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This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f . The family of second-order methods recently received a large amount of attention, until the famous contribution of Nesterov [Nes83], leading to the explosion of large-scale optimization problems. This work provides an in-depth description of the stochastic heavy-ball method, which is an adaptation of the deterministic one when only unbiased evalutions of the gradient are available and used throughout the iterations of the algorithm. We first describe some almost sure convergence results in the case of general non-convex coercive functions f . We then examine the situation of convex and strongly convex potentials and derive some non-asymptotic results about the stochastic heavy-ball method. We end our study with limit theorems on several rescaled algorithms., 39 pages, 3 pages

This paper is a continuation of [2], [9] and [10]. Its aim is to characterize non-removable ideals in the class ${\cal M}$ of all multiplicatively convex bornological algebras. Let $A$ be in ${\cal M}$. We show that an ideal $I\subset A$ is ${\cal M}$-non-removable if and only if it consists of joint m-bounding elements (D\'ef. 9). Such an ideal doesn't consist, in general, of joint bounding elements (D\'ef. 8). This shows that the concept of ${\cal M}$-non-removable ideals is of relative character (D\'ef. 4).

cote interne IRCAM: Mignot11b; None / None; National audience; [Français]: Ce travail porte sur la modélisation physique de tubes acoustiques convexes pour la simulation numérique en temps-réel. Le propos principal de ce papier est l’utilisation de l’automatique pour la résolution de problème de stabilité dans les systèmes mis en oeuvre. [English] This work deals with the physical modelling of convex acoustic pipes for real-time simulation. The main purpose of this paper is the use of automatic control to solve some stability problems.

The advanced methods of Automatic control often yield high order controllers, which have to be reduced. However the reduction generally yields a lack of performance in such a way that the set of specifications is no longer satisfied. To solve this problem, a method for retuning the controller is proposed which allows to reduce the reduction error on the frequency response, in frequency domains considered as crucial for satisfying the specifications. The constraints on the the reduction error are expressed using LMIs, so that the problem is relevant to convex optimisation. The application to the reduction of a controller designed for a space launcher underlines the efficiency of the proposed approach.

Characterizations of Pasch-Peano spaces. We show that complete convexity spaces (Bryant and Webster, Cantwell) of dimension at least three are convex subsets of real affine spaces. The proof uses an infinite-dimensional analogue of a result of de Groot and de Vries on convex sets in projective spaces., Doignon Jean-Paul. Caractérisations d'espaces de Pasch-Peano. In: Bulletin de la Classe des sciences, tome 62, 1976. pp. 679-699.