Level sets and non Gaussian integrals of positively homogeneous functions - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue International Game Theory Review Année : 2015

Level sets and non Gaussian integrals of positively homogeneous functions

Résumé

We investigate various properties of the sublevel set $\{x \,:\,g(x)\leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(-g)$ on the whole space $R^n$ (a non Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of $g$. In fact, whenever $h$ is nonnegative, the functional $\int \phi(g(x))h(x)dx$ is a convex function of $g$ for a large class of functions $\phi:R_+\to R$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set $\{x \,:\,g(x)\leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.
Fichier principal
Vignette du fichier
levelset-new.pdf (276.46 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-00637049 , version 1 (29-10-2011)
hal-00637049 , version 2 (02-11-2011)
hal-00637049 , version 3 (28-02-2012)

Licence

Copyright (Tous droits réservés)

Identifiants

Citer

Jean-Bernard Lasserre. Level sets and non Gaussian integrals of positively homogeneous functions. International Game Theory Review, 2015, 17 (1), pp.10.1142/S0219198915400010. ⟨10.1142/S0219198915400010⟩. ⟨hal-00637049v3⟩
207 Consultations
437 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More