816 research outputs found
Measures with zeros in the inverse of their moment matrix
We investigate and discuss when the inverse of a multivariate truncated
moment matrix of a measure has zeros in some prescribed entries. We
describe precisely which pattern of these zeroes corresponds to independence,
namely, the measure having a product structure. A more refined finding is that
the key factor forcing a zero entry in this inverse matrix is a certain
conditional triangularity property of the orthogonal polynomials associated
with .Comment: Published in at http://dx.doi.org/10.1214/07-AOP365 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Semidefinite approximations of projections and polynomial images of semialgebraic sets
Given a compact semialgebraic set S of R^n and a polynomial map f from R^n to R^m, we consider the problem of approximating the image set F = f(S) in R^m. This includes in particular the projection of S on R^m for n greater than m. Assuming that F is included in a set B which is simple (e.g. a box or a ball), we provide two methods to compute certified outer approximations of F. Method 1 exploits the fact that F can be defined with an existential quantifier, while Method 2 computes approximations of the support of image measures.The two methods output a sequence of superlevel sets defined with a single polynomial that yield explicit outer approximations of F. Finding the coefficients of this polynomial boils down to computing an optimal solution of a convex semidefinite program. We provide guarantees of strong convergence to F in L^1 norm on B, when the degree of the polynomial approximation tends to infinity. Several examples of applications are provided, together with numerical experiments
A prolongation-projection algorithm for computing the finite real variety of an ideal
We provide a real algebraic symbolic-numeric algorithm for computing the real
variety of an ideal , assuming it is finite while may not
be. Our approach uses sets of linear functionals on , vanishing on a
given set of polynomials generating and their prolongations up to a given
degree, as well as on polynomials of the real radical ideal of , obtained
from the kernel of a suitably defined moment matrix assumed to be positive
semidefinite and of maximum rank. We formulate a condition on the dimensions of
projections of these sets of linear functionals, which serves as stopping
criterion for our algorithm. This algorithm, based on standard numerical linear
algebra routines and semidefinite optimization, combines techniques from
previous work of the authors together with an existing algorithm for the
complex variety. This results in a unified methodology for the real and complex
cases.Comment: revised versio
New approximations for the cone of copositive matrices and its dual
We provide convergent hierarchies for the cone C of copositive matrices and
its dual, the cone of completely positive matrices. In both cases the
corresponding hierarchy consists of nested spectrahedra and provide outer
(resp. inner) approximations for C (resp. for its dual), thus complementing
previous inner (resp. outer) approximations for C (for the dual). In
particular, both inner and outer approximations have a very simple
interpretation. Finally, extension to K-copositivity and K-complete positivity
for a closed convex cone K, is straightforward.Comment: 8
Exploiting symmetries in SDP-relaxations for polynomial optimization
In this paper we study various approaches for exploiting symmetries in
polynomial optimization problems within the framework of semi definite
programming relaxations. Our special focus is on constrained problems
especially when the symmetric group is acting on the variables. In particular,
we investigate the concept of block decomposition within the framework of
constrained polynomial optimization problems, show how the degree principle for
the symmetric group can be computationally exploited and also propose some
methods to efficiently compute in the geometric quotient.Comment: (v3) Minor revision. To appear in Math. of Operations Researc
On representations of the feasible set in convex optimization
We consider the convex optimization problem where is convex, the feasible set K is convex and Slater's
condition holds, but the functions are not necessarily convex. We show
that for any representation of K that satisfies a mild nondegeneracy
assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely
every KKT point is a minimizer. That is, the KKT optimality conditions are
necessary and sufficient as in convex programming where one assumes that the
are convex. So in convex optimization, and as far as one is concerned
with KKT points, what really matters is the geometry of K and not so much its
representation.Comment: to appear in Optimization Letter
Insulin and Glucagon Impairments in Relation with Islet Cells Morphological Modifications Following Long Term Pancreatic Duct Ligation in the Rabbit – A Model of Non-insulin-dependent Diabete
Plasma levels of glucose, insulin and glucagon were
measured at various time intervals after pancreatic
duct ligation (PDL) in rabbits. Two hyperglycemic
periods were observed: one between 15–90 days
(peak at 30 days of 15.1 ± 1.2mmol/l, p < 0.01), and
the other at 450 days (11.2 ± 0.5 mmol/l, p < 0.02). The
first hyperglycemic episode was significantly correlated
with both hypoinsulinemia (41.8 ± 8pmol/l,
r= –0.94, p < 0.01) and hyperglucagonemia (232 ±
21ng/l, r=0.95, p < 0.01). However, the late hyperglycemic
phase (450 days), which was not accompanied
by hypoinsulinemia, was observed after the
hyperglucagonemia (390 days) produced by abundant
immunostained A-cells giving rise to a 3-fold
increase in pancreatic glucagon stores. The insulin
and glucagon responses to glucose loading at 180,
270 and 450 days reflected the insensitivity of B- and
A-cells to glucose. The PDL rabbit model with
chronic and severe glycemic disorders due to the
predominant role of glucagon mimicked key features
of the NIDDM syndrome secondary to
exocrine disease
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