L'Optimisation fête le printemps 21 mars 2003

09h00-09h30 Accueil des participants.

09h30-10h15 Darinka DENTCHEVA

10h15-11h00 Thomas LACHAND-ROBERT

11h00-11h30 Pause Café

11h30-12h15 Samir ADLY

12h15-14h30 Déjeuner

14h30-15h15 Roberto LUCCHETTI

15h15-16h00 Patrick SAINT-PIERRE

16h00-16h30 Pause Café

16h30-17h15 Constantin ZALINESCU

17h15-18h00 Andrzej RUSZCZYNSKI

A Stability Theory for second order Nonsmooth Dynamical Systems with Applications to Friction Problems.

In this talk, a LaSalle's Invariance Theory for a class of first order evolution variational inequalities will be developped. Using this approach, stability and asymptotic properties of important classes of second order dynamical systems is studied. The theoretical result will be supported by examples in nonsmooth Mechanics and some numerical simulations.

Darinka DENTCHEVA, Stevens, Rutgers

Regular selections of multifunctions

We consider set-valued mappings defined on a topological space with closed convex images in a finite dimensional space. The measurability of a multifunction is characterized by the existence of a Castaing representation for it: a countable set of measurable selections that pointwise fills up the graph of the multifunction. A Castaing representation will be constructed that characterizes the continuity of the multifunction, and inherits certain regularity properties of the multifunction. The construction is based on a generalization of the Steiner center. All generalized Steiner selections are measurable, continuous, Hoelder-continuous, or directionally differentiable, if the multifunction has the corresponding properties.
The results are applied to obtain statements about the asymptotic behavior of random sets and their selections via the delta-method. In particular, various multifunctions arising naturally in stochastic optimization problems will be duscussed.
 

Thomas LACHAND-ROBERT, Université de Chambéry

Un problème d'optimisation de forme résultant d'un modèle de glissement de terrain

Nous consid*rons le probl*me de l'*coulement d'un fluide viscoplastique. Intervenant dans la mod*lisation des glissements de terrain, le mod*le choisi est de type Bingham avec un seuil de plasticit* non homog*ne. Nous nous int*ressons notamment aux conditions n*cessaires et suffisantes de blocage du fluide. Le probl*me antiplan est consid*r* dans les cas bidimensionnel et monodimensionnel et une formulation variationnelle en contraintes est obtenue et utilis*e. En dimension deux, ce probl*me se formule par une propri*t* de minimisation sur un ensemble, ce qui le classe dans la cat*gorie des probl*mes d'optimisation de formes.

Roberto LUCCHETTI,  Como, Italie

Porosity of ill posed problems in some classes of optimization problems

The notion of porosity is interesting as a $\sigma$-porous set is small in the Baire category sense, and also of measure zero on Euclidean spaces. On the other hand, the notion of well posedness we consider is a strong one. Thus it is interesting to produce results that in classes of optimization problems, the ill posed are a $\sigma$-porous set. We shall illustrate some of these results, mainly in the convex setting.

Andrzej RUSZCZYNSKI, Rutgers

Stochastic Dominance and Mean-Risk Models

Decision problems involving random outcomes require the application of a decision principle to make the corresponding optimization problems well-defined. One of the established principles is the relation of stochastic dominance. It defines a partial order among random outcomes, but it is very difficult to use in practice.
The practice frequently resorts to mean--risk models which use two measures of quality: the expected outcome and some measure of the uncertainty of the outcome, called the risk.
In this talk we shall discuss relations between stochastic dominance and mean--risk models. We shall show that central semi-deviations used as risk measures are in harmony with the stochastic dominance orders of the corresponding degrees.
Next, by exploiting duality relations of convex analysis we show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation of the second degree.
We provide stochastic linear programming formulations of these models and we illustrate our results on a portfolio problem involving 719 securities and return data from the last 12 years.

Patrick SAINT-PIERRE, Université Paris-Dauphine

Lorenz and Rossler Attractors, Julia Sets and Viability Kernels: Viability Kernels and Capture Basins as Tools for Analyzing the Dynamic Behavior of Systems

Some issues on chaotic evolution are related to viability kernels of subsets under continuous time systems - like attractors of Lorenz and Rossler systems - or discrete time systems - like Julia and Mandelbrot sets or Cantor nature subsets under quadratic or inverses of Hutchinson maps.
Viability kernels and capture basins of targets provide tools for analysing the local behavior around equilibria as, for instance, local stable and unstable manifolds, fluctuation between two areas of a domain or for approximating heteroclines. Using algorithms designed for computing viability kernels and capture basins, we show through the well known Lorenz system how to localize attractors going beyond the mere simulation of trajectories of evolutions and how to give a full explanation of their fluctuations.
We also investigate the properties of viability kernels of discrete systems. We show that filled-in Julia and Mandelbrot sets are related to viability kernels of discrete systems on $\mathbb{R}^{2}$, allowing us to compute them without determining the initial states from which the evolution starts and leaves a bounded set in finite time. We also use the viability property of the boundary of a viability kernel to approximate the Julia subsets themselves that are the boundaries of the filled-in Julia sets and that are numerically unstable.

Constantin ZALINESCU, Iasi

On the necessity of some constraint qualification conditions in convex programming

We realize a study of various constraint qualification conditions for the existence of Lagrange multipliers for convex minimization problems in general normed vector spaces; it is based on a new formula for the normal cone to the constraint set, on local metric regularity and a metric regularity property on bounded subsets. As a by-product we obtain a characterization of the metric regularity of a finite family of closed convex sets.