Section: New Results

Applications

Geometry of the Loewner order and application to the synthesis of quadratic invariants in static analysis of program

Participants : Xavier Allamigeon, Stéphane Gaubert, Nikolas Stott.

This section presents the PhD work of Nikolas Stott. An essential part of the present work is in collaboration with Éric Goubault and Sylvie Putot (from LIX).

We develop a numerical abstract domain based on ellipsoids designed for the formal verification of switched linear systems. The novelty of this domain does not consist in the use of ellipsoids as abstractions, but rather in the fact that we overcome two key difficulties which so far have limited the use of ellipsoids in abstract interpretation. The first issue is that the ordered set of ellipsoids does not constitute a lattice. This implies that there is a priori no canonical choice of the abstraction of the union of two sets, making the analysis less predictable as it relies on the selection of good upper bounds. The second issue is that most recent works using on ellipsoids rely on LMI methods. The latter are efficient on moderate size examples but they are inherently limited by the complexity of interior point algorithms, which, in the case of matrix inequality problems, do not scale as well as for linear programming or second order cone programming problems.

We developed a new approach, in which we reduce the computation of an invariant to the determination of a fixed point, or eigenvector, of a non-linear map that provides a safe upper-approximation of the action induced by the program on the space of quadratic forms. This allows one to obtain invariants of systems of sized inaccessible by LMI methods, at the price of a limited loss of precision. A key ingredient here is the fast computation of least upper bounds in Löwner ordering, by an algebraic algorithm. This relies on the study of the geometry of the space of quadratic forms (Section 7.2.2).

The initial part of this work was described in the article  [57], in which we obtained a single ellipsoidal invariant. In [16], we showed that finer disjunctive invariants, expressed as union of ellipsoids, can still be obtained by nonlinear fixed point methods in a scalable way. In [30], we developed a dual approach, which we applied to the problem of computing the joint spectral radius. We showed that an approximation of a Barabanov norm by a supremum of quadratic forms can be obtained by solving a nonlinear eigenvalue problem involving “tropical Kraus maps”. The latter can be thought of as the tropical analogues of the completely positive maps appearing in quantum information. The fixed point methods in [30] allow one to solve large scale instances, unaccessible by earlier (LMI based) methods.

Performance evaluation of an emergency call center

Participants : Xavier Allamigeon, Vianney Boeuf, Stéphane Gaubert.

This work arose from a question raised by Régis Reboul from Préfecture de Police de Paris (PP), regarding the analysis of the projected evolution of the treatment of emergency calls (17-18-112). This work benefited from the help of LtL Stéphane Raclot, from Brigade de Sapeurs de Pompiers de Paris (BSPP), now with PP. It is part of the PhD work of Vianney Bœuf, carried out in collaboration with BSPP.

We introduced an algebraic approach which allows to analyze the performance of systems involving priorities and modeled by timed Petri nets. Our results apply to the class of Petri nets in which the places can be partitioned in two categories: the routing in certain places is subject to priority rules, whereas the routing at the other places is free choice.

We initially introduced a discrete model in  [54], showing that the counter variables, which determine the number of firings of the different transitions as a function of time, are the solutions of a piecewise linear dynamical system. We showed the stationary regimes are precisely the solutions of a set of lexicographic piecewise linear equations, which constitutes a polynomial system over a tropical (min-plus) semifield of germs. However, the convergence to a stationnary regime may not occur in the discrete model. We developed in [15] a continuous time analogue of this model, involving a piecewise linear dynamical systems, and showed that it has the same stationnary regimes, avoiding some pathologies of the discrete model.

In essence, this result shows that computing stationary regimes reduces to solving tropical polynomial systems. Solving tropical polynomial systems is one of the most basic problems of tropical geometry. The latter provides insights on the nature of solutions, as well as algorithmic tools. In particular, the tropical approach allows one to determine the different congestion phases of the system. This analysis has been recovered by a probabilistic model in [42].

We applied this approach to a case study relative to the project led by Préfecture de Police de Paris, involving BSPP, of a new organization to handle emergency calls to Police (number 17), Firemen (number 18), and untyped emergency calls (number 112), in the Paris area. We combined explicit analytic computations of the different congestion phases of a simplified model and extensive simulations of a realistic an detailed model, to evaluate the performance of the center as a function of the number of operators. This analysis also suggested some ways to monitor early signs of potential congestions as well as possible correcting measures to avoid congestion.

Tropical models of fire propagation in urban areas

Participants : Stéphane Gaubert, Daniel Jones.

As part of the team work in the ANR project Democrite, we developed a model of fire propagation in urban areas, involving a discrete analogue of a Hamilton-Jacobi PDE. This models indicates that the fire propagates according to polyhedral ball, which is in accordance from data from historical fires (London, Chicago, or more recently Kobe).

Smart Data Pricing

Participants : Marianne Akian, Jean-Bernard Eytard.

This work is in collaboration with Mustapha Bouhtou (Orange Labs).

The PhD work of Jean-Bernard Eytard concerns the optimal pricing of data trafic in mobile networks. We developed a bilevel programming approach, allowing to an operator to balance the load in the network through price incentives. We showed that a subclass of bilevel programs can be solved in polynomial time, by combining methods of tropical geometry and of discrete convexity. This work is presented in [28]. In a followup work (collaboration with Gleb Koshevoy), we managed to extend these results to wider classes of bilevel problems, and to relate them to competitive equilibria problems.

Game theory models of decentralized mechanisms of pricing of the smart grid

Participants : Stéphane Gaubert, Paulin Jacquot.

This work is in collaboration with Nadia Oudjane and Olivier Beaude (EDF).

The PhD work of Paulin Jacquot concerns the application of game theory techniques to pricing of energy. We are developing a game theory framework for demand side management in the smart grid, in which users have movable demands (like charging an electric vehicle). We compared in particular the daily and hourly billing mechanisms. The latter, albeit more complex to analyse, has a merit in terms of incitatives, as it leads the user to move his or her consumption at off peak hours. We showed the Nash equilibrium is unique, under some assumptions, and gave theoretical bounds of the price of anarchy of the game with a hourly billing, showing this mechanism remains efficient while being more “fair” than the daily billing. We proposed and tested decentralized algorithms to compute the Nash equilibrium. These contriutions are presented In [31], [32], [44].