Fonction quasi-convexe

Résumé

En mathématiques, une fonction quasi-convexe est une fonction à valeurs réelles, définie sur un ensemble convexe d'un espace vectoriel réel, telle que l' de tout ensemble de la forme \left]-\infty,a\right] est convexe ou encore telle que, sur tout segment, la plus grande valeur de la fonction est atteinte à l'une des extrémités. L'opposée d'une fonction quasi-convexe est dite quasi-concave. Toute fonction convexe est quasi-convexe mais la réciproque est fausse : par exemple, toute fonction monotone sur un intervalle réel est quasi-linéaire, c'est-à-dire à la fois quasi-convexe et quasi-concave. Définition Une fonction f:C\to\R définie sur une partie convexe C d'un espace vectoriel réel E est dite : *quasi-convexe si pour tout réel a, l'ensemble de sous-niveau {x\in C\mid f(x)\le a} est convexe ou, ce qui est équivalent, si\forall x,y\in C\quad\forall z\in\left[x,y\right]\quad f(z)\le\max\left(f(x),f(y)\right)

Source officielle

https://fr.wikipedia.org/wiki/Fonction_quasi-convexe

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Inclusions différentielles et problèmes variationnels

Ana Margarida Fernandes Ribeiro

In this thesis we deal with three different but connected questions. Firstly (cf. Chapter 2) we make a systematic study of the generalized notions of convexity for sets. We study the notions of polyconvex, quasiconvex and rank one convex set. We remark that these notions are nowadays well known in the context of functions, but not in the context of sets. Following the classical approach, we give precise definitions of generalized convex sets and we study several issues, in this generalized sense, as the concept of convex hull, Carathéodory and separation theorems and the notion of extremal point. Secondly we have studied some differential inclusions of the form The method we have used to solve this kind of problems is the Baire categories method developed by Dacorogna-Marcellini [14]. Known sufficient conditions for this problem are connected to the generalized convex hull of the set E. In Chapter 3, we compute the rank one convex hull of some matrix sets to obtain, in Chapter 4, existence results. Namely, we have considered the problem of finding u : Ω ⊂ Rn → RN with Dirichlet boundary condition such that Φ (Du(x)) ∈ {α, β}, a.e. x ∈ Ω, Φ being an arbitrary quasi-affine function. We have also considered the problem of finding u : Ω ⊂ Rn → Rn such that where λ1(Du) ≤...≤ λn(Du) are the singular values of Du ∈ Rn×n. Finally, in Chapter 5, we deal with several minimizing problems of the form Denoting by Qf the quasiconvex envelope of f, we verify that solving the equation Qf(Du(x)) = f(Du(x)), a.e. x ∈ Ω is, under some conditions, sufficient to ensure the existence of solution of (P). The differential inclusions that we consider in Chapter 4 are helpful to solve some equations of the form (2) and thus, it allows us to solve problems of type (P). In particular, we have considered the problem (P) with f(ξ) = g(Φ(ξ)), ∀ ξ ∈ RN×n Φ being an arbitrary quasi-affine function.

EPFL2006

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On non quasiconvex problems of the calculus of variations

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We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective functional. We show that an under-damped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.

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Inclusions différentielles et problèmes variationnels

Ana Margarida Fernandes Ribeiro

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