1,346 research outputs found
Equivariant semidefinite lifts and sum-of-squares hierarchies
A central question in optimization is to maximize (or minimize) a linear
function over a given polytope P. To solve such a problem in practice one needs
a concise description of the polytope P. In this paper we are interested in
representations of P using the positive semidefinite cone: a positive
semidefinite lift (psd lift) of a polytope P is a representation of P as the
projection of an affine slice of the positive semidefinite cone
. Such a representation allows linear optimization problems
over P to be written as semidefinite programs of size d. Such representations
can be beneficial in practice when d is much smaller than the number of facets
of the polytope P. In this paper we are concerned with so-called equivariant
psd lifts (also known as symmetric psd lifts) which respect the symmetries of
the polytope P. We present a representation-theoretic framework to study
equivariant psd lifts of a certain class of symmetric polytopes known as
orbitopes. Our main result is a structure theorem where we show that any
equivariant psd lift of size d of an orbitope is of sum-of-squares type where
the functions in the sum-of-squares decomposition come from an invariant
subspace of dimension smaller than d^3. We use this framework to study two
well-known families of polytopes, namely the parity polytope and the cut
polytope, and we prove exponential lower bounds for equivariant psd lifts of
these polytopes.Comment: v2: 30 pages, Minor changes in presentation; v3: 29 pages, New
structure theorem for general orbitopes + changes in presentatio
On the origin of nonclassicality in single systems
In the framework of certain general probability theories of single systems,
we identify various nonclassical features such as incompatibility, multiple
pure-state decomposability, measurement disturbance, no-cloning and the
impossibility of certain universal operations, with the non-simpliciality of
the state space. This is shown to naturally suggest an underlying simplex as an
ontological model. Contextuality turns out to be an independent nonclassical
feature, arising from the intransitivity of compatibility.Comment: Close to the published versio
Igusa's p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure
For p prime, we give an explicit formula for Igusa's local zeta function
associated to a polynomial mapping f=(f_1,...,f_t): Q_p^n -> Q_p^t, with
f_1,...,f_t in Z_p[x_1,...,x_n], and an integration measure on Z_p^n of the
form |g(x)||dx|, with g another polynomial in Z_p[x_1,...,x_n]. We treat the
special cases of a single polynomial and a monomial ideal separately. The
formula is in terms of Newton polyhedra and will be valid for f and g
sufficiently non-degenerated over F_p with respect to their Newton polyhedra.
The formula is based on, and is a generalization of results of Denef -
Hoornaert, Howald et al., and Veys - Zuniga-Galindo.Comment: 20 pages, 5 figures, 2 table
The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems
Since its inception as a student project in 2001, initially just for the
handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library
has been continuously improved and extended by joining scrupulous research on
the theoretical foundations of (possibly non-convex) numerical abstractions to
a total adherence to the best available practices in software development. Even
though it is still not fully mature and functionally complete, the Parma
Polyhedra Library already offers a combination of functionality, reliability,
usability and performance that is not matched by similar, freely available
libraries. In this paper, we present the main features of the current version
of the library, emphasizing those that distinguish it from other similar
libraries and those that are important for applications in the field of
analysis and verification of hardware and software systems.Comment: 38 pages, 2 figures, 3 listings, 3 table
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