198 research outputs found
Small Extended Formulations for Cyclic Polytopes
We provide an extended formulation of size O(log n)^{\lfloor d/2 \rfloor} for
the cyclic polytope with dimension d and n vertices (i,i^2,\ldots,i^d), i in
[n]. First, we find an extended formulation of size log(n) for d= 2. Then, we
use this as base case to construct small-rank nonnegative factorizations of the
slack matrices of higher-dimensional cyclic polytopes, by iterated tensor
products. Through Yannakakis's factorization theorem, these factorizations
yield small-size extended formulations for cyclic polytopes of dimension d>2
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
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
On the Linear Extension Complexity of Regular n-gons
In this paper, we propose new lower and upper bounds on the linear extension
complexity of regular -gons. Our bounds are based on the equivalence between
the computation of (i) an extended formulation of size of a polytope ,
and (ii) a rank- nonnegative factorization of a slack matrix of the polytope
. The lower bound is based on an improved bound for the rectangle covering
number (also known as the boolean rank) of the slack matrix of the -gons.
The upper bound is a slight improvement of the result of Fiorini, Rothvoss and
Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp.
658-668, 2012]. The difference with their result is twofold: (i) our proof uses
a purely algebraic argument while Fiorini et al. used a geometric argument, and
(ii) we improve the base case allowing us to reduce their upper bound by one when for some integer . We conjecture that this new upper bound
is tight, which is suggested by numerical experiments for small . Moreover,
this improved upper bound allows us to close the gap with the best known lower
bound for certain regular -gons (namely, and ) hence allowing for the first time to determine their extension
complexity.Comment: 20 pages, 3 figures. New contribution: improved lower bound for the
boolean rank of the slack matrices of n-gon
Robust H
A new version of delay-dependent bounded real lemma for singular systems with state delay is established by parameterized Lyapunov-Krasovskii functional approach. In order to avoid generating nonconvex problem formulations in control design, a strategy that introduces slack matrices and decouples the system matrices from the Lyapunov-Krasovskii parameter matrices is used. Examples are provided to demonstrate that the results in this paper are less conservative than the existing corresponding ones in the literature
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