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Containment problems for polytopes and spectrahedra

Kai Kellner (kellner***at***math.uni-frankfurt.de)
Thorsten Theobald (theobald***at***math.uni-frankfurt.de)
Christian Trabandt (trabandt***at***math.uni-frankfurt.de)

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/poly\-tope-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 $\nu S_A$ in $S_B$.

Keywords: polyhedron, spectrahedron, containment, relaxation, semidefinite programming, polytope

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Combinatorial Optimization (Polyhedra )

Citation: Submitted to SIAM Journal on Optimization

Download: [PDF]

Entry Submitted: 04/25/2012
Entry Accepted: 04/25/2012
Entry Last Modified: 10/22/2012

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