Explicit Sensor Network Localization using Semidefinite Representations and Facial Reductions
SIAM Journal on Optimization
The sensor network localization (SNL) problem in embedding dimension r consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming (SDP) completion problem by using the linear mapping between Euclidean distance matrices (EDM) and semidefinite matrices. The
... trices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances, we are able to efficiently solve many huge instances of this NP-hard problem to high accuracy by finding a representation of the minimal face of the SDP cone that contains the SDP matrix representation of the EDM. The main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the SDP cone that correspond to cliques of the SNL problem. Key words. sensor network localization, Euclidean distance matrix completions, semidefinite programming, loss of the Slater constraint qualification AMS subject classifications. 90C35, 90C22, 90C26, 90C06 N. KRISLOCK AND H. WOLKOWICZ  that there are advantages for handling the SNL problem as an EDMC and ignoring the distinction between the anchors and the other sensors until after the EDMC is solved. In this paper we use this framework and derive an algorithm that locates the sensors by exploiting the structure and implicit degeneracy in the SNL problem. In particular, we solve the SDP problems explicitly (exactly) without using any p-d i-p techniques. We do so by repeatedly viewing SNL in three equivalent forms: as a graph realization problem, as an EDMC, and as a rank restricted SDP. A common approach to solving the EDMC problem is to relax the rank constraint and solve a weighted, nearest, positive semidefinite completion problem (like problem (1.1)) using SDP. The resulting SDP is, implicitly, highly degenerate in the sense that the feasible semidefinite matrices have low rank. In particular, cliques in the graph of the SNL problem reduce the ranks of these feasible semidefinite matrices. This means that the Slater constraint qualification (strict feasibility) implicitly fails for the SDP. Our algorithm is based on exploiting this degeneracy. We characterize the face of the SDP cone that corresponds to a given clique in the graph, thus reducing the size of the SDP problem. Then we characterize the intersection of two faces that correspond to overlapping cliques. This allows us to explicitly grow/increase the size of the cliques by repeatedly finding the intersection of subspaces that represent the faces of the SDP cone that correspond to these cliques. Equivalently, this corresponds to completing overlapping blocks of the EDM. In this way, we further reduce the dimension of the faces until we get a completion of the entire EDM. The intersection of the subspaces can be found using a singular value decomposition or by exploiting the special structure of the subspaces. No SDP solver is used. Thus we solve the SDP problem in a finite number of steps, where the work of each step is to find the intersection of two subspaces (or, equivalently, each step is to find the intersection of two faces of the SDP cone). Though our results hold for general embedding dimension r, our preliminary numerical tests involve sensors with embedding dimensions r = 2 and r = 3. The sensors are in the region [0, 1] r . There are n sensors, m of which are anchors. The radio range is R units. Related work/applications. The number of applications for distance geometry problems is large and increasing in number and importance. The particular case of SNL has applications to environmental monitoring of geographical regions, as well as tracking of animals and machinery; see, for example, [5, 12] . There have been many algorithms published recently that solve the SNL problem. Many of these involve SDP relaxations and use SDP solvers; see, for example, [5, 6, 7, 8, 9, 13] and more recently [20, 28] . Heuristics are presented in, for example,  . SNL is closely related to the EDMC problem; see, for example, [3, 12] and the survey . Carter, Jin, Saunders, and Ye  and Jin  propose the SpaseLoc heuristic. It is limited to r = 2 and uses an SDP solver for small localized subproblems. They then sew these subproblems together. So and Ye  show that the problem of solving a noiseless SNL that is uniquely localizable 1 can be phrased as an SDP and thus can be solved in polynomial time. They also give an efficient criterion for checking whether a given instance has a unique solution for r = 2. Two contributions of this paper are as follows: we do not use iterative p-d i-p techniques to solve the SDP but rather we solve it with a finite number of explicit 1 An SNL problem is uniquely localizable in dimension r if it has a unique solution in R r and it does not have any solution whose affine span is R h , where h > r; see  .