Some developments in spectrahedral computation

and provide a central link between real algebraic geometry and
convex optimization. In this talk, we first provide some general
insights into recent developments on effective methods for handling
spectrahedra. This includes the algorithmic problems of deciding
emptiness of a spectrahedron S_A (as given by the positive
semidefiniteness region of a linear matrix pencil A(x)):
or its boundedness. Then we study the computational question
whether a given polytope or spectrahedron S_A is contained
in another one S_B. All these problems can profitably be
approached by combinations of methods from real algebra and
optimization.