{"paper":{"title":"Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Etienne de Klerk, Monique Laurent, Roxana Hess","submitted_at":"2016-03-10T16:56:42Z","abstract_excerpt":"We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of given degree $r$) that minimizes the expectation $\\int_{[-1,1]^n} f(x)h(x)d\\mu(x)$, where $d\\mu(x)$ is a fixed, finite Borel measure supported on $[-1,1]^n$. It is known that, for the Lebesgue measure $d\\mu(x) = dx$, one may show an error bound $O(1/\\sqrt{r})$ if $h$ is a sum-of-squares density, and an $O(1/r)$ error bound if $h$ is the density of a beta distribution. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03329","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}