15 research outputs found
Convex Hulls, Oracles, and Homology
This paper presents a new algorithm for the convex hull problem, which is
based on a reduction to a combinatorial decision problem
POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a
simplicial homology computation. Like other convex hull algorithms, our
algorithm is polynomial (in the size of input plus output) for simplicial or
simple input. We show that the ``no''-case of
POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in
polynomial time (if integrity of the input is guaranteed).Comment: 11 pages, 2 figure
Sharpening Geometric Inequalities using Computable Symmetry Measures
Many classical geometric inequalities on functionals of convex bodies depend
on the dimension of the ambient space. We show that this dimension dependence
may often be replaced (totally or partially) by different symmetry measures of
the convex body. Since these coefficients are bounded by the dimension but
possibly smaller, our inequalities sharpen the original ones. Since they can
often be computed efficiently, the improved bounds may also be used to obtain
better bounds in approximation algorithms.Comment: This is a preprint. The proper publication in final form is available
at journals.cambridge.org, DOI 10.1112/S002557931400029
Computational Aspects of the Hausdorff Distance in Unbounded Dimension
We study the computational complexity of determining the Hausdorff distance
of two polytopes given in halfspace- or vertex-presentation in arbitrary
dimension. Subsequently, a matching problem is investigated where a convex body
is allowed to be homothetically transformed in order to minimize its Hausdorff
distance to another one. For this problem, we characterize optimal solutions,
deduce a Helly-type theorem and give polynomial time (approximation) algorithms
for polytopes
A Semidefinite Hierarchy for Containment of Spectrahedra
A spectrahedron is the positivity region of a linear matrix pencil and thus
the feasible set of a semidefinite program. We propose and study a hierarchy of
sufficient semidefinite conditions to certify the containment of a
spectrahedron in another one. This approach comes from applying a moment
relaxation to a suitable polynomial optimization formulation. The hierarchical
criterion is stronger than a solitary semidefinite criterion discussed earlier
by Helton, Klep, and McCullough as well as by the authors. Moreover, several
exactness results for the solitary criterion can be brought forward to the
hierarchical approach. The hierarchy also applies to the (equivalent) question
of checking whether a map between matrix (sub-)spaces is positive. In this
context, the solitary criterion checks whether the map is completely positive,
and thus our results provide a hierarchy between positivity and complete
positivity.Comment: 24 pages, 2 figures; minor corrections; to appear in SIAM J. Opti