[论文解读] Set-Membership Localization via Range Measurements

Giuseppe C. Calafiore|arXiv (Cornell University)|Mar 5, 2026

Advanced Optimization Algorithms Research被引用 0

一句话总结

该论文为从对已知锚点的嘈杂距离测量中本地化一个点，建立了集合成员身份框架，利用凸规划得到保证的外部包络集合（盒子或椭圆），具有边界性。

ABSTRACT

In this paper we discuss a classical geometrical problem of estimating an unknown point's location in $\Real{n}$ from several noisy measurements of the Euclidean distances from this point to a set of known reference points (anchors). We approach the problem via a set-mem\-ber\-ship methodology, in which we assume the distance measurements to be affected by unknown-but-bounded errors, and we characterize the set of all points that are consistent with the measurements and their assumed error model. This set is nonconvex, but we show in the paper that it is contained in a region given by the intersection of certain closed balls and a polytope, which we call the {\em localization set}. Then, we develop efficient methods, based on convex programming, for computing a tight outer-bounding set of simple structure (a box, or an ellipsoid) for the localization set, which then acts as a guaranteed set-valued location estimate. % The center of the bounding set also serves as a point location estimate. Related problems of inner approximation of the localization set via balls and ellipsoids are also posed as convex programming problems. Different from existing methods based on semidefinite programming relaxations of a nonconvex cost minimization problem, our approach is direct, geometric and based on a polyhedral set of points that satisfy pairwise differences of the measurement equations.

研究动机与目标

• 在未知但有界误差下，从对已知锚点的嘈杂距离测量中估计Rn中的未知点。
• 将定位集合表征为闭球与多边形的交集。
• 开发基于凸规划的程序以计算定位集的紧的外部边界集合（盒子/椭圆）。
• 提供通过球和椭圆的定位集内近似方法。
• 为集合值定位提供直接、几何性的替代 SDP 松弛的方法。

提出的方法

• 将定位集合表征为来自测量方程差的对偶闭球和多边形的交集。
• 推导凸规划问题，以简单形状（盒子或椭圆）计算包含定位集的紧外边界。
• 通过凸规划求解定位集的球和椭圆的内近似问题。
• 提供来自边界集的中心点估计。
• 通过采用直接、几何、多面体方法，与基于 SDP 的非凸松弛进行对比。

实验结果

研究问题

• RQ1 如何表征与距离测量及有界误差一致的所有点的集合？
• RQ2 是否可以使用简单的凸形状计算定位集的紧的保证外边界？
• RQ3 定位集的有效内近似策略有哪些？
• RQ4 与 SDP 松弛相比，该方法在可处理性与保证性方面如何？

主要发现

• 定位集被闭球与多面体所形成的交集区域（定位集合）包含。
• 高效的凸规划工作流能够为定位集提供紧的外部包络盒或椭圆。
• 边界集的中心作为点定位估计。
• 通过球/椭圆的内近似可以作为凸规划来表述。
• 该方法为非凸定位问题提供了一个直接的几何替代 SDP 松弛的方法。