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

David de Laat(mail***at***daviddelaat.nl)

Abstract: 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 the 2-sphere. This is the first time a 4-point bound has been computed for a continuous problem.

Keywords: Thomson problem, Riesz s-energy, 4-point bounds, semidefinite programming, Lasserre hierarchy, harmonic analysis on spaces of subsets, invariant polynomials

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Infinite Dimensional Optimization (Semi-infinite Programming )

Category 3: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: 10/2016

Download: [PDF]

Entry Submitted: 10/18/2016
Entry Accepted: 10/20/2016
Entry Last Modified: 10/18/2016

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