Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Overview of Convergence Rates for Sum of Squares Hierarchies in Polynomial Optimization

Monique Laurent, Lucas Slot

TL;DR

This survey analyzes convergence rates for sum-of-squares hierarchies in polynomial optimization, focusing on bounds that approximate the global minimum of a polynomial $f$ over a compact semialgebraic set $\mathbf{X}$. It surveys upper bounds $\mathrm{ub}(f,\mathbf{X},\mu)_r$ obtained via SOS densities and lower bounds $\mathrm{lb}(f,\mathcal{Q}(\mathbf{g}))_r$ and $\mathrm{lb}(f,\mathcal{T}(\mathbf{g}))_r$ provided by Positivstellensätze, detailing rates that depend on geometry, measure, and certificates. Core techniques include eigenvalue reformulations, links to orthogonal polynomials, the polynomial kernel method (PKM), and algebro-geometric reductions, with extensions to sparse POPs and generalized moment problems. The work highlights that upper-bound rates are essentially tight (up to log factors) for many sets, while lower-bound rates are more nuanced and exhibit both robust sublinear rates and, under additional conditions, potential exponential convergence. These insights connect to cubature rules and orthogonal-polynomial theory, offering a framework for analyzing and improving SOS-based relaxations in practical polynomial optimization tasks.

Abstract

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex) optimization problems. Various hierarchies of (lower and upper) bounds have been introduced, having the remarkable property that they converge asymptotically to the global minimum. These bounds exploit algebraic representations of positive polynomials in terms of sums of squares and can be computed using semidefinite optimization. Our focus lies in the performance analysis of these hierarchies of bounds, namely, in how far the bounds are from the global minimum as the degrees of the sums of squares they involve tend to infinity. We present the main state-of-the-art results and offer a gentle introductory overview over the various techniques that have been recently developed to establish them, stemming from the theory of orthogonal polynomials, approximation theory, Fourier analysis, and more.

An Overview of Convergence Rates for Sum of Squares Hierarchies in Polynomial Optimization

TL;DR

This survey analyzes convergence rates for sum-of-squares hierarchies in polynomial optimization, focusing on bounds that approximate the global minimum of a polynomial over a compact semialgebraic set . It surveys upper bounds obtained via SOS densities and lower bounds and provided by Positivstellensätze, detailing rates that depend on geometry, measure, and certificates. Core techniques include eigenvalue reformulations, links to orthogonal polynomials, the polynomial kernel method (PKM), and algebro-geometric reductions, with extensions to sparse POPs and generalized moment problems. The work highlights that upper-bound rates are essentially tight (up to log factors) for many sets, while lower-bound rates are more nuanced and exhibit both robust sublinear rates and, under additional conditions, potential exponential convergence. These insights connect to cubature rules and orthogonal-polynomial theory, offering a framework for analyzing and improving SOS-based relaxations in practical polynomial optimization tasks.

Abstract

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex) optimization problems. Various hierarchies of (lower and upper) bounds have been introduced, having the remarkable property that they converge asymptotically to the global minimum. These bounds exploit algebraic representations of positive polynomials in terms of sums of squares and can be computed using semidefinite optimization. Our focus lies in the performance analysis of these hierarchies of bounds, namely, in how far the bounds are from the global minimum as the degrees of the sums of squares they involve tend to infinity. We present the main state-of-the-art results and offer a gentle introductory overview over the various techniques that have been recently developed to establish them, stemming from the theory of orthogonal polynomials, approximation theory, Fourier analysis, and more.
Paper Structure (24 sections, 3 theorems, 43 equations, 1 figure, 2 tables)

This paper contains 24 sections, 3 theorems, 43 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Polynomial Optimization
  3. Examples and Applications
  4. Nonnegative Polynomials and Sums of Squares
  5. Sums of Squares and Semidefinite Optimization
  6. Upper Bounds
  7. Lower Bounds
  8. Link to Cubature Rules
  9. Performance Analysis of the Upper Bounds
  10. Some Useful Tricks for the Analysis
  11. Reformulation as an Eigenvalue Problem
  12. Univariate Case: Links to Roots of Orthogonal Polynomials
  13. Another Analysis Technique using Needle Polynomials
  14. Performance Analysis of the Lower Bounds
  15. Analysis via the Polynomial Kernel Method
  16. ...and 9 more sections

Key Result

Lemma 1

Let $f, g\in \mathbb{R}[\mathbf{x}]$ satisfy (i) $f(\mathbf{x})\le g(\mathbf{x})$ for all $\mathbf{x} \in \mathbf{X}$, and (ii) $f$ and $g$ take the same minimum value on $\mathbf{X}$, i.e., $f_{\text{\rm min},\mathbf{X}}=g_{\text{\rm min},\mathbf{X}}$. Then, for any $r\in \mathbb{N}$, we have $\tex

Figures (1)

  • Figure 1: (a) on the left: the Motzkin polynomial (\ref{['eqMotzinpol']}); (b) on the right: the optimal degree 16 sum-of-squares polynomial $\sigma$ in program (\ref{['eqfupr']}).

Theorems & Definitions (3)

  • Lemma 1: de Klerk, Laurent, Sun deKlerkLaurentSun2017
  • Lemma 2: Slot, Laurent SlotLaurent2022a
  • Lemma 3