Sum-of-squares hierarchies for binary polynomial optimization - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Mathematical Programming Année : 2023

Sum-of-squares hierarchies for binary polynomial optimization

Lucas Slot
Monique Laurent
  • Fonction : Auteur
  • PersonId : 856623

Résumé

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube B n = {0, 1} n. This hierarchy provides for each integer r ∈ N a lower bound f (r) on the minimum f min of f , given by the largest scalar λ for which the polynomial f − λ is a sum-of-squares on B n with degree at most 2r. We analyze the quality of these bounds by estimating the worstcase error f min − f (r) in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed t ∈ [0, 1/2], we can show that this worst-case error in the regime r ≈ t • n is of the order 1/2 − √ t(1 − t) as n tends to ∞. Our proof combines classical Fourier analysis on B n with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds f (r) and another hierarchy of upper bounds f (r) , for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the q-ary cube (Z/qZ) n. Furthermore, our results apply to the setting of matrix-valued polynomials.
Fichier principal
Vignette du fichier
2011.04027.pdf (437.79 Ko) Télécharger le fichier

Dates et versions

hal-04088837 , version 1 (15-05-2023)

Identifiants

Citer

Lucas Slot, Monique Laurent. Sum-of-squares hierarchies for binary polynomial optimization. Mathematical Programming, 2023, 197 (2), pp.621-660. ⟨10.1007/s10107-021-01745-9⟩. ⟨hal-04088837⟩
4 Consultations
80 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More