Approximate global minimizers to pairwise interaction problems via convex relaxation

TL;DR

This paper introduces a convex relaxation method to efficiently approximate global minimizers for non-local pairwise interaction problems, providing guarantees and insights into the structure of solutions across various models.

Contribution

The authors develop a convex relaxation approach that predicts approximate global minimizers with recovery guarantees, applicable to a broad class of pairwise interaction problems, and demonstrate its effectiveness in multiple scenarios.

Findings

The method often finds near-optimal solutions within a few percent of the true minimum.

It exactly recovers the global minimizer when the relaxation corresponds to a lattice of Dirac masses.

The approach decomposes the energy into convex and non-convex parts, aiding in understanding the solution structure.

Abstract

We present a new approach for computing approximate global minimizers to a large class of non-local 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 non-convex quadratic variational problem, in the form of a linear, convex, optimization problem for the auto-correlation 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 non-convex functional. We observe that in some cases, the non-convex component of the decomposition can be used to characterize the support of the recovered minimizers.