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-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 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  70 (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. In particular, we introduced a generalization of fiber polytopes, which characterize the value function. We deduce polynomial-time solvability results in fixed dimension, both for exact and approximated versions of the problem. Further results are presented in the PhD thesis  71.

8.1.3 Tropical numerical methods for stochastic control problems

Participants: Marianne Akian.

In a work presented in 19, 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 102, 87, 111, which includes in particular SARSOP algorithm.

With Luz Pascal (QUT & CSIRO, Australia), we are also studying the convergence of SARSOP algorithm using the same techniques as in  33.

8.1.4 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  65, 108, 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, presented in particular in 20, 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.

More recently, 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  35.

8.1.5 Universal complexity bounds for value iteration applied to zero-sum stochastic games and entropy games

Participants: Xavier Allamigeon, Stéphane Gaubert.

In a joint work with Ricardo Katz (CONICET) and Mateusz Skomra (LAAS, CNRS),  23, we have been investigating complexity bounds, based on value iteration, for stochastic zero-sum games problems. This leads for parametrized tractability results for classes of games, including ergodic turn-based stochastic games with a fixed number of random positions, and entropy games. The latter are a class of games in which one player wishes to maximize a topological entropy, whereas the other player wishes to minimize it.

8.2.1 Volume in Hilbert and Funk geometries

Participants: Constantin Vernicos, Cormac Walsh.

Some of this work is joint work with Dmitry Faifman (Tel Aviv).

In a recent paper  116, 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, 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 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.

We are also studying the following refinement of the volume entropy. We can show that for any Hilbert geometry $K$,

The limit on the left-hand-side is called the entropy coefficient. It can be thought of as the constant on front of the highest order term $e^{(n-1)r}$ in the growth of the volume. The quantity $A_p\left(K\right)$ is the centro-projective area of the convex body $K$. This quantity can be expressed as an integral over the boundary, and is invariant under projective transformations that leave the origin unchanged. The equation above had previously been established by Berck–Bernig–Vernicos in the case where the boundary of $K$ is $C^{1,1}$, that is, where it is differentiable with Lipschitz continuous derivative. The novelty is that we can now prove it for completely general convex bodies. The new argument builds on work of Tholozan relating the Hilbert metric to the Blaschke metric on the convex body. This allows us to control the contribution to the volume of regions close to the parts of the boundary where the curvature is singular. The argument can also be modified to work for the Funk geometry. Here, the constant that appears on front of the entropy term is the centro-affine area, a well-known invariant from convexity theory.

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 investigated 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 18, 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 (Meta), 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 the journal Logical Methods in Computer Science 14.

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 been accepted for publication in the proceedings of the conference CPP'23.

8.3.2 Linear algebra over systems

Participants: Marianne Akian, Stéphane Gaubert.

In a joint work with Louis Rowen (Univ. Bar Ilan), 27, 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 27, we characterize the semiring systems which arise from hyperrings.

We are now studying linear algebra properties over “systems”.

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 Approximation by small rank tropical tensors

Participants: Marianne Akian, Stéphane Gaubert, Jiawei He, Yang Qi.

During the internship of Jiawei He (of 3rd year of Polytechnique), we have studied and compared experimentally several approximations of functions by tropical tensors of small rank, obtained by sampling or optimization.

8.3.6 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  103 (see also 51), and the signed valuation which associates to any series its valuation together with its sign.

In a work 28 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 52 (for multiplicities) and on 50.

We now study with these tools the asymptotics of eigenvalues and eigenvectors of symmetric positive definite matrices over the field of Puiseux series.

8.3.7 Tropical Systems of Polynomial Equations

Participants: Marianne Akian, Antoine Bereau, Stéphane Gaubert.

The PhD thesis of Antoine Bereau, started in September 2021, deals with systems of polynomial equations over tropical semifields. 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.

8.4.1 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  41,6 that the answer is negative.

In a subsequent work  40 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  41,6.

In the work 24, 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), this is the object of a current work, with Mateusz Skomra.

In a joint work 22 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  48 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 the tropical analogue of the notion of polar of a cone over the symmetrized tropical semiring (see for instance  103, 51). 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 Topology of tropical ranks

Participants: Yang Qi.

The primary goal of 62 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 89 (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  43, 44, 55. 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  45.

In 21, 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  57.

8.6.2 Optimal control of energy flexibilities in a stochastic environment

Participants: Maxime Grangereau, Stéphane Gaubert.

The PhD thesis of Maxime Grangereau  80 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  72, including the modelling of storage resources and decentralized aspects  73. Another contribution concerns a version of the Newton method in stochastic control  74. 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 17.

8.6.3 Optimal pricing of energy contracts

Participants: 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  85 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  85.

In 25, 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 29 develops a mean field game model to model an incentive pricing scheme, in which agents compete to get the best reward.

10.1.1 STIC/MATH/CLIMAT AmSud project

10.1.2 Participation in other International Programs

10.2.1 Visits of international scientists

10.2.2 Visits to international teams

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.

10.3.4 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 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”.
• A. Bigel
  • 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  56.
• S. Liu
  • Exercises classes for the first year of Bachelor program of Ecole polytechnique 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: Marin Boyet, registered at Univ. Paris Saclay since October 2018, thesis supervisor: Stéphane Gaubert, cosupervision: Xavier Allamigeon. The defense took place on May 25, 2022.
• PhD: Maël Forcier, registered at ENPC since September 2019, thesis supervisor: Vincent Leclère, cosupervision Stéphane Gaubert. The defense took place on December 14, 2022.
• PhD: Tom Ferragut, on the horosphérical products of Gromov-hyperbolic and Busemann convex metric spaces, registered at U. Montpellier, the defense took place in 2022. Supervisor: C. Vernicos, co-supervisor Jérémie Brieussel.
• 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: 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.

11.2.3 Juries

• M. Akian
  • Jury (and reviewer) of the PhD thesis of Eloise Berthier (ENS PSL and Inria Paris), October 27, 2022.
  • Jury of the PhD thesis of Cyrille Vessaire (CERMICS), December 16, 2022.
• X. Allamigeon
  • Jury of the PhD of Marin Boyet, Ecole polytechnique, May 25, 2022.
• S. Gaubert
  • Jury of the PhD of Marin Boyet, Ecole polytechnique, May 25, 2022.
  • Jury (and reviewer) of the PhD of Manuel Radons, TU-Berlin, October 26, 2022.
  • Jury of the PhD of Eugenie Marescaux, Ecole polytechnique, November 21, 2022.
  • Jury (and reviewer) of the PhD of Guilherme Espindola-Winck, Université d'Angers, December 9, 2022.
  • Jury of the PhD of Maël Forcier, CERMICS, December 14, 2022.
  • Jury of the PhD of Othmane Jerhaoui, ENSTA, December 15, 2022.

11.2.4 Communications at conferences and seminars

• M. Akian
  • 2nde Journée MAS-MODE 2022, 7 mars 2022, INRIA Paris. Talk: “Jeux stochastiques à somme nulle: ergodicité, complexité et théorie de Perron-Frobenius non linéaire”.
  • Séminaire commun MATHRISK / LPSM, Paris Diderot, 7 avril 2022. Talk: “Tropical numerical methods for solving stochastic control problems”.
  • Applied Mathematics Colloquium at the Mathematics Institute, Universidade Federal do Rio de Janeiro (UFRJ). Talk: “Tropical geometry, zero-sum games and nonlinear Perron-Frobenius theory”.
  • 9th French-Chilean Meeting of Optimization, June 29, July 1st, 2022, Perpignan. Talk: “Tropical linear regression and mean payoff games”.
  • 25th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2022), 12-16 September 2022, Bayreuth, Germany. Session “Hamilton-Jacobi Equations and Mean Field Games”, Talk: “Tropical Numerical Methods for Solving Stochastic Control Problems” (work with J.P. Chancelier, L. Pascal, and B. Tran).
• X. Allamigeon:
  • MIND-SoDA joint seminar, Inria Saclay, February 2022. Talk: “Recent progress in the complexity analysis of linear programming”.
  • DGeCo seminar, IMJ, Paris, February 2022. Talk: “Tropical complementarity problems and Nash equilibria”.
  • STOC 2022: 54th Annual ACM Symposium on Theory of Computing, Rome, Italy, June 20-24, 2022. Talk: “No Self-Concordant Barrier Interior Point Method Is Strongly Polynomial”.
• A. Bereau:
  • Combinatorial, Computational, and Applied Algebraic Geometry (CCAAGS-22), June 27-July 1, 2022, Seattle. Poster: “The Nullstellensatz for Sparse Tropical Polynomial Systems”.
  • ARGO2022: Workshop on Algebraic Real Geometry and Optimization, Aug 30-Sept-2, 2022, Santiago, Chile. Talk: “The Nullstellensatz for Sparse Tropical Polynomial Systems”.
  • Workshop on Solving Polynomial Equations and Applications, CWI, Amsterdam, 5-7 october, 2022. Poster: “The Nullstellensatz for Sparse Tropical Polynomial Systems”.
• Q. Canu:
  • DGeCo seminar, IMJ, Paris. April 2022. Talk: “Calcul Polyédral efficace et formellement prouvé”.
  • Gallinette seminar, Inria Nantes, December 2022. Talk: “A formal disproof of the Hirsch conjecture”.
• M. Forcier:
  • ECSO-CMS, June 2022, Venice: Exact quantization methods for Multistage Stochastic Linear Problems, best student paper prize.
  • SMAI-MODE, June 2022, Limoges: Secondary simplex method for two-stage stochastic linear programs with general cost distribution.
  • ROADEF , February 2022, Lyon: Generalized Adaptive Partition-based method for two-stage stochastic linear programs. Slides.
• S. Gaubert:
  • “Polytopes in Paris and more: Geometry, Combinatorics and Optimization”, 27-29 June, 2022. Talk: “Ambitropical convexity, Hyperconvexity, and zero-sum games”.
  • “ICALP 2022 conference”, IRIF, Paris, 4-8 July, 2022. “Universal complexity bounds based on value iteration and application to entropy games”.
  • 9th French-Chilean Meeting of Optimization, June 29, July 1st, 2022, Perpignan. Talk: “Solving mean-payoff problems by value iteration: universal complexity bounds and application to entropy games” (work with X. Allamigeon, R. Katz and M. Skomra).
  • ARGO2022: Workshop on Algebraic Real Geometry and Optimization, Aug 30-Sept-2, 2022, Santiago, Chile. Talk: “Convexity, Mean Payoff Games and Nonarchimedean Convex Programming”.
• Q. Jacquet:
  • CDC 2022, Cancun, (hybrid conference), “Ergodic control of a heterogeneous population and application to electricity pricing”.
  • PGMO Days 2022, EDF Labs, “A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings”.
• S. Liu:
  • “Journées SMAI MODE 2022”, May 30, 2022 to June 3, 2022. Poster: “A Multilevel Fast-Marching Method for the Minimum Time Problem” (work with M. Akian and S. Gaubert).
  • 25th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2022), 12-16 September 2022, Bayreuth, Germany. Talk: “A Multilevel Fast-Marching Method” (work with M. Akian and S. Gaubert).
• N. Vandame:
  • “PGMO days”, Nov. 29-30, 2022, EDF Labs Paris Saclay, “No Self-Concordant Barrier Interior Point Method Is Strongly Polynomial” (work with X. Allamigeon and S. Gaubert).
• Y. Qi:
  • “Tensors from the physics viewpoint”, Institute of Mathematics of the Polish Academy of Sciences, Warsaw, October 2022, “Tropical linear regression and low-rank approximation — a first step in tropical data analysis”.
  • “Workshop on tensor theory and methods”, Lagrange Mathematics and Computing Research Center, Paris, November 2022, “Tropical linear regression and low-rank approximation — a first step in tropical data analysis”.
• C. Walsh:
  • Conference “Géométrie, topologie et dynamique en basses dimensions”, Sète, 23-27 May 2022. Talk: “Volume growth in Funk geometry”.
  • “Annual meeting of the German Mathematical Society”, Berlin, 12-16 Sept. 2022. Talk: “Order isomorphisms and antimorphisms in partially-ordered vector spaces”.

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 Applications4312022
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• 14 articleX.Xavier Allamigeon, R.Ricardo Katz and P.-Y.Pierre-Yves Strub. Formalizing the Face Lattice of Polyhedra.Logical Methods in Computer ScienceVolume 18, Issue 2May 2022
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• 15 articleP.-C.Pierre-Cyril Aubin-Frankowski and S.Stéphane Gaubert. Tropical reproducing kernels and optimization.Integral Equations and Operator Theory2022
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• 16 articleT.Thierry Garaix, S.Stéphane Gaubert, J.Julie Josse, N.Nicolas Vayatis and A.Amandine Véber. Decision-making tools for healthcare structures in times of pandemic.Anaesthesia Critical Care & Pain Medicine412March 2022, 101052
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• 17 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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• 18 articleA. W.Armando W. Gutiérrez and C.Cormac Walsh. Firm non-expansive mappings in weak metric spaces.Archiv der Mathematik 1192022, 389-400
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International peer-reviewed conferences

• 19 inproceedingsM.Marianne Akian, J.-P.Jean-Philippe Chancelier, L.Luz Pascal and B.Benoît Tran. Tropical numerical methods for solving stochastic control problems.MTNS 2022 - 25th International Symposium on Mathematical Theory of Networks and SystemsBayreuth (DE), GermanySeptember 2022
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• 20 inproceedingsM.Marianne Akian, S.Stéphane Gaubert and S.Shanqing Liu. A multilevel fast-marching method.MTNS 2022 - 25th International Symposium on Mathematical Theory of Networks and SystemsBayreuth (DE), GermanySeptember 2022
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• 21 inproceedingsX.Xavier Allamigeon, M.Marin Boyet and S.Stephane Gaubert. Computing Transience Bounds of Emergency Call Centers: a Hierarchical Timed Petri Net Approach.PETRI NETS 2022: Application and Theory of Petri Nets and Concurrency13288PETRI NETS 2022: Application and Theory of Petri Nets and Concurrency, Springer Lecture Notes in Computer SciencesBergen, NorwaySpringer2022, 90-112
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• 22 inproceedingsX.Xavier Allamigeon, D.Daniel Dadush, G.Georg Loho, B.Bento Natura and L.Laszlo Vegh. Interior point methods are not worse than Simplex.2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)Denver, United StatesIEEEOctober 2022, 267-277
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• 23 inproceedingsX.Xavier Allamigeon, S.Stéphane Gaubert, R. D.Ricardo David Katz and M.Mateusz Skomra. Universal Complexity Bounds Based on Value Iteration and Application to Entropy Games.49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)Paris, FranceJuly 2022
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• 24 inproceedingsX.Xavier Allamigeon, S.Stéphane Gaubert and N.Nicolas Vandame. No self-concordant barrier interior point method is strongly polynomial.STOC '22: 54th Annual ACM SIGACT Symposium on Theory of ComputingRome Italy, FranceACMJune 2022, 515-528
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• 25 inproceedingsQ.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 CDC 2022Cancun, MexicoDecember 2022
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• 26 inproceedingsQ.Quentin Jacquet, W.Wim van Ackooij, C.Clémence Alasseur and S.Stéphane Gaubert. Une régularisation quadratique pour la tarification de contrats d'électricité.23ème congrès annuel de la Société Française de Recherche Opérationnelle et d'Aide à la DécisionVilleurbanne - Lyon, FranceFebruary 2022
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Reports & preprints

• 27 miscM.Marianne Akian, S.Stephane Gaubert and L.Louis Rowen. Semiring systems arising from hyperrings.September 2022
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• 28 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 2023
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• 29 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.September 2022
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• 30 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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• 31 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.November 2022
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• 32 miscQ.Quentin Jacquet and R.Riadh Zorgati. Tight Bound for Sum of Heterogeneous Random Variables: Application to Chance Constrained Programming.November 2022
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