  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.
