This paper explores the distance-based relative state estimation problem in large-scale systems, which is hard to solve effectively due to its high-dimensionality and non-convexity. In this paper, we alleviate this inherent hardness to simultaneously achieve scalability and robustness of inference on this problem. Our idea is launched from a universal geometric formulation, called \emph{generalized graph realization}, for the distance-based relative state estimation problem. Based on this formulation, we introduce two collaborative optimization models, one of which is convex and thus globally solvable, and the other enables fast searching on non-convex landscapes to refine the solution offered by the convex one. Importantly, both models enjoy \emph{multiconvex} and \emph{decomposable} structures, allowing efficient and safe solutions using \emph{block coordinate descent} that enjoys scalability and a distributed nature. The proposed algorithms collaborate to demonstrate superior or comparable solution precision to the current centralized convex relaxation-based methods, which are known for their high optimality. Distinctly, the proposed methods demonstrate scalability beyond the reach of previous convex relaxation-based methods. We also demonstrate that the combination of the two proposed algorithms achieves a more robust pipeline than deploying the local search method alone in a continuous-time scenario.

翻译：本文探讨大规模系统中的距离相对状态估计问题，该问题因其高维性和非凸性而难以有效求解。本文通过缓解这一固有难度，同时实现对该问题推理的可扩展性和鲁棒性。我们的思路源于一个称为广义图实现的通用几何框架，该框架为距离相对状态估计问题提供了统一表述。基于此框架，我们引入了两种协同优化模型：其中一种是凸的，因而可全局求解；另一种则能在非凸空间上进行快速搜索，以优化凸模型提供的解。重要的是，这两种模型均具有多重凸和可分解结构，允许使用块坐标下降法进行高效且安全的求解，该方法兼具可扩展性和分布式特性。所提出的算法通过协作，在求解精度上展现出优于或相当于当前基于集中式凸松弛方法的性能，后者以高最优性著称。显著不同的是，所提方法展现出的可扩展性超越了以往基于凸松弛方法的范围。我们还证明，在连续时间场景中，结合使用两种所提算法比单独部署局部搜索方法能实现更鲁棒的流程。