317 research outputs found
Lifts of convex sets and cone factorizations
In this paper we address the basic geometric question of when a given convex
set is the image under a linear map of an affine slice of a given closed convex
cone. Such a representation or 'lift' of the convex set is especially useful if
the cone admits an efficient algorithm for linear optimization over its affine
slices. We show that the existence of a lift of a convex set to a cone is
equivalent to the existence of a factorization of an operator associated to the
set and its polar via elements in the cone and its dual. This generalizes a
theorem of Yannakakis that established a connection between polyhedral lifts of
a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts
of convex sets can also be characterized similarly. When the cones live in a
family, our results lead to the definition of the rank of a convex set with
respect to this family. We present results about this rank in the context of
cones of positive semidefinite matrices. Our methods provide new tools for
understanding cone lifts of convex sets.Comment: 20 pages, 2 figure
Orbitopes
An orbitope is the convex hull of an orbit of a compact group acting linearly
on a vector space. These highly symmetric convex bodies lie at the crossroads
of several fields, in particular convex geometry, optimization, and algebraic
geometry. We present a self-contained theory of orbitopes, with particular
emphasis on instances arising from the groups SO(n) and O(n). These include
Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese
orbitopes and Grassmann orbitopes. We study their face lattices, their
algebraic boundary hypersurfaces, and representations as spectrahedra or
projected spectrahedra.Comment: 37 pages. minor revisions of origina
The Convex Hull of a Variety
We present a characterization, in terms of projective biduality, for the
hypersurfaces appearing in the boundary of the convex hull of a compact real
algebraic variety.Comment: 12 pages, 2 figure
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
Support Sets in Exponential Families and Oriented Matroid Theory
The closure of a discrete exponential family is described by a finite set of
equations corresponding to the circuits of an underlying oriented matroid.
These equations are similar to the equations used in algebraic statistics,
although they need not be polynomial in the general case. This description
allows for a combinatorial study of the possible support sets in the closure of
an exponential family. If two exponential families induce the same oriented
matroid, then their closures have the same support sets. Furthermore, the
positive cocircuits give a parameterization of the closure of the exponential
family.Comment: 27 pages, extended version published in IJA
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