Degree Bounds for Putinar's Positivstellensatz on the hypercube - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2023

Degree Bounds for Putinar's Positivstellensatz on the hypercube

Lorenzo Baldi
  • Fonction : Auteur
  • PersonId : 1086466
Lucas Slot
  • Fonction : Auteur
  • PersonId : 1231062

Résumé

The Positivstellensätze of Putinar and Schmüdgen show that any polynomial f positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of these results, namely to show bounds on the required degree of the sums of squares in such representations. These effective Positivstellensätze have direct implications for the convergence rate of the celebrated moment-SOS hierarchy in polynomial optimization. In this paper, we restrict to the fundamental case of the hypercube B^n = [−1, 1]^n. We show an upper degree bound for Putinar-type representations on B n of the order O(fmax/fmin), where fmax, f min are the maximum and minimum of f on B^n , respectively. Previously, specialized results of this kind were available only for Schmüdgentype representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound in Ω((fmax/fmin)^1/8). This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities. Contents
Fichier principal
Vignette du fichier
PutinarHypercube_v1.pdf (495.51 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04003633 , version 1 (24-02-2023)
hal-04003633 , version 2 (22-09-2023)

Identifiants

  • HAL Id : hal-04003633 , version 1

Citer

Lorenzo Baldi, Lucas Slot. Degree Bounds for Putinar's Positivstellensatz on the hypercube. 2023. ⟨hal-04003633v1⟩
50 Consultations
46 Téléchargements

Partager

Gmail Facebook X LinkedIn More