TY - JOUR A1 - Designolle, Sébastien A1 - Iommazzo, Gabriele A1 - Besançon, Mathieu A1 - Knebel, Sebastian A1 - Gelß, Patrick A1 - Pokutta, Sebastian T1 - Improved local models and new Bell inequalities via Frank-Wolfe algorithms T2 - Physical Review Research N2 - In Bell scenarios with two outcomes per party, we algorithmically consider the two sides of the membership problem for the local polytope: Constructing local models and deriving separating hyperplanes, that is, Bell inequalities. We take advantage of the recent developments in so-called Frank-Wolfe algorithms to significantly increase the convergence rate of existing methods. First, we study the threshold value for the nonlocality of two-qubit Werner states under projective measurements. Here, we improve on both the upper and lower bounds present in the literature. Importantly, our bounds are entirely analytical; moreover, they yield refined bounds on the value of the Grothendieck constant of order three: 1.4367⩽KG(3)⩽1.4546. Second, we demonstrate the efficiency of our approach in multipartite Bell scenarios, and present local models for all projective measurements with visibilities noticeably higher than the entanglement threshold. We make our entire code accessible as a julia library called BellPolytopes.jl. Y1 - 2023 UR - https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/9415 VL - 5 SP - 043059 ER -