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Sum-of-squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel

Lucas Slot

Abstract

Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set ${\mathbf{X} \subseteq \mathbb{R}^n}$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schmüdgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of $f$ at a rate in $O(1/r^2)$, matching recently obtained convergence rates for the hypersphere and hypercube $[-1,1]^n$. For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel-Darboux kernel, and make use of closed form expressions for this kernel derived by Xu.

Sum-of-squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel

Abstract

Consider the problem of minimizing a polynomial over a compact semialgebraic set . Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schmüdgen. When is the unit ball or the standard simplex, we show that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of at a rate in , matching recently obtained convergence rates for the hypersphere and hypercube . For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel-Darboux kernel, and make use of closed form expressions for this kernel derived by Xu.

Paper Structure

This paper contains 29 sections, 20 theorems, 114 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Positivity certificates and sum-of-squares hierarchies
  3. Hierarchies of lower bounds
  4. Hierarchies of upper bounds.
  5. Related work
  6. Lower bounds.
  7. Upper bounds.
  8. Our contributions
  9. Outline of the proof technique for the main results
  10. Preliminaries
  11. Notations
  12. Quadratic modules and preorderings
  13. Gegenbauer polynomials.
  14. Construction of the linear operator
  15. Closed forms of the Christoffel-Darboux kernel
  16. ...and 14 more sections

Key Result

Theorem 1

Let $\mathbf{X} \subseteq \mathbb{R}^n$ be a semialgebraic set, and assume that $R - \|\mathbf{x}\|^2 \in {\mathcal{Q}}(\mathbf{X})$ for some $R > 0$. Then for any polynomial $f \in \mathcal{P}_+(\mathbf{X})$ and $\eta > 0$, we have $f + \eta \in {\mathcal{Q}}(\mathbf{X})$.

Figures (1)

  • Figure 1: Overview of the construction of a a linear operator ${\mathbf{K} : \mathbb{R}[\mathbf{x}]_d \to \mathbb{R}[\mathbf{x}]_d}$ satisfying the properties \ref{['PROPERTY:normalization']}, \ref{['PROPERTY:incone']}, \ref{['PROPERTY:infnorm']} of Section \ref{['SEC:outline']} for the unit ball. The construction for the standard simplex is analogous.

Theorems & Definitions (29)

  • Theorem 1: Putinar's Positvistellensatz
  • Theorem 2: Schmüdgen's Positvistellensatz
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Example 8
  • Theorem 9: Xu Xu:ball, Theorem 3.1
  • Theorem 10: Xu Xu:simplex, Corollary 2.4
  • ...and 19 more