arXiv subjects

Lucas Slot

Publications and source records attributed to Lucas Slot.

20 records · Page 2Linked to original sources

arXiv2020

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

We consider a hierarchy of upper approximations for the minimization of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$ 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 $O(\log^2 r / r^2)$ whenever $K$ satisfies a mild geometric condition, which holds, e.g., 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 $Ω(1/r^2)$ for a class of polynomials on $K=[-1,1]$, obtained by exploiting a connection to orthogonal polynomials.

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arXiv2020

Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets

We consider the problem of computing the minimum value $f_{\min,K}$ of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$, which can be reformulated as finding a probability measure $ν$ on $K$ minimizing $\int_K f dν$. Lasserre showed that it suffices to consider such measures of the form $ν= qμ$, where $q$ is a sum-of-squares polynomial and $μ$ is a given Borel measure supported on $K$. By bounding the degree of $q$ by $2r$ one gets a converging hierarchy of upper bounds $f^{(r)}$ for $f_{\min,K}$. When $K$ is the hypercube $[-1, 1]^n$, equipped with the Chebyshev measure, the parameters $f^{(r)}$ are known to converge to $f_{\min,K}$ at a rate in $O(1/r^2)$. We extend this error estimate to a wider class of convex bodies, while also allowing for a broader class of reference measures, including the Lebesgue measure. Our analysis applies to simplices, balls and convex bodies that locally look like a ball. In addition, we show an error estimate in $O(\log r / r)$ when $K$ satisfies a minor geometrical condition, and in $O(\log^2 r / r^2)$ when $K$ is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in $O(1 / \sqrt{r})$ and $O(1/r)$ for these two respective cases.

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