Related papers: A New Linear Programming Method in Sphere Packing
A novel algorithm to solve the quadratic programming problem over ellipsoids is proposed. This is achieved by splitting the problem into two optimisation sub-problems, quadratic programming over a sphere and orthogonal projection. Next, an…
We consider continuous linear programs over a continuous finite time horizon $T$, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions, where we search for optimal solutions in the space…
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic…
Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains…
In this article we obtain linear programming bounds for the maximal sphere packing density of commutative spaces. A special case of our results solves a conjecture by Cohn and Zhao on linear programming bounds for sphere packings in…
We propose a new class of space-filling designs called rotated sphere packing designs for computer experiments. The approach starts from the asymptotically optimal positioning of identical balls that covers the unit cube. Properly scaled,…
Linear programming is the seminal optimization problem that has spawned and grown into today's rich and diverse optimization modeling and algorithmic landscape. This article provides an overview of the recent development of first-order…
The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by…
The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of…
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…
Many of the classic problems of coding theory are highly symmetric, which makes it easy to derive sphere-packing upper bounds and sphere-covering lower bounds on the size of codes. We discuss the generalizations of sphere-packing and…
The linear model uses the space defined by the input to project the target or desired signal and find the optimal set of model parameters. When the problem is nonlinear, the adaption requires nonlinear models for good performance, but it…
Based on smoothing techniques, we propose two new methods to solve linear complementarity problems (LCP) called TLCP and Soft-Max. The idea of these two new methods takes inspiration from interior-point methods in optimization. The…
The linear coupling method was introduced recently by Allen-Zhu and Orecchia for solving convex optimization problems with first order methods, and it provides a conceptually simple way to integrate a gradient descent step and mirror…
Polyhedral projection is a main operation of the polyhedron abstract domain.It can be computed via parametric linear programming (PLP), which is more efficient than the classic Fourier-Motzkin elimination method.In prior work, PLP was done…
Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for…
How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in…
We propose a unified framework that synthesizes advances in high-dimensional lattice theory with novel computational algorithms for the shortest vector problem (SVP) to model pure root lattices and compute sphere packing densities. Building…
The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi proved there are finitely many inequivalent local optima and presented…
This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are…