Talk: “The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems, and Parametric Mean-Payoff Games” (joint work with Marianne Akian and Antoine Béreau).
• Yang Qi :
  • Optimization Seminar, Telecom Paris, April 2023, Talk: “Tropical linear regression and low-rank approximation — a first step in tropical data analysis”.
  • MC2 Seminar, ENS Lyon, May 2023, Talk: “Tropical linear regression and low-rank approximation — a first step in tropical data analysis”.
  • Frontiers of computational mathematics, Kunming, China, August 2023, Talk: “Tropical data analysis”.
• Nicolas Vandame :
  • Minisymposium “Computational Tropical Geometry” at SIAM Conference on Applied Algebraic Geometry (AG23), July 2023, Eindhoven, The Netherlands. Talk: “No Self-Concordant Barrier Interior Point Method Is Strongly Polynomial” (work with Stéphane Gaubert and Xavier Allamigeon).
• Cormac Walsh :
  • “Workshop on Ordered Vector Spaces and Positive Operators”, Wuppertal, Germany, 29 March–1 April 2023. Talk: “Order and metric structures on cones”.

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.
  • Co-responsabilité du programme d'approfondissement en mathématiques appliquées (troisième année) à l'École Polytechnique (jusqu'à septembre 2023).
• Matìas Bender
  • Exercises classes for the second year of Bachelor program of Ecole polytechnique in the framework of a “Chargé d'ensegnement”.
  • Course “Computational commutative algebra” at the CIMPA summer school Algebraic and Tropical Methods for Solving Differential Equations, Oaxaca, Mexico.
• 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”.
• Stéphane Gaubert
  • Co-head of the Master “Optimization” of University Paris-Saclay and IPP.
  • 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  61.
• Shanqing Liu
  • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”.
• Nicolas Vandame
  • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”.

11.2.2 Supervision

• Matìas Bender directed the M2 thesis of Tom Baumbach at the Technische Universität Berlin.
• PhD: Quentin Jacquet, registered at IPP (EDMH) since November 2020, thesis supervisor, Stéphane Gaubert , co-supervised by Clémence Alasseur and Wim van Ackooij. The defense took place on October 24, 2023.
• PhD: Shanqing Liu, registered at IPP (EDMH) since September 2020, thesis supervisor, Marianne Akian , co-supervised by Stéphane Gaubert . The defense took place on December 22, 2023.
• 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 in progress: Antoine Bereau, registered at IPP (EDMH) since September 2021. Thesis supervisor: Stéphane Gaubert, cosupervision: Marianne Akian.
• PhD in progress: Nicolas Vandame, registered at IPP (EDMH) since September 2021, thesis 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: Loic 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.

11.2.3 Juries

• Marianne Akian
  • Jury (and reviewer) of the HDR of Vincent Leclere, Ecole des Ponts ParisTech, April 17, 2023.
  • Jury of the PhD thesis of Christina Katsamaki, Sorbonne Univerity, June 21, 2023.
  • Jury of the PhD thesis of Manh Hung LE, Université de Limoges, Dec. 20, 2023.
  • Jury of the PhD thesis of Shanqing Liu, Ecole polytechnique, Dec. 22, 2023.
• Stéphane Gaubert
  • Member of the HCERES visiting committee of CEREMADE, October 2023.
  • Member of “Comité de Sélection” (MdC, Optimization) at Nice, Spring 2023.
  • Jury of the PhD thesis of Quentin Jacquet, Ecole polytechnique, October 24, 2023.
  • Jury the PhD thesis of Francisco Venegas, University of Chile, July 2023.

11.2.4 Communications at conferences and seminars

• Marianne Akian
  • Minisymposium “Équations d’Hamilton-Jacobi-Bellman et de jeux à champ moyen” à SMAI 2023, May 22-26, 2023, Le Gosier en Guadeloupe. Exposé: “A multi-level fast-marching method for the minimum time problem” (travail avec Stéphane Gaubert et Shanqing Liu).
  • 13th ISDG Workshop, June 7-9, 2023, Naples, Italy, Session “Numerics and Theory for Differential Games, in Memory of Maurizio Falcone”, Talk : “Solving irreducible stochastic mean-payoff games and entropy games by relative Krasnoselskii-Mann iteration”.
  • 17th International Conference on Reachability Problems (RP 2023). Talk : “Solving irreducible stochastic mean-payoff games and entropy games by relative Krasnoselskii-Mann iteration”.
• Antoine Béreau :
  • Journées nationales de calcul formel, Marseille, March 2023. Talk: “The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems”.
  • Rencontres Doctorales Lebesgue 2023, Nantes, April 2023. Exposé: “Un tour d’horizon des mathématiques tropicales”.
  • International Symposium on Symbolic and Algebraic Computation 2023, July 24-27, 2023. Talk: “The Tropical Nullstenllensatz and Positivstellensatz for Sparse Polynomial Systems” (work with Marianne Akian and Stéphane Gaubert).
• Stéphane Gaubert :
  • 48TH International symposium on mathematical foundations of computer science (MFCS 2023). Talk : “Solving irreducible stochastic mean-payoff games and entropy games by relative Krasnoselskii-Mann iteration”.
  • Seminar of of the Dyogene INRIA Team, “Nonarchimedean convexity and applications”, April 2023.
  • Seminar of the verification group at IRIF, “Entropy games”, May 2023.
• Shanqing Liu :
  • IFAC World Congress 2023, Yokohama, Japan. Talk: “An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems”.
  • PGMO days, Palaiseau, November 2023. Talk: “Semiconcave Dual Dynamic Programming and Its Application to N-body Problems” (work with Marianne Akian, Stéphane Gaubert and Yang Qi).
• Cormac Walsh :
  • “Positivity XI”, Ljubljana, Slovenia, 10–14 July 2023. Talk: “Hilbert geometries isometric to Banach spaces”.

International journals

• 13 articleM.Marianne Akian, S.Stéphane Gaubert, Y.Yang Qi and O.Omar Saadi. Tropical linear regression and mean payoff games: or, how to measure the distance to equilibria.SIAM Journal on Discrete Mathematics372May 2023, 632-674HALDOIback to text
• 14 articleM.Marianne Akian, S.Stephane Gaubert and L.Louis Rowen. Semiring systems arising from hyperrings.Journal of Pure and Applied Algebra2286June 2024, 107584HALDOIback to textback to text
• 15 articleX.Xavier Allamigeon, S.Stéphane Gaubert and F.Frédéric Meunier. Tropical Complementarity Problems and Nash Equilibria.SIAM Journal on Discrete Mathematics373August 2023, 1645-1665HALDOI
• 16 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

• 17 inproceedingsM.Marianne Akian, A.Antoine Béreau and S.Stéphane Gaubert. The Tropical Nullstellensatz and Positivstellensatz for Sparse Polynomial Systems.ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic ComputationISSAC 2023 - International Symposium on Symbolic and Algebraic ComputationTromsø, NorwayACM2023HALDOIback to text
• 18 inproceedingsM.Marianne Akian, S.Stéphane Gaubert and S.Shanqing Liu. An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems.Proceedings of the 22nd IFAC World CongressIFAC World Congress 2023Yokohama, JapanJuly 2023HALback to text
• 19 inproceedingsM.Marianne Akian, S.Stéphane Gaubert, U.Ulysse Naepels and B.Basile Terver. Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration.MFCS 2023 - 48th International Symposium on Mathematical Foundations of Computer Science272Bordeaux, FranceSchloss Dagstuhl – Leibniz-Zentrum für InformatikAugust 2023HALDOIback to text
• 20 inproceedingsX.Xavier Allamigeon, Q.Quentin Canu and P.-Y.Pierre-Yves Strub. A Formal Disproof of Hirsch Conjecture.Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and ProofsCPP 2023 - 12th ACM SIGPLAN International Conference on Certified Programs and ProofsBoston, United StatesACMJanuary 2023, 17-29HALDOIback to text
• 21 inproceedingsQ.Quentin Jacquet, W.Wim van Ackooij, C.Clémence Alasseur and S.Stéphane Gaubert. A Quantization Procedure for Nonlinear Pricing with an Application to Electricity Markets.Proceedings of IEEE CDC 202362nd IEEE Conference on Decision and ControlSingapore (SG), SingaporeDecember 2023HALback to text

Reports & preprints

• 22 miscM.Marianne Akian, X.Xavier Allamigeon, S.Stéphane Gaubert and S.Sergei Sergeev. Signed tropicalization of polar cones.December 2023HALback to text
• 23 miscM.Marianne Akian, A.Antoine Béreau and S.Stéphane Gaubert. The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems.December 2023HALback to text
• 24 miscM.Marianne Akian, S.Stéphane Gaubert and S.Shanqing Liu. A Multi-Level Fast-Marching Method For The Minimum Time Problem.March 2023HALback to text
• 25 miscM.Marianne Akian, S.Stephane Gaubert and L.Louis Rowen. Linear algebra over T-pairs.October 2023HALback to text
• 26 miscM.Marianne Akian, S.Stephane Gaubert and H.Hanieh Tavakolipour. Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields.January 2023HALback to text
• 27 miscM.Marianne Akian, S.Stéphane Gaubert and S.Sara Vannucci. Ambitropical geometry, hyperconvexity and zero-sum games.July 2023HALback to text
• 28 miscC.Clémence Alasseur, E.Erhan Bayraktar, R.Roxana Dumitrescu and Q.Quentin Jacquet. A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings.July 2023HALback to text
• 29 miscX.Xavier Allamigeon, Q.Quentin Canu, C.Cyril Cohen, K.Kazuhiko Sakaguchi and P.-Y.Pierre-Yves Strub. Design patterns of hierarchies for order structures.February 2023HALback to text
• 30 miscM. R.Matías R Bender and P.-J.Pierre-Jean Spaenlehauer. Dimension results for extremal-generic polynomial systems over complete toric varieties.May 2023HALback to text
• 31 miscP.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.June 2023HAL
• 32 miscD.Dmitry Faifman, C.Constantin Vernicos and C.Cormac Walsh. Volume growth of Funk geometry and the flags of polytopes.June 2023HALback to text
• 33 miscS.Shmuel Friedland and S.Stéphane Gaubert. Bit-complexity estimates in geometric programming, and application to the polynomial-time computation of the spectral radius of nonnegative tensors.January 2023HAL
• 34 miscK.Khazhgali Kozhasov, A.Alan Muniz, Y.Yang Qi and L.Luca Sodomaco. On the minimal algebraic complexity of the rank-one approximation problem for general inner products.September 2023HALback to text