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
• News of the Year:

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

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

• URL:
  https://­github.­com/­nhojem/­Coq-Polyhedra
• Publications:
  hal-01673390, hal-03151656, hal-03505724, 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, Baptiste Colin

8.1.1 Multiply Accelerated Value Iteration Algorithms For Classes of Markov Decision Processes

Participants: Marianne Akian, Stéphane Gaubert, Omar Saadi.

8.1.2 Polyhedral representation of multi-stage stochastic linear problems

Participants: Maël Forcier, Stéphane Gaubert.

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

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

Participants: Marianne Akian, Stéphane Gaubert, Shanqing Liu.

Hamilton-Jacobi-Bellman equations arise as the dynamic programming equations of deterministic or stochastic optimal control problems. They allow to obtain the global optimum of these problems and to synthetize an optimal feedback control, leading to a solution robust against system perturbations. Several methods have been proposed in the litterature to bypass the obstruction of curse of dimensionality of such equations, assuming a certain structure of the problem, and/or using “unstructured discretizations”, that are not based on given grids. Among them, one may cite tropical numerical method, and probabilistic numerical method. On another direction, “highway hierarchies”, developped by Sanders, Schultes and coworkers  60, 90, initially for applications to on-board GPS systems, are a computational method that allows one to accelerate Dijkstra algorithm for discrete time and state shortest path problems.

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

In a first work, we have developped a multilevel fast-marching method, extending to the PDE case the idea of “highway hierarchies”. Given the problem of finding an optimal trajectory between two given points, the method consists in refining the grid only in a neighborhood of the optimal trajectory, which is itself computed using an approximation of the value function on a coarse grid. A first account of this work has been presentend to the PGMO days in December 2021.

8.2.1 Volume in Hilbert and Funk geometries

Participants: Constantin Vernicos, Cormac Walsh.

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

where $B(x,r)$ is the metric ball with center $x$ and radius $r$, and $Vol$ denotes the Holmes–Thompson volume. Note that the volume entropy does not depend on the particular choice of $x$. We showed that the volume entropy is exactly twice the flag-approximability of the convex body. This is a new notion of approximability we introduced that measures the complexity of a polytope by counting its number of flags rather than its number of vertices. A corollary is that the Euclidean ball has the maximal volume entropy among Hilbert geometries of a given dimension, a fact that was recently proved by Tholozan by other means. we also showed that the rate of growth of the volume is minimised when the convex body is a simplex.

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

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

8.2.2 Intrinsic charaterization of Hilbert geometries

Participants: Constantin Vernicos.

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

In a Hilbert geometry, if one knows the shape of a metric ball of radius $R$ at some point, then one knows the geometry it lives in. However knowing the tangent unit ball at one point is not enough. In dimension one a simple computation shows that one needs to know the tangent ball at two points. Hence we conjecture that in dimension $n$ one needs to know the shape of tangent ball at $n+1$ points to be able to characterize the convex sets defining the geometry. We were able to prove the conjecture for polytopes in dimension 2, and we are currently looking at smooth convex sets and higher dimension.

8.2.3 Reflexion and refraction in Finsler geometry

Participants: Constantin Vernicos.

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

8.2.4 Barycentres in Funk/Hilbert geometries

Participants: Constantin Vernicos.

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

8.2.5 Funk and Hilbert geometries of tropical convex sets

Participants: Constantin Vernicos, Stéphane Gaubert.

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

8.2.6 Firm non-expansive mappings

Participants: Armando Gutiérrez, Cormac Walsh.

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

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

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

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

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

8.2.7 Metric functionals and invariant spaces

Participants: Armando Gutiérrez.

More precisely, the first result is that for every non-expansive mapping $T$ in ${\ell }_1$, there exists a non-trivial continuous linear functional $f$ such that $f\left(T^{n}0\right)\ge 0$ for all $n\ge 0$. The second result is that for every affine non-expansive mapping $T=U+T0$ in a Hilbert space, if 0 is not an element of the norm closure of the set Ran(I-T), then there exists a co-dimension one closed invariant subspace for the linear operator $U$.

8.3.1 Formalizing convex polyhedra in Coq

Participants: Xavier Allamigeon, Quentin Canu.

In a joint work with Ricardo Katz (Conicet, Argentina) and Pierre-Yves Strub (LIX, Ecole Polytechnique), we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library providing the basic constructions and operations over polyhedra, including projections, convex hulls and images under linear maps. Moreover, we design a special mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. We demonstrate the usability of this approach by establishing some of the most important combinatorial properties of faces, namely that they constitute a family of graded atomistic and coatomistic lattices closed under sublattices. We also prove a theorem due to Balinski on the $d$-connectedness of the adjacency graph of polytopes of dimension $d$. This is implemented in the CoqPolyhedra library (we refer to the software session for more details). This work has been published in 10th International Joint Conference on Automated Reasoning  47, and is currently accepted (under minor revisions) in a special issue of Logical Methods in Computer Science.

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

8.3.2 Linear algebra over systems

Participants: Marianne Akian, Stéphane Gaubert.

In a joint work with Louis Rowen (Univ. Bar Ilan), we study linear algebra and convexity properties over “systems”. The latter provide a general setting encompassing extensions of the tropical semifields and hyperfields. Moreover, they have the advantage to be well adapted to the study of linear or polynomial equations.

8.3.3 Ambitropical convexity and Shapley retracts

Participants: Marianne Akian, Stéphane Gaubert.

8.3.4 Tropical linear regression and applications

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

8.3.5 Eigenvalues of Tropical Symmetric Matrices

Participants: Marianne Akian, Stéphane Gaubert, Hanieh Tavakolipour.

The tropical semifield can be thought of as the image of a field with a non-archimedean valuation. It allows in this way to study the asymptotics of Puiseux series with complex coefficients. When dealing with Puiseux series with real coefficients and with its associated order, it is convenient to use the symmetrized tropical semiring introduced in  86 (see also 48), and the signed valuation which associates to any series its valuation together with its sign.

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

8.3.6 Tropical Systems of Polynomial Equations

Participants: Marianne Akian, Antoine Bereau, Stéphane Gaubert.

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 6, 14 that the answer is negative.

In a joint work  38 with Abdellah Aznag (Columbia University) and Yassine Hamdi (Ecole Polytechnique), we have studied the tropicalization of the central path associated with the entropic barrier studied by Bubeck and Eldan (Proc. Mach. Learn. Research, 2015), i.e., the logarithmic limit of this central path for a parametric family of linear programs defined over the field of Puiseux series. Our main result is that the tropicalization of the entropic central path is a piecewise linear curve which coincides with the tropicalization of the logarithmic central path studied by Allamigeon et al. in 6.

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

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  44 that the analogue of this problem in tropical algebra can be solved in polynomial time. Moreover, we prove that the Lemke–Howson algorithm carries over the tropical setting and performs a linear number of pivots in the worst case. A consequence of this result is a new class of (classical) bimatrix games for which Nash equilibria computation can be done in polynomial time. This is joint work with Frédéric Meunier (Cermics, ENPC).

8.5.1 Topology of tropical ranks

Participants: Yang Qi.

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

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

8.5.2 Approximation theory of neural networks

Participants: Yang Qi.

In 18 (joint work with Lek-Heng Lim and Mateusz Michałek) we show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. More precisely, given a training set $s_1,\cdots ,s_n\in {ℝ}^p$ with corresponding responses $t_1,\cdots ,t_n\in {ℝ}^q$, fitting a $k$-layer neural network ${\nu }_{\theta }:{ℝ}^p\to {ℝ}^q$ involves estimation of the weights $\theta \in {ℝ}^m$ via an ERM:

We show that even for $k=2$, this infimum is not attainable in general for common activations like ReLU, hyperbolic tangent, and sigmoid functions. In addition, we show that for smooth activations $\sigma \left(x\right)=1/\left(1+exp(-x)\right)$ and $\sigma \left(x\right)=tanh\left(x\right)$, such failure to attain an infimum can happen on a positive-measured subset of responses. For the ReLU activation $\sigma \left(x\right)=max(0,x)$, we completely classify cases where the ERM for a best two-layer neural network approximation attains its infimum. In recent applications of neural networks, where overfitting is commonplace, the failure to attain an infimum is avoided by ensuring that the system of equations $t_i={\nu }_{\theta }\left(s_i\right)$, $i=1,\cdots ,n$, has a solution. For a two-layer ReLU-activated network, we show when such a system of equations has a solution generically, i.e., when can such a neural network be fitted perfectly with probability one.

8.6.1 Performance evaluation of emergency call centers

Participants: Xavier Allamigeon, Marin Boyet, Baptiste Colin, Stéphane Gaubert.

Since 2014, we have been collaborating with Préfecture de Police (Régis Reboul and LcL Stéphane Raclot), more specifically with Brigade de Sapeurs de Pompiers de Paris (BSPP) and Direction de Sécurité de Proximité de l'agglomération parisienne (DSPAP), on the performance evaluation of the new organization (PFAU, “Plate forme d'appels d'urgence”) to handle emergency calls to firemen and policemen in the Paris area. We developed analytical models, based on Petri nets with priorities, and fluid limits, see   40, 41, 51. In 2019, with four students of École polytechnique, Céline Moucer, Julia Escribe, Skandère Sahli and Alban Zammit, we performed case studies, showing the improvement brought by the two level filtering procedure.

Moreover, in 2019, this work has been extended to encompass the handling of health emergency calls, with a new collaboration, involving responsibles from the four services of medical emergency aid of Assistance Publique – Hôpitaux de Paris (APHP), i.e., with SAMU75, 92, 93, 94, in the framework of a project coordinated by Dr. Christophe Leroy from APHP. As part of his PhD work, Marin Boyet have developed Petri net models capturing the characteristic of the centers (CRRA) handling emergency calls the SAMU, in order to make dimensioning recommendations. Following this, we have been strongly solicited by APHP during the pandemic of Covid-19 in order to determine crisis dimensioning of SAMU. Besides, we have initiated a new collaboration, with SAMU69, also on dimensioning.

In parallel, we have further investigated the theoretical properties of timed Petri nets with preselection and priority routing. We represent the behavior of these systems by piecewise affine dynamical systems. We use tools from the theory of nonexpansive mappings to analyze these systems. We establish an equivalence theorem between priority-free fluid timed Petri nets and semi-Markov decision processes, from which we derive the convergence to a periodic regime and the polynomial-time computability of the throughput. More generally, we develop an approach inspired by tropical geometry, characterizing the congestion phases as the cells of a polyhedral complex. These results are illustrated by the application to the performance evaluation of emergency call centers of SAMU in the Paris area. These results have been published in 15.

8.6.2 Optimal control of energy flexibilities in a stochastic environment

Participants: Maxime Grangereau, Stéphane Gaubert.

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

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

8.6.3 Bilevel programming applied to optimal pricing of energy contracts

Stéphane Gaubert , Quentin Jacquet

The PhD thesis of Quentin Jacquet, is cosupervised by Stéphane Gaubert, Clémence Alasseur (EDF Labs), and Wim van Ackooij (EDF Labs). It concerns the application of bilevel programming methods to the pricing of electricity contracts. We investigated in 31 a new model of customer's response, based on a quadratic regularization. We showed that this model has qualitative properties and a realism similar to the classical models based on the logit-response, while being amenable to mathematical programming and polyhedral techniques, and so to exact solutions, via a reduction to quadratic complementary problems. An application to a set of instances representative of French electricity contracts was also developed 31.

8.6.4 POMDP and adaptive management

Participants: Marianne Akian.

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

10.1.1 STIC/MATH/CLIMAT AmSud project

Participants: Marianne Akian, Stéphane Gaubert.

• Math AmSud Project ARGO, “Algebraic Real Geometry and Optimization” (2020-2021) between CMM (Chile), Univ. Buenos Aires (Argentina), Univ. Fed. Rio and Univ. Fed. Ceara (Brasil), Univ Savoie and CMAP, Ecole polytechnique (France), including in particular M. Akian and S. Gaubert of Tropical team.

10.1.2 Bilateral project FACCTS with the University of Chicago

Participants: Stéphane Gaubert.

• Bilateral project FACCTS, between the University of Chicago (Statistics) – Lek-Heng Lim– and Ecole polytechnique – Stéphane Gaubert– “Tropical geometry of deep learning”. This project was postoned owing to the Covid-19 crisis.

10.2.1 Visits to international teams

Participants: Xavier Allamigeon.

10.3.1 ANR

Participants: Xavier Allamigeon.

• Project ANR JCJC CAPPS (“Combinatorial Analysis of Polytopes and Polyhedral Subdivisions”). Responsable: Arnau Padrol (IMJ-PRG, Sorbonne Université). Partners : IMJ-PRG (Sorbonne Université), INRIA Saclay (Tropical), LIGM (Université Paris-Est Marne-la-Vallée), LIF (Université Aix-Marseille), CERMICS (École Nationale des Ponts et Chaussées), LIX (École Polytechnique).

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

Participants: Xavier Allamigeon, Stéphane Gaubert.

• Méthodes tropicales pour le dimensionnement de centres d'appels : application à un centre de supervision EDF. Participants : X. Allamigeon S. Gaubert, P. Bendotti (EDF) et T. Triboulet (EDF).

10.3.3 Centre des Hautes Études du Ministère de l'Intérieur

• Project “Optimisation de la performance de centres de traitement d'appels d'urgence en cas d’événements planifiés ou imprévus”, coordinated by X. Allamigeon, involving M. Boyet, B. Colin and S. Gaubert.

11.1.1 Scientific events: organisation

11.1.2 Scientific events: selection

11.1.3 Journal

11.1.4 Invited talks

• M. Akian:
  • MCA 2021 (Mathematical Congress of the Americas), July 12-16, 2021, online. Session “Mathematics of Planet Earth’’, Talk: “Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area”.
  • Online talk: “Tropical linear regression and mean payoff games: or, how to measure the distance to equilibria” at Workshop “Tropical geometry and the geometry of linear programming”, Hausdorff Institute for Mathematics, Sep. 2021, Bonn, Germany.
• S. Gaubert:
  • “Ambitropical convexity, mean payoff games and nonarchimedean convex programming”. Combinatorics and Arithmetic for Physics: special days, IHES, Dec. 2021.
  • Online talk: “Tropical convexity and its relation with mean payoff games and linear programming” at Workshop “Tropical geometry and the geometry of linear programming”, Hausdorff Institute for Mathematics, Sep. 2021, Bonn, Germany.
  • Talk on “Ambitropical convexity, mean payoff games and nonarchimedean convex programming”, Combinatorics and Arithmetic for Physics: special days, IHES, Dec. 2021.
• C. Vernicos:
  • Talk for the "Séminaire virtuel francophone Groupes et Géométrie”, 05/06/2021, "Entropie volumique des géométries de Hilbert et approximabilité".
  • Conference in Cortona, Italy, on ""Gromov hyperbolicity and negative curvature in Complex Analysis" 09/6-09/10 2021. Talk on "Flag appoximability and volume Entropy of Hilbert geometries".
• Y. Qi:
  • “On best approximations of high-dimensional semialgebraic data sets”, HCM Symposium, Hausdorff Center for Mathematics, August 26, 2021.
  • “Tropical linear regression and low-rank approximation – a first step in tropical data analysis”, TATERS, Boise State University, December 3, 2021.

11.1.5 Leadership within the scientific community

See Section 11.1.1.

11.1.6 Scientific expertise

• Marianne Akian: Scientific expertise for Datastorm and Air Liquide on the research and development works of Air Liquide in optimization of Air Separation Units (dec. 2020-jan. 2021).

11.1.7 Research administration

11.2.1 Teaching

• M. 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.
• X. 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 du M2 “Optimisation” de l'Université Paris Saclay, cours partagé avec Céline Gicquel (LRI, Université Paris Sud).
  • Co-responsabilité du programme d'approfondissement en mathématiques appliquées (troisi-ème année) à l'École Polytechnique.
• A. Bereau
  • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”.
• S. 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  52.
• S. Liu
  • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”.
• O. Saadi
  • Exercises classes in the framework of a “Monitorat”.
• N. Vandame
  • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”.

11.2.2 Supervision

• PhD: Maxime Grangereau, registered at Univ. Paris Saclay, thesis supervisor: Emanuel Gobet, cosupervision: Stéphane Gaubert. The defense took place on March 23, 2021.
• PhD: Omar Saadi, registered at Univ. Paris Saclay thesis supervisor: Stéphane Gaubert, cosupervision: Marianne Akian. The defense took place on December 17, 2021.
• PhD in progress: Marin Boyet, registered at Univ. Paris Saclay since October 2018, thesis supervisor: Stéphane Gaubert, cosupervision: Xavier Allamigeon.
• PhD in progress: Maël Forcier, registered at ENPC since September 2019, thesis supervisor: Vincent Leclère, cosupervision Stéphane Gaubert.
• 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: Shanqing Liu, registered at IPP (EDMH) since September 2020, thesis supervisor, M. Akian, co-supervised by S. Gaubert.
• PhD in progress: Quentin Jacquet, registered at IPP (EDMH) since November 2020, thesis supervisor, S. Gaubert, co-supervised by Clémence Alasseur and Wim van Ackooij.
• PhD in progress: Tom Ferragut, on the horosphérical products of Gromov-hyperbolic and Busemann convex metric spaces, registered at U. Montpellier since 2018, thesis supervisor: C. Vernicos, co-supervisor Jérémie Brieussel.
• 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.

11.2.3 Juries

• M. Akian
  • Jury of the 2021 competition for a Maître de Conférence position in Applied Mathematics at Limoges University.
  • Jury of the 2021 competition for a Maître de Conférence position in Computer Science (Optimization) at Clermont University.
  • Jury of the PhD thesis of Dylan Dronnier (U. Paris Est, ENPC), Nov. 26, 2021.
  • Review and Jury of the PhD thesis of Adrien Lefranc (U. Paris Est, ENPC), Dec. 8, 2021.
  • Jury of the PhD thesis of Aurélien Desoeuvres (U. Montpellier), Dec. 9, 2021.
  • Jury of the PhD of Omar Saadi, Ecole polytechnique, Dec. 17, 2020.
• S. Gaubert
  • Jury of the PhD of Maxime Grangerau, Ecole polytechnique, March 23, 2021.
  • Jury of the PhD of Sebastian Tapia, Université de Bordeaux et Universida del Chile, November 8, 2021.
  • Jury of the PhD of Omar Saadi, Ecole polytechnique, Dec. 17, 2020.

International journals

• 13 articleM.Marianne Akian, S.Stéphane Gaubert, Z.Zheng Qu and O.Omar Saadi. Multiply Accelerated Value Iteration for Non-Symmetric Affine Fixed Point Problems and application to Markov Decision Processes.SIAM Journal on Matrix Analysis and Applications2021
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• 14 articleX.Xavier Allamigeon, P.Pascal Benchimol, S.Stéphane Gaubert and M.Michael Joswig. What Tropical Geometry Tells Us about the Complexity of Linear Programming.SIAM Review631February 2021, 123-164
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• 15 articleX.Xavier Allamigeon, M.Marin Boyet and S.Stéphane Gaubert. Piecewise Affine Dynamical Models of Timed Petri Nets -- Application to Emergency Call Centers.Fundamenta Informaticae1833-42021, 169-201
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• 16 articleM.Maxime Grangereau, W.Wim Van Ackooij and S.Stéphane Gaubert. Multi-stage Stochastic Alternating Current Optimal Power Flow with Storage: Bounding the Relaxation Gap.Electric Power Systems Research206January 2022, 107774
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• 17 articleA. W.Armando W. Gutiérrez and A.Anders Karlsson. Comments on the cosmic convergence of nonexpansive maps.Journal of Fixed Point Theory and Applications234November 2021
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• 18 articleL.-H.Lek-Heng Lim, M.Mateusz Michalek and Y.Yang Qi. Best $k$-layer neural network approximations.Constructive ApproximationJune 2021
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• 19 articleC.Constantin Vernicos and C.Cormac Walsh. Flag-approximability of convex bodies and volume growth of Hilbert geometries.Annales Scientifiques de l'École Normale Supérieure542021, 1297-1315
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International peer-reviewed conferences

• 20 inproceedingsL. V.Luz V. Pascal, M.Marianne Akian, S.Sam Nicol and I.Iadine Chades. A Universal 2-state n-action Adaptive Management Solver.Proceedings of the AAAI Conference on Artificial Intelligence3517virtual conference, United StatesMay 2021, 14884-14892
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Conferences without proceedings

• 21 inproceedingsR.Reynald Affeldt, X.Xavier Allamigeon, Y.Yves Bertot, Q.Quentin Canu, C.Cyril Cohen, P.Pierre Roux, K.Kazuhiko Sakaguchi, E.Enrico Tassi, L.Laurent Théry and A.Anton Trunov. Porting the Mathematical Components library to Hierarchy Builder.the COQ Workshop 2021virtuel- Rome, ItalyJuly 2021
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Reports & preprints

• 22 miscM.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.June 2021
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• 23 miscM.Marianne Akian, S.Stephane Gaubert and S.Sara Vannucci. Ambitropical convexity: The geometry of fixed point sets of Shapley operators.August 2021
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• 24 miscX.Xavier Allamigeon, R. D.Ricardo D. Katz and P.-Y.Pierre-Yves Strub. Formalizing the Face Lattice of Polyhedra.December 2021
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• 25 miscM.Maël Forcier, S.Stéphane Gaubert and V.Vincent Leclère. Exact quantization of multistage stochastic linear problems.July 2021
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• 26 miscM.Maël Forcier and V.Vincent Leclère. Generalized adaptive partition-based method for two-stage stochastic linear programs : convergence and generalization.January 2022
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• 27 miscE.Emmanuel GOBET and M.Maxime Grangereau. Extended McKean-Vlasov optimal stochastic control applied to smart grid management.January 2021
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• 28 miscE.Emmanuel GOBET and M.Maxime Grangereau. Federated stochastic control of numerous heterogeneous energy storage systems.January 2021
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• 29 miscE.Emmanuel GOBET and M.Maxime Grangereau. Newton method for stochastic control problems.January 2021
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• 30 miscA. W.Armando W. Gutiérrez and C.Cormac Walsh. Firm non-expansive mappings in weak metric spaces.December 2021
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• 31 miscQ.Quentin Jacquet, W.Wim Van Ackooij, C.Clémence Alasseur and S.Stéphane Gaubert. A Quadratic Regularization for the Multi-Attribute Unit-Demand Envy-Free Pricing Problem.October 2021
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