1,324 research outputs found
On the Closedness of the Linear Image of a Closed Convex Cone
When is the linear image of a closed convex cone closed? We present very simple, and intuitive necessary conditions, which unify, and generalize seemingly disparate, classical sufficient conditions: polyhedrality of the cone, and “Slater” type conditions are necessary and sufficient, when the dual cone belongs to a class, that we call nice cones. Nice cones subsume all cones amenable to treatment by efficient optimization algorithms: for instance, polyhedral, semidefinite, and p-cones. provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones
Coordinate shadows of semi-definite and Euclidean distance matrices
We consider the projected semi-definite and Euclidean distance cones onto a
subset of the matrix entries. These two sets are precisely the input data
defining feasible semi-definite and Euclidean distance completion problems. We
classify when these sets are closed, and use the boundary structure of these
two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. In
particular, we show that under a chordality assumption, the "minimal cones" of
these problems admit combinatorial characterizations. As a byproduct, we record
a striking relationship between the complexity of the general facial reduction
algorithm (singularity degree) and facial exposedness of conic images under a
linear mapping.Comment: 21 page
Bad semidefinite programs: they all look the same
Conic linear programs, among them semidefinite programs, often behave
pathologically: the optimal values of the primal and dual programs may differ,
and may not be attained. We present a novel analysis of these pathological
behaviors. We call a conic linear system {\em badly behaved} if the
value of is finite but the dual program has no
solution with the same value for {\em some} We describe simple and
intuitive geometric characterizations of badly behaved conic linear systems.
Our main motivation is the striking similarity of badly behaved semidefinite
systems in the literature; we characterize such systems by certain {\em
excluded matrices}, which are easy to spot in all published examples.
We show how to transform semidefinite systems into a canonical form, which
allows us to easily verify whether they are badly behaved. We prove several
other structural results about badly behaved semidefinite systems; for example,
we show that they are in in the real number model of computing.
As a byproduct, we prove that all linear maps that act on symmetric matrices
can be brought into a canonical form; this canonical form allows us to easily
check whether the image of the semidefinite cone under the given linear map is
closed.Comment: For some reason, the intended changes between versions 4 and 5 did
not take effect, so versions 4 and 5 are the same. So version 6 is the final
version. The only difference between version 4 and version 6 is that 2 typos
were fixed: in the last displayed formula on page 6, "7" was replaced by "1";
and in the 4th displayed formula on page 12 "A_1 - A_2 - A_3" was replaced by
"A_3 - A_2 - A_1
On the connection of facially exposed and nice cones
A closed convex cone K is called nice, if the set K^* + F^\perp is closed for
all F faces of K, where K^* is the dual cone of K, and F^\perp is the
orthogonal complement of the linear span of F. The niceness property is
important for two reasons: it plays a role in the facial reduction algorithm of
Borwein and Wolkowicz, and the question whether the linear image of a nice cone
is closed also has a simple answer.
We prove several characterizations of nice cones and show a strong connection
with facial exposedness. We prove that a nice cone must be facially exposed; in
reverse, facial exposedness with an added condition implies niceness.
We conjecture that nice, and facially exposed cones are actually the same,
and give supporting evidence
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
The convex real projective orbifolds with radial or totally geodesic ends: a survey of some partial results
A real projective orbifold has a radial end if a neighborhood of the end is
foliated by projective geodesics that develop into geodesics ending at a common
point. It has a totally geodesic end if the end can be completed to have the
totally geodesic boundary.
The purpose of this paper is to announce some partial results. A real
projective structure sometimes admits deformations to parameters of real
projective structures. We will prove a homeomorphism between the deformation
space of convex real projective structures on an orbifold with
radial or totally geodesic ends with various conditions with the union of open
subspaces of strata of the corresponding subset of Lastly, we will talk about the
openness and closedness of the properly (resp. strictly) convex real projective
structures on a class of orbifold with generalized admissible ends.Comment: 36 pages, 2 figure. Corrected a few mistakes including the condition
(NA) on page 22, arXiv admin note: text overlap with arXiv:1011.106
Approximate cone factorizations and lifts of polytopes
In this paper we show how to construct inner and outer convex approximations
of a polytope from an approximate cone factorization of its slack matrix. This
provides a robust generalization of the famous result of Yannakakis that
polyhedral lifts of a polytope are controlled by (exact) nonnegative
factorizations of its slack matrix. Our approximations behave well under
polarity and have efficient representations using second order cones. We
establish a direct relationship between the quality of the factorization and
the quality of the approximations, and our results extend to generalized slack
matrices that arise from a polytope contained in a polyhedron
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