1,694 research outputs found
A semidefinite programming hierarchy for packing problems in discrete geometry
Packing problems in discrete geometry can be modeled as finding independent
sets in infinite graphs where one is interested in independent sets which are
as large as possible. For finite graphs one popular way to compute upper bounds
for the maximal size of an independent set is to use Lasserre's semidefinite
programming hierarchy. We generalize this approach to infinite graphs. For this
we introduce topological packing graphs as an abstraction for infinite graphs
coming from packing problems in discrete geometry. We show that our hierarchy
converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in
Mathematical Programming Series B special issue on polynomial optimizatio
Upper bounds for packings of spheres of several radii
We give theorems that can be used to upper bound the densities of packings of
different spherical caps in the unit sphere and of translates of different
convex bodies in Euclidean space. These theorems extend the linear programming
bounds for packings of spherical caps and of convex bodies through the use of
semidefinite programming. We perform explicit computations, obtaining new
bounds for packings of spherical caps of two different sizes and for binary
sphere packings. We also slightly improve bounds for the classical problem of
packing identical spheres.Comment: 31 page
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
A second order cone formulation of continuous CTA model
The final publication is available at link.springer.comIn this paper we consider a minimum distance Controlled Tabular Adjustment (CTA) model for statistical disclosure limitation (control) of tabular data. The goal of the CTA model is to find the closest safe table to some original tabular data set that contains sensitive information. The measure of closeness is usually measured using l1 or l2 norm; with each measure having its advantages and disadvantages. Recently, in [4] a regularization of the l1 -CTA using Pseudo-Huber func- tion was introduced in an attempt to combine positive characteristics of both l1 -CTA and l2 -CTA. All three models can be solved using appro- priate versions of Interior-Point Methods (IPM). It is known that IPM in general works better on well structured problems such as conic op- timization problems, thus, reformulation of these CTA models as conic optimization problem may be advantageous. We present reformulation of Pseudo-Huber-CTA, and l1 -CTA as Second-Order Cone (SOC) op- timization problems and test the validity of the approach on the small example of two-dimensional tabular data set.Peer ReviewedPostprint (author's final draft
Matrix convex functions with applications to weighted centers for semidefinite programming
In this paper, we develop various calculus rules for general smooth matrix-valued functions and for the class of matrix convex (or concave) functions first introduced by Loewner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function -log X to study a new notion of weighted convex centers for semidefinite programming (SDP) and show that, with this definition, some known properties of weighted centers for linear programming can be extended to SDP. We also show how the calculus rules for matrix convex functions can be used in the implementation of barrier methods for optimization problems involving nonlinear matrix functions
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