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

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.Comment: 17 pages, written for the SIAG/OPT Views-and-News, (v2) some updates and correction

Moment methods in energy minimization: New bounds for Riesz minimal energy problems

We use moment methods to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate the infinite dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and we develop symmetric sum-of-squares techniques. We compute the second step of our hierarchy for Riesz $s$-energy problems with five particles on the $2$-dimensional unit sphere, where the $s=1$ case known as the Thomson problem. This yields new sharp bounds (up to high precision) and suggests the second step of our hierarchy may be sharp throughout a phase transition and may be universally sharp for $5$-particles on $S^2$. This is the first time a $4$-point bound has been computed for a continuous problem

Complete positivity and distance-avoiding sets

We introduce the cone of completely-positive functions, a subset of the cone of positive-type functions, and use it to fully characterize maximum-density distance-avoiding sets as the optimal solutions of a convex optimization problem. As a consequence of this characterization, it is possible to reprove and improve many results concerning distance-avoiding sets on the sphere and in Euclidean space.Comment: 57 pages; minor corrections in comparison to the previous versio