Containment Problems for Polytopes and Spectrahedra

We study the computational question whether a given polytope or spectrahedron SA (as given by the positive semidefiniteness region of a linear matrix pencil A(x)) is contained in another one SB . First we classify the computational complexity, extending results on the polytope/polytope-case by Gritzmann and Klee to the polytope/spectrahedron-case. For various restricted containment problems, NP-hardness is shown. We then study in detail semidefinite conditions to certify containment, building upon work by Ben-Tal, Nemirovski and Helton, Klep, McCullough. In particular, we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron. It is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases, including containment of a spectrahedron in a polyhedron. Moreover, in the case of bounded SA the criteria will always succeed in certifying containment of some scaled spectrahedron νSA in SB .