Frank Vallentin 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 … Read more
We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical … Read more
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 … Read more
Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: A difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constrained is dropped. For instance, the integrality gap … Read more
In the last years many results in the area of semidefinite programming were obtained for invariant (finite dimensional, or infinite dimensional) semidefinite programs – SDPs which have symmetry. This was done for a variety of problems and applications. The purpose of this handbook chapter is to give the reader the necessary background for dealing with … Read more
Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint is (SDP_n) maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, …, x_m \in S^{n-1}. In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation … Read more
The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for … Read more
The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for … Read more
Lecture notes for the tutorial at the workshop HPOPT 2008 – 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands. CitationarXiv:0809.2017v1 [math.OC]ArticleDownload View PDF
In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions … Read more