Approximate global minimizers to pairwise interaction problems via convex relaxation

Document Type

Article

Publication Date

1-1-2018

Abstract

We present a new approach for computing approximate global minimizers to a large class of nonlocal pairwise interaction problems defined over probability distributions. The approach predicts candidate global minimizers, with a recovery guarantee, that are sometimes exact, and often within a few percent of the optimum energy (under appropriate normalization of the energy). The procedure relies on a convex relaxation of the pairwise energy that exploits translational symmetry, followed by a recovery procedure that minimizes a relative entropy. Numerical discretizations of the convex relaxation yield a linear programming problem over convex cones that can be solved using well-known methods. One advantage of the approach is that it provides sufficient conditions for global minimizers to a nonconvex quadratic variational problem, in the form of a linear, convex, optimization problem for the autocorrelation of the probability density. We demonstrate the approach in a periodic domain for examples arising from models in materials, social phenomena, and flocking. The approach also exactly recovers the global minimizer when a lattice of Dirac masses solves the convex relaxation. An important by-product of the relaxation is a decomposition of the pairwise energy functional into the sum of a convex functional and nonconvex functional. We observe that in some cases, the nonconvex component of the decomposition can be used to characterize the support of the recovered minimizers.

Identifier

85046619099 (Scopus)

Publication Title

SIAM Journal on Applied Dynamical Systems

External Full Text Location

https://doi.org/10.1137/16M1069146

e-ISSN

15360040

First Page

417

Last Page

456

Issue

1

Volume

17

Grant

1719693

Fund Ref

National Science Foundation

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