Related papers: Three-point bounds for energy minimization
In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems,…
We show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes (the new cases are $56$ points in $20$ dimensions, $50$ points in $21$ dimensions, and $77$ points in $21$ dimensions), as…
We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
A spherical three-distance set is a finite collection $X$ of unit vectors in $\mathbb{R}^{n}$ such that for each pair of distinct vectors has three inner product values. We use the semidefinite programming method to improve the upper bounds…
We study a discrete model of repelling particles, and we show using linear programming bounds that many familiar families of error-correcting codes minimize a broad class of potential energies when compared with all other codes of the same…
We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m…
Universal bounds for the potential energy of weighted spherical codes are obtained by linear programming. The universality is in the sense of Cohn-Kumar -- every attaining code is optimal with respect to a large class of potential functions…
We use linear programming techniques to find points of absolute minimum over the unit sphere $S^{d}$ in $\mathbb R^{d+1}$ of the total potential of a point configuration $\omega_N\subset S^{d}$ which is a spherical $(2m-1)$-design contained…
We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex quadratic programming (CQP), and a semidefinite programming (SDP) bound. Since the last two bounds were not compared directly before, we…
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…
We derive general linear programming bounds for spherical $(k,k)$-designs. This includes lower bounds for the minimum cardinality and lower and upper bounds for minimum and maximum energy, respectively. As applications we obtain a universal…
In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions…
Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show…
We introduce a linear programming framework for obtaining upper bounds for the potential energy of spherical codes of fixed cardinality and minimum distance. Using Hermite interpolation we construct polynomials to derive corresponding…
This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point…
We define three-point bounds for sphere packing that refine the linear programming bound, and we compute these bounds numerically using semidefinite programming by choosing a truncation radius for the three-point function. As a result, we…
In this article we present a unified treatment for obtaining bounds on the potential energy of codes in the general context of polynomial metric spaces (PM-spaces). The lower bounds we derive via the linear programming (LP) techniques of…
We use moment techniques to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate (from below) the infinite dimensional optimization problems in this…
We develop lower bounds for the energy of configurations in $\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial…