2,424 research outputs found
The Quadratic Graver Cone, Quadratic Integer Minimization, and Extensions
We consider the nonlinear integer programming problem of minimizing a
quadratic function over the integer points in variable dimension satisfying a
system of linear inequalities. We show that when the Graver basis of the matrix
defining the system is given, and the quadratic function lies in a suitable
{\em dual Graver cone}, the problem can be solved in polynomial time. We
discuss the relation between this cone and the cone of positive semidefinite
matrices, and show that none contains the other. So we can minimize in
polynomial time some non-convex and some (including all separable) convex
quadrics.
We conclude by extending our results to efficient integer minimization of
multivariate polynomial functions of arbitrary degree lying in suitable cones
Volumes of Nonnegative Polynomials, Sums of Squares and Powers of Linear Forms
We study the quantitative relationship between the cones of nonnegative
polynomials, cones of sums of squares and cones of sums of powers of linear
forms. We derive bounds on the volumes (raised to the power reciprocal to the
ambient dimension) of compact sections of the three cones. We show that the
bounds are asymptotically exact if the degree is fixed and number of variables
tends to infinity. When the degree is larger than two it follows that there are
significantly more non-negative polynomials than sums of squares and there are
significantly more sums of squares than sums of powers of linear forms.
Moreover, we quantify the exact discrepancy between the cones; from our bounds
it follows that the discrepancy grows as the number of variables increases.Comment: 24 page
Do Sums of Squares Dream of Free Resolutions?
We associate to a real projective variety two convex cones which are
fundamental in real algebraic geometry: the cone of quadratic forms
nonnegative on , and the cone of sums of squares of linear forms.
The dual cone is a spectrahedron and we show that its convexity
properties are closely related to homological properties of . For instance,
we show that all extreme rays of have rank one if and only if X
has Castelnuovo-Mumford regularity two. More generally, if has
an extreme ray of rank , then does not satisfy the property
. We show that the converse also holds in a wide variety of
situations: the smallest for which property does not hold is
equal to the smallest rank of an extreme ray of greater than
one. These results allow us to generalize the work of Blekherman-Smith-Velasco
on equality of nonnegative polynomials and sums of squares from irreducible
varieties to reduced schemes and to classify all spectrahedral cones with only
rank one extreme rays. Our results have applications to the positive
semidefinite matrix completion problem and to the truncated moment problem on
projective varieties.Comment: 26 pages, comments welcom
Amenable cones: error bounds without constraint qualifications
We provide a framework for obtaining error bounds for linear conic problems
without assuming constraint qualifications or regularity conditions. The key
aspects of our approach are the notions of amenable cones and facial residual
functions. For amenable cones, it is shown that error bounds can be expressed
as a composition of facial residual functions. The number of compositions is
related to the facial reduction technique and the singularity degree of the
problem. In particular, we show that symmetric cones are amenable and compute
facial residual functions. From that, we are able to furnish a new H\"olderian
error bound, thus extending and shedding new light on an earlier result by
Sturm on semidefinite matrices. We also provide error bounds for the
intersection of amenable cones, this will be used to provided error bounds for
the doubly nonnegative cone.Comment: 36 pages, 1 figure. This version was significantly revised. A
discussion on the relation between amenability and related concepts was
added. In particular, there is a proof that amenable cones are nice and,
therefore, facially exposed. Also, gathered the results on symmetric cones in
a single section. Several typos and minor issues were fixe
Asymptotical flatness and cone structure at infinity
We investigate asymptotically flat manifolds with cone structure at infinity.
We show that any such manifold M has a finite number of ends. For simply
connected ends we classify all possible cones at infinity, except for the
4-dimensional case where it remains open if one of the theoretically possible
cones can actually arise.
This result yields in particular a complete classification of asymptotically
flat manifolds with nonnegative curvature: The universal covering of an
asymptotically flat manifold with nonnegative sectional curvature is isometric
to a product of Euclidean space and an asymptotically flat surface.Comment: 20 pages 1 pic, old paper with minor correction
A Novel Unified Approach to Invariance for a Dynamical System
In this paper, we propose a novel, unified, general approach to investigate
sufficient and necessary conditions under which four types of convex sets,
polyhedra, polyhedral cones, ellipsoids and Lorenz cones, are invariant sets
for a linear continuous or discrete dynamical system. In proving invariance of
ellipsoids and Lorenz cones for discrete systems, instead of the traditional
Lyapunov method, our novel proofs are based on the S-lemma, which enables us to
extend invariance conditions to any set represented by a quadratic inequality.
Such sets include nonconvex and unbounded sets. Finally, according to the
framework of our novel method, sufficient and necessary conditions for
continuous systems are derived from the sufficient and necessary conditions for
the corresponding discrete systems that are obtained by Euler methods
LP-based Tractable Subcones of the Semidefinite Plus Nonnegative Cone
The authors in a previous paper devised certain subcones of the semidefinite
plus nonnegative cone and showed that satisfaction of the requirements for
membership of those subcones can be detected by solving linear optimization
problems (LPs) with variables and constraints. They also
devised LP-based algorithms for testing copositivity using the subcones. In
this paper, they investigate the properties of the subcones in more detail and
explore larger subcones of the positive semidefinite plus nonnegative cone
whose satisfaction of the requirements for membership can be detected by
solving LPs. They introduce a {\em semidefinite basis (SD basis)} that is a
basis of the space of symmetric matrices consisting of
symmetric semidefinite matrices. Using the SD basis, they devise two new
subcones for which detection can be done by solving LPs with variables
and constraints. The new subcones are larger than the ones in the
previous paper and inherit their nice properties. The authors also examine the
efficiency of those subcones in numerical experiments. The results show that
the subcones are promising for testing copositivity as a useful application.Comment: 26 pages, 5 figure
Thom polynomials of invariant cones, Schur functions, and positivity
We generalize the notion of Thom polynomials from singularities of maps
between two complex manifolds to invariant cones in representations, and
collections of vector bundles. We prove that the generalized Thom polynomials,
expanded in the products of Schur functions of the bundles, have nonnegative
coefficients. For classical Thom polynomials associated with maps of complex
manifolds, this gives an extension of our former result for stable
singularities to nonnecessary stable ones. We also discuss some related aspects
of Thom polynomials, which makes the article expository to some extent.Comment: 14 page
Positive Gorenstein Ideals
We introduce positive Gorenstein ideals. These are Gorenstein ideals in the
graded ring \RR[x] with socle in degree 2d, which when viewed as a linear
functional on \RR[x]_{2d} is nonnegative on squares. Equivalently, positive
Gorenstein ideals are apolar ideals of forms whose differential operator is
nonnegative on squares. Positive Gorenstein ideals arise naturally in the
context of nonnegative polynomials and sums of squares, and they provide a
powerful framework for studying concrete aspects of sums of squares
representations. We present applications of positive Gorenstein ideals in real
algebraic geometry, analysis and optimization. In particular, we present a
simple proof of Hilbert's nearly forgotten result on representations of ternary
nonnegative forms as sums of squares of rational functions. Drawing on our
previous work, our main tools are Cayley-Bacharach duality and elementary
convex geometry
Generating functions and triangulations for lecture hall cones
We investigate the arithmetic-geometric structure of the lecture hall cone We show that is isomorphic to the cone
over the lattice pyramid of a reflexive simplex whose Ehrhart -polynomial
is given by the st Eulerian polynomial, and prove that lecture hall
cones admit regular, flag, unimodular triangulations. After explicitly
describing the Hilbert basis for , we conclude with observations and a
conjecture regarding the structure of unimodular triangulations of ,
including connections between enumerative and algebraic properties of and
cones over unit cubes
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