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Frank Vallentin

Publications and source records attributed to Frank Vallentin.

At least 37 records · Page 2Linked to original sources
arXiv2020

On the integrality gap of the maximum-cut semidefinite programming relaxation in fixed dimension

We describe a factor-revealing convex optimization problem for the integrality gap of the maximum-cut semidefinite programming relaxation: for each $n \geq 2$ we present a convex optimization problem whose optimal value is the largest possible ratio between the value of an optimal rank-$n$ solution to the relaxation and the value of an optimal cut. This problem is then used to compute lower bounds for the integrality gap.

math.OC↗
arXiv2020

A simplex algorithm for rational cp-factorization

In this paper we provide an algorithm, similar to the simplex algorithm, which determines a rational cp-factorization of a given matrix, whenever the matrix allows such a factorization. This algorithm can be used to show that every integral completely positive $2 \times 2$ matrix has an integral cp-factorization.

math.OC↗
arXiv2019

$k$-point semidefinite programming bounds for equiangular lines

We give a hierarchy of $k$-point bounds extending the Delsarte-Goethals-Seidel linear programming $2$-point bound and the Bachoc-Vallentin semidefinite programming $3$-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute~$4$, $5$, and $6$-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.

math.OC↗
arXiv2019

Semidefinite programming bounds for error-correcting codes

This chapter is written for the forthcoming book "A Concise Encyclopedia of Coding Theory" (CRC press), edited by W. Cary Huffman, Jon-Lark Kim, and Patrick Solé. This book will collect short but foundational articles, emphasizing definitions, examples, exhaustive references, and basic facts on the model of the Handbook of Finite Fields. The target audience of the Encyclopedia is upper level undergraduates and graduate students.

cs.IT↗
arXiv2018

New dense superball packings in three dimensions

In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p \in (1, 1.58]$. We conjecture that the family also exists for $p \in (1.58, \log_2 3 = 1.5849625\ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.

math.MG↗
arXiv2018

Covering compact metric spaces greedily

A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of Chvatal's analysis of the greedy algorithm for the weighted set cover problem. The approach is demonstrated in an exemplary manner to construct efficient coverings of the n-dimensional sphere and n-dimensional Euclidean space to give short and transparent proofs of several best known bounds obtained from deterministic constructions in the literature on sphere coverings.

math.MG↗
arXiv2017

New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.

math.MG↗
arXiv2016

Computing upper bounds for the packing density of congruent copies of a convex body

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We illustrate its use by computing an upper bound for the maximum density of packings of regular pentagons in the plane. Our computational approach is numerical and uses a combination of semidefinite programming, sums of squares, and the harmonic analysis of the Euclidean motion group. We show how, with some extra work, the bounds so obtained can be made rigorous.

math.MG↗
arXiv2015

Mathematical optimization for packing problems

During the last few years several new results on packing problems were obtained using a blend of tools from semidefinite optimization, polynomial optimization, and harmonic analysis. We survey some of these results and the techniques involved, concentrating on geometric packing problems such as the sphere-packing problem or the problem of packing regular tetrahedra in R^3.

math.OC↗
arXiv2015

A copositive formulation for the stability number of infinite graphs

In the last decade, copositive formulations have been proposed for a variety of combinatorial optimization problems, for example the stability number (independence number). In this paper, we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite-dimensional copositive program. For this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures. We determine the extreme rays of the latter cone, and we illustrate this theory with the help of the kissing number problem.

math.OC↗
arXiv2015

On the Turing model complexity of interior point methods for semidefinite programming

It is known that one can solve semidefinite programs to within fixed accuracy in polynomial time using the ellipsoid method (under some assumptions). In this paper it is shown that the same holds true when one uses the short-step, primal interior point method. The main idea of the proof is to employ Diophantine approximation at each iteration to bound the intermediate bit-sizes of iterates.

math.OC↗
arXiv2013

Fourier analysis on finite groups and the Lovász theta-number of Cayley graphs

We apply Fourier analysis on finite groups to obtain simplified formulations for the Lovász theta-number of a Cayley graph. We put these formulations to use by checking a few cases of a conjecture of Ellis, Friedgut, and Pilpel made in a recent article proving a version of the Erdős-Ko-Rado theorem for $k$-intersecting families of permutations. We also introduce a $q$-analog of the notion of $k$-intersecting families of permutations, and we verify a few cases of the corresponding Erdős-Ko-Rado assertion by computer.

math.OC↗
arXiv2013

Spectral bounds for the independence ratio and the chromatic number of an operator

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 framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the unit sphere.

math.FA↗
arXiv2013

A quantitative version of Steinhaus' theorem for compact, connected, rank-one symmetric spaces

Let $d_1$, $d_2$, ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus' theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances $d_1$, $d_2$, ... from each other, then it has to have measure zero. We present a quantitative version of this result for compact, connected, rank-one symmetric spaces, by showing how to choose distances so that the measure of a subset not containing pairs of points at these distances decays exponentially in the number of distances.

math.CO↗