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.

翻译：本文讨论了一个经典的几何问题：在已知若干参考点（锚点）的情况下，通过测量未知点到这些参考点的欧几里得距离（含噪声）来估计该未知点在 $\Real{n}$ 中的位置。我们采用集合成员方法处理该问题，假设距离测量受到未知但有界误差的影响，并刻画所有与测量值及其假设误差模型一致的点所构成的集合。该集合是非凸的，但我们在文中证明它包含于由若干闭球与一个多面体相交所确定的区域内，我们称该区域为{\em 定位集}。随后，我们基于凸规划开发了高效的方法，用于计算结构简单（如盒型或椭球型）的紧致外包围集来近似定位集，该包围集即可作为有保证的集值定位估计。与定位集相关的内近似问题（通过球或椭球）也被表述为凸规划问题。与现有基于非凸代价最小化问题的半定规划松弛方法不同，我们的方法是直接的、几何的，并且基于满足测量方程两两差值的一个多面体点集。