  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.
As an application, 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: $1.4367\leqslant
K_G(3)\leqslant1.4546$. We also demonstrate the efficiency of our approach in
multipartite Bell scenarios, and present the first 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.
