Comité local d’organisation
- Jérôme Bolte (UT1 et TSE)
- Sonia Cafieri (ENAC)
- Olivier Cots (INP-ENSEEIHT et IRIT)
- Victor Magron (LAAS-CNRS)
- Pierre Maréchal (UPS et IMT)
- Emmanuel Soubies (IRIT et CNRS)
- Edouard Pauwels (UPS et IRIT)
- Aude Rondepierre (INSA et IMT)
SPOT 75 – Lundi 4 Juillet 2022 (salle des thèses de l’ENSEEIHT)
14h – Alain Rapaport (UMR MISTEA Montpellier)
Equivalent formulations of the min peak problem and applications
We consider optimal control problems which consist in minimizing the maximum over a time interval [0,T] of a scalar function y(.) for a dynamics in dimension n+1 of the form \dot x = f(x,y,u), \dot y = g(x,y,u) with u in U. This problem is not in the usual Mayer, Lagrange or Bolza forms of the optimal control theory, and thus does not allow to use directly numerical software based on direct or Hamilton-Jacobi Bellman methods.
We propose and discuss several reformulations of this problem in Mayer form
- with state or mixed constraint ,
or
- with upper semi-continuous differential inclusion without state constraint
We consider also a particular class of problems for which we are able to give an explicit optimal solution. This allows us to compare numerical solutions obtained for the reformulations that we propose on an example in this class of problems, and to show their potential merits as practical methods to determine optimal solution of L\infty optimal control problems. This problem has been motivated by epidemiological questions, when one looks for minimizing the peak of an epidemic playing with restrictions as a control variable.
This is a joined work Emilio Molina and Hector Ramirez from University of Chile.
SPOT 74 – Mardi 7 Juin 2022 (salle des thèses de l’ENSEEIHT)
14h – Leo Liberti (LIX Ecole Polyechnique Palaiseau)
Random projections in Mathematical Programming: Survey and new directions
Random projections are random matrices that decrease the dimensionality of a finite set of vectors while guaranteeing approximate congruence of the high and low dimensional point sets. Their application to Mathematical Programming yield projected formulations with fewer constraints or variables (or, occasionally, both), which can be solved faster than their full-dimensional counterparts, and provide: reasonable bounds on the optimal value, and approximately feasible solutions. I am going to provide a summary of the work done so far in LP, SDP, QP, then discuss current work.
15h – Sixin Zhang (IRIT and ENSEEIHT Toulouse)
On the Nash equilibrium of moment-matching GANs for stationary Gaussian processes
Generative Adversarial Networks (GANs) learn an implicit generative model from data samples through a two-player game. We study the existence of Nash equilibrium of the game which is consistent as the number of data samples grows to infinity. In a realizable setting where the goal is to estimate the ground-truth generator of a stationary Gaussian process, we show that the existence of consistent Nash equilibrium depends crucially on the choice of the discriminator family. The discriminator defined from second-order statistical moments can result in non-existence of Nash equilibrium, existence of consistent non-Nash equilibrium, or existence and uniqueness of consistent Nash equilibrium, depending on whether symmetry properties of the generator family are respected. We further study empirically the local stability and global convergence of gradient descent-ascent methods towards consistent equilibrium.
SPOT 73 – Lundi 16 Mai 2022 (salle des thèses de l’ENSEEIHT)
14h – David Salas (Universidad de O’Higgins, Chili)
The Bayesian approach for bilevel games: new advances and challenges
In 1996, Mallozzi and Morgan proposed a new model for bilevel games (with one leader) which they called Intermediate Stackelberg games. The leader has only partial information about how followers select their response among possible multiple optimal ones. This partial information is modeled as a decision-dependent distribution, the so-called belief of the leader. In this talk, we will explore new existence results for such model. In particular, we will show that when the lower-level problem is fully linear, existence of solutions can be assured for a large family of beliefs (which are somehow constructed from the lebesgue measure). We will also discuss some partial results in terms of computation of such solutions and some challenges to validate this model as a viable alternative to classic optimistic-pessimistic approaches. Due to the similarity of the approach to Bayesian games, we propose to rename it as the Bayesian approach for bilevel games.
15h – Rodolfo Rios-Zeruche (LAAS-CNRS)
Long term dynamics of non-smooth non-convex optimization schemes
Large scale artificial intelligence applications, notably Deep Learning, require the optimization of non-smooth, non-convex functions in very high dimensional domains. The lack of structure of these problems has revived interest in a variety of optimization algorithms, like the subgradient method, that are often simple in their conception, but produce very complicated dynamics, to the point that their success has so far been impossible to explain. In this talk we will describe how closed measures, a statistical method borrowed from other areas of dynamics, has led to substantial success in the description of the long term behavior of several algorithms. This is joint work with Jérôme Bolte and Edouard Pauwels.
SPOT 72 – Lundi 4 Avril 2022 (salle des thèses de l’ENSEEIHT)
14h – Antoine Oustry (LIX-CNRS, Ecole Polytechnique, Palaiseau)
ACOPF: Nonsmooth optimization to improve the computation of SDP bounds
The Alternating-Current Optimal Power Flow (ACOPF) problem models the optimization of power dispatch in an AC electrical network. In the quest for global optimality certificates, the semidefinite programming (SDP) relaxation of the ACOPF problem is a major asset since it is known to produce tight lower bounds. To improve the scalability of the SDP relaxation, state-of-the-art approaches exploit the sparse structure of the power grid by using a clique decomposition technique. Despite this advance, the clique-based SDP relaxation remains difficult to solve for large-scale instances: numerical instabilities may arise when solving this convex optimization problem. These difficulties cause two issues (i) inaccuracy and (ii) lack of certification. We tackle both issues with an original approach. We reformulate the Lagrangian dual of the ACOPF, whose value equals the value of the SDP relaxation, as a concave maximization problem with the following properties: (i) it is unconstrained (ii) the objective function is partially separable. Based on this new formulation, we present how to obtain a certified lower bound from any dual vector, whether feasible or not in the classical dual SDP problem. Our new formulation is solved with a tailored polyhedral bundle method exploiting the structure of the problem. We use this algorithm as a post-processing step, after solving the SDP relaxation with the state-of-the-art commercial interior point solver MOSEK. For many instances from the PGLib-OPF library, this post-processing significantly reduces the computed optimality gap.
Joint work with Claudia D’Ambrosio (LIX-CNRS), Leo Liberti (LIX-CNRS) and Manuel Ruiz (RTE).