Near-optimal analysis of Lasserre’s univariate measure-based bounds for multivariate polynomial optimization

Abstract

We consider a hierarchy of upper approximations for the minimization of a polynomial f over a compact set $$K \subseteq \mathbb {R}^n$$ K ⊆ R n proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward measure of the Lebesgue measure on K by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r. Hence it is weaker, but cheaper to compute, than an earlier hierarchy by Lasserre (SIAM Journal on Optimization 21(3), 864–885, 2011), which uses multivariate sums of squares. We show that this new hierarchy converges to the global minimum of f at a rate in $$O(\log ^2 r / r^2)$$ O ( log 2 r / r 2 ) whenever K satisfies a mild geometric condition, which holds, eg., for convex bodies and for compact semialgebraic sets with dense interior. As an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares, which improves and extends earlier convergence results to a wider class of compact sets. Furthermore, we show that our analysis is near-optimal by proving a lower bound on the convergence rate in $$\varOmega (1/r^2)$$ Ω ( 1 / r 2 ) for a class of polynomials on $$K=[-1,1]$$ K = [ - 1 , 1 ] , obtained by exploiting a connection to orthogonal polynomials.

An effective version of Schmüdgen's Positivstellensatz for the hypercube

“…We refer to the works [5,12] (and further references therein) for the analysis of hierarchies of upper bounds (obtained by minimizing the expected value of f on S with respect to a sum-of-squares density).…”

Sum-of-Squares Hierarchies for Binary Polynomial Optimization

“…It has been shown that, on the hypersphere [8], the unit ball and the simplex [40], and the unit box [9], the bound f (r ) converges at a rate in O(1/r 2 ). A slightly weaker convergence rate in O(log 2 r /r 2 ) is known for general (full-dimensional) semi-algebraic sets [40,42]. Again, these results are all asymptotic in r , and thus hard to compare directly to our analysis on B n .…”

The Moment-SOS hierarchy and the Christoffel-Darboux kernel

“…In a series of papers, de Klerk, Laurent an co-workers have provided several rates of convergence of u t ↓ f * for several examples of sets B. For more details and results, the interested reader is referred to [1,11,12,13] and references therein.…”

“…At last but not least, this interpretation establishes another (and rather surprising) simple link between polynomial optimization (here the Moment-SOS hierarchy), the Christoffel-Darboux kernel and the Christoffel function, fundamental tools in the theory of orthogonal polynomials and the theory of approximation. Previous contributions in this vein include [8] to characterize upper bounds (1.3), [1,11,12] to analyze their rate of convergence to f * . The more recent contribution [13] provides rates of convergence of both upper and lower bounds on B = {0, 1} n , by following the strategy in [4] where the authors also relate the Moment-SOS hierarchy with the upper bound hierarchy (as also in [2]) to establish rates of convergence of the former on the hypersphere.…”