Euclidean distance matrices, semidefinite programming and sensor network localization

Abstract

The fundamental problem of distance geometry, F P DG , involves the characterization and study of sets of points based only on given values of (some of) the distances between pairs of points. This problem has a wide range of applications in various areas of mathematics, physics, chemistry, and engineering. Euclidean Distance Matrices, EDM , play an important role in F P DG . They use the squared distances and provide elegant and powerful convex relaxations for F P DG . These EDM problems are closely related to graph realization, GRL ; and graph rigidity, GRD , plays an important role. Moreover, by relaxing the embedding dimension restriction, EDM problems *

The Many Faces of Degeneracy in Conic Optimization

“…Suppose now that ω[α] has rank r < k, i.e., the principal submatrix of ω indexed by α is singular. Then the right-hand-side 2 If some elements ii do not lie in E, contrary to our simplifying assumption, then the image PE(S n + ) can fail to be closed. A precise characterization is given in [47].…”

“…Jonathon Borwein passed away unexpectedly on Aug. 2,2016. Jon was an extraordinary mathematician who made significant contributions in an amazing number of very diverse areas.…”

“…An excellent online reference for EDM is the book by Dattorro [37]. In addition, there are many excellent survey articles, e.g., [77,43,87,2,97]. The survey [86] contains many open problems in EDMC and references for application areas.…”

Euclidean Distance Matrices and Applications

“…Further theoretical results are given in [10,13]. Books and survey papers containing a treatment of Euclidean distance matrices include, for example, [31,44,87], and most recently [3]. The topic of rank minimization for Euclidean distance matrix problems is discussed in, for example, [34,35,55,56,99,100].…”

Robust all-source positioning of UAVs based on belief propagation

“…Existing algorithms such as iterative least square and Kalman filters can be extended to cooperative positioning, which leads to cooperative least square and cooperative Kalman filtering algorithm [21]. In recent years, convex optimizations, including semidefinite programming (SDP) [22][23][24][25], sum of squares (SOS) relaxation [26,27], and distributed gradient algorithm, also found their place in cooperative positioning. Another category of positioning algorithms are Bayesian approaches with belief propagation as an outstanding representative.…”