Containment Problems for Polytopes and Spectrahedra

Abstract

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 νS A in S B .

Linear Encodings for Polytope Containment Problems

“…By left multiplying both sides in (17) by Y , X ⊆ Y is established. Before moving to the necessary conditions, we remark that (16) is a generalized form of the conditions reported in [21], where the authors considered containment problems for orthogonal projections of polytopes. Note that u is restricted to a cone that is the intersection of positive orthant and the linear subspace given by {u ∈ R qy |H y u ∈ range(Y )}.…”

“…Corollary 2 is a known result in the literature. A version was derived in [21]. It can also be proved using other techniques in basic convex analysis [12].…”

“…The proof relied on the fact that such a decision problem must involve vertex/facet enumeration. More recently, Kellner [21] proved that polytope containment problem for projections of H-polytopes is also NP-complete, and cast (2) as a bilinear optimization problem, which can be solved using sequential semidefinite programs. This paper provides linear sufficient conditions for this problem.…”

“…One can verify Z x ⊆ Z y through checking all the vertices of Z x . However, Theorem 3 fails to establish containment as (19) is infeasible. However, feasibility is gained by scaling Z x by 0.9915.…”

A Matrix Positivstellensatz with Lifting Polynomials

“…If S A has nonempty interior (this is the case if A 0 = I d , the d × d identity matrix), then (1.2) holds if and only if the matricial relaxation of S A is contained in the matricial relaxation of S B [HKM12,HKM13]. When B(x) is the normal form of an ellipsoid or polytope, the certificate (1.2) is necessary and sufficient for S A ⊆ S B , as shown by Kellner, Theobald and Trabandt [KTT13]. More general spectrahedral containment is also addressed by the same authors in [KTT15].…”