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Les effets de l'exposant de la fonction barrière multiplicative dans les méthodes de points intérieurs

Adama Coulibaly, Jean-Pierre Crouzeix (2010)

RAIRO - Operations Research

Les méthodes de points intérieurs en programmation linéaire connaissent un grand succès depuis l'introduction de l'algorithme de Karmarkar. La convergence de l'algorithme repose sur une fonction potentielle qui, sous sa forme multiplicative, fait apparaître un exposant p. Cet exposant est, de façon générale, choisi supérieur au nombre de variables n du problème. Nous montrons dans cet article que l'on peut utiliser des valeurs de p plus petites que n. Ceci permet d'améliorer le conditionnement...

Level sets of continuous functions increasing with respect to each variable

Katarzyna Sajbura (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We are going to prove that level sets of continuous functions increasing with respect to each variable are arcwise connected (Theorem 3) and characterize those of them which are arcs (Theorem 2). In [3], we will apply the second result to the classical linear functional equation φ∘f = gφ + h (cf., for instance, [1] and [2]) in a case not studied yet, where f is given as a pair of means, that is so-called mean-type mapping.

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Jean Van Schaftingen (2013)

Journal of the European Mathematical Society

The estimate D k - 1 u L n / ( n - 1 ) ≤ A ( D ) u L 1 is shown to hold if and only if A ( D ) is elliptic and canceling. Here A ( D ) is a homogeneous linear differential operator A ( D ) of order k on ℝ n from a vector space V to a vector space E . The operator A ( D ) is defined to be canceling if ⋂ ξ ∈ ℝ n ∖ { 0 } A ( ξ ) [ V ] = { 0 } . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential...

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