Multiscale semidefinite programming approach to positioning problems with pairwise structure

TL;DR

This paper introduces a multiscale semidefinite programming method for optimizing pairwise functions, improving global solutions in complex problems like sensor network localization and Lennard-Jones potential minimization.

Contribution

It proposes a novel multiscale convex relaxation approach that leverages smoothness to better navigate complex energy landscapes in pairwise optimization problems.

Findings

Outperforms existing methods in sensor network localization with high noise

Effectively explores near-optimal configurations in Lennard-Jones potential

Provides theoretical advantages over traditional approaches

Abstract

We consider the optimization of pairwise objective functions, i.e., objective functions of the form $H(\mathbf{x}) = H(x_1,\ldots,x_N) = \sum_{1\leq i<j \leq N} H_{ij}(x_i,x_j)$ for $x_i$ in some continuous state spaces $\mathcal{X}_i$. Global optimization in this setting is generally confounded by the possible existence of spurious local minima and the impossibility of global search due to the curse of dimensionality. In this paper, we approach such problems via convex relaxation of the marginal polytope considered in graphical modeling, proceeding in a multiscale fashion which exploits the smoothness of the cost function. We show theoretically that, compared with existing methods, such an approach is advantageous even in simple settings for sensor network localization (SNL). We successfully apply our method to SNL problems, particularly difficult instances with high noise. We also validate performance on the optimization of the Lennard-Jones potential, which is plagued by the existence of many near-optimal configurations. We demonstrate that in MMR allows us to effectively explore these configurations.