Sums of squares representations on singular loci - LAAS - Laboratoire d'Analyse et d'Architecture des Systèmes Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2023

Sums of squares representations on singular loci

Résumé

The problem of characterizing a real polynomial $f$ as a sum of squares of polynomials on a real algebraic variety $V$ dates back to the pioneering work of Hilbert in [Mathematische Annalen 32.3 (1888): 342-350]. In this paper, we investigate this problem with a focus on cases where the real zeros of $f$ on $V$ are singular points of $V$. By using optimality conditions and irreducible decomposition, we provide a positive answer to the following essential question of polynomial optimization: Are there always exact semidefinite programs to compute the minimum value attained by a given polynomial over a given real algebraic variety? Our answer implies that Lasserre's hierarchy, which is known as a bridge between convex and non-convex programs with algebraic structures, has finite convergence not only in the generic case but also in the general case. As a result, we constructively prove that each hyperbolic program is equivalent to a semidefinite program.

Dates et versions

hal-04023980 , version 1 (10-03-2023)

Identifiants

Citer

Ngoc Hoang Anh Mai, Victor Magron. Sums of squares representations on singular loci. 2023. ⟨hal-04023980⟩
9 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More