Realizability and inscribability for simplicial polytopes via nonlinear optimization

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a complete classification of neighborly polytopes of dimension $4$, $6$ and $7$ with $11$ vertices, of neighborly $5$-polytopes with $10$ vertices, as well as a complete classification of simplicial $3$-spheres with $10$ vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable.