10 research outputs found
Facially exposed cones are not always nice
We address the conjecture proposed by Gabor Pataki that every facially
exposed cone is nice. We show that the conjecture is true in the
three-dimensional case, however, there exists a four-dimensional counterexample
of a cone that is facially exposed but is not nice
Geometry of amenable cones
ISM Online Open House, 2021.6.18統計数理研究所オープンハウス(オンライン開催)、R3.6.18ポスター発
Hyperbolicity cones are amenable
Amenability is a notion of facial exposedness for convex cones that is
stronger than being facially dual complete (or "nice") which is, in turn,
stronger than merely being facially exposed. Hyperbolicity cones are a family
of algebraically structured closed convex cones that contain all spectrahedra
(linear sections of positive semidefinite cones) as special cases. It is known
that all spectrahedra are amenable. We establish that all hyperbolicity cones
are amenable. As part of the argument, we show that any face of a hyperbolicity
cone is a hyperbolicity cone. As a corollary, we show that the intersection of
two hyperbolicity cones, not necessarily sharing a common relative interior
point, is a hyperbolicity cone.Comment: 8 pages. Comments welcom
Amenable cones are particularly nice
Amenability is a geometric property of convex cones that is stronger than
facial exposedness and assists in the study of error bounds for conic
feasibility problems. In this paper we establish numerous properties of
amenable cones, and investigate the relationships between amenability and other
properties of convex cones, such as niceness and projectional exposure.
We show that the amenability of a compact slice of a closed convex cone is
equivalent to the amenability of the cone, and prove several results on the
preservation of amenability under intersections and other convex operations. It
then follows that homogeneous, doubly nonnegative and other cones that can be
represented as slices of the cone of positive semidefinite matrices, are
amenable.
It is known that projectionally exposed cones are amenable and that amenable
cones are nice, however the converse statements have been open questions. We
construct an example of a four-dimensional cone that is nice but not amenable.
We also show that amenable cones are projectionally exposed in dimensions up to
and including four.
We conclude with a discussion on open problems related to facial structure of
convex sets that we came across in the course of this work, but were not able
to fully resolve.Comment: 29 pages, 3 figures, comments welcom
Self-dual polyhedral cones and their slack matrices
We analyze self-dual polyhedral cones and prove several properties about
their slack matrices. In particular, we show that self-duality is equivalent to
the existence of a positive semidefinite (PSD) slack. Beyond that, we show that
if the underlying cone is irreducible, then the corresponding PSD slacks are
not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of
DNN matrices, which correspond to a family of extreme rays not previously
described. More surprisingly, we show that, unless the cone is simplicial, PSD
slacks not only fail to be completely positive matrices but they also lie
outside the cone of completely positive semidefinite matrices. Finally, we show
how one can use semidefinite programming to probe the existence of self-dual
cones with given combinatorics. Our results are given for polyhedral cones but
we also discuss some consequences for negatively self-polar polytopes.Comment: 26 pages, 4 figures. Some minor fixes and simplification
Facially exposed cones are not always nice
Abstract We address the conjecture proposed by Gábor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the three-dimensional case, however, there exists a four-dimensional counterexample of a cone that is facially exposed but is not nice