Containment problems for polytopes and spectrahedra
Abstract
We study the computational question whether a given polytope or spectrahedron
S
A
(as given by the positive semidefiniteness region of a linear matrix pencil
A(x)
) is contained in another one
S
B
.
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
S
A
the criteria will always succeed in certifying containment of some scaled spectrahedron
ν
S
A
in
S
B
.