Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming

Euclidean Distance Matrices and Applications

“…The cone of Euclidean distance matrices and its geometry is described in, for example, [11,59,71,111,112]. Using semidefinite optimization to solve Euclidean distance matrix problems is studied in [2,4]. Further theoretical results are given in [10,13].…”

“…Since int(S n C ∩ S n + ) = ∅, we can have problems with constraint qualifications and unbounded optimal sets. To avoid this ( [2,4]), we define…”

Sensor network localization, euclidean distance matrix completions, and graph realization

“…[14,10] and more recently in [2,1] and the references therein. The latter two references studied algorithms based on SDP formulations of the EDM completion problem.…”

A Semismooth Newton Method for the Nearest Euclidean Distance Matrix Problem

“…Given (−D) being used in (2), the matrix D in (3) should be −D. This change of sign has been widely adopted to reformulate (3) (see., e.g., [19,17,1]) and it reminds us that the objective is to minimize a distance.…”

“…The same method was independently studied by Gaffke and Mathar [17], but based on a different projection formula on K n + (see (15) and (17)). However, MAP does not apply to (4) unless H = E. Problem (3) (and in general (4)) can also be solved by Semi-Definite Programming (SDP) initiated by Alfakih et al [1] (see also [2]). We note that the dimension of S n h is n(n − 1)/2, as is the dimension of the cone E n of the Euclidean distance matrices, where E n := S n h ∩(−K n + ) (see [26]).…”