The molecule problem: determining conformation from pairwise distances
The molecule problem: determining conformation from pairwise distances
Conditions for unique graph realizations
SIAM Journal on Computing
The simplest semidefinite programs are trivial
Mathematics of Operations Research
Matrix computations (3rd ed.)
Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming
Computational Optimization and Applications - Special issue on computational optimization—a tribute to Olvi Mangasarian, part I
Finding and certifying a large hidden clique in a semirandom graph
Random Structures & Algorithms
Scalable sensor localization algorithms for wireless sensor networks
Scalable sensor localization algorithms for wireless sensor networks
Semidefinite programming based algorithms for sensor network localization
ACM Transactions on Sensor Networks (TOSN)
Theory of semidefinite programming for Sensor Network Localization
Mathematical Programming: Series A and B
Second-Order Cone Programming Relaxation of Sensor Network Localization
SIAM Journal on Optimization
SpaseLoc: An Adaptive Subproblem Algorithm for Scalable Wireless Sensor Network Localization
SIAM Journal on Optimization
SIAM Journal on Scientific Computing
Further Relaxations of the Semidefinite Programming Approach to Sensor Network Localization
SIAM Journal on Optimization
Exploiting Sparsity in SDP Relaxation for Sensor Network Localization
SIAM Journal on Optimization
(Robust) Edge-based semidefinite programming relaxation of sensor network localization
Mathematical Programming: Series A and B
On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming
Mathematics of Operations Research
On the number of solutions of the discretizable molecular distance geometry problem
COCOA'11 Proceedings of the 5th international conference on Combinatorial optimization and applications
Universal Rigidity and Edge Sparsification for Sensor Network Localization
SIAM Journal on Optimization
Protein structure by semidefinite facial reduction
RECOMB'12 Proceedings of the 16th Annual international conference on Research in Computational Molecular Biology
The discretizable molecular distance geometry problem
Computational Optimization and Applications
Computational Optimization and Applications
Strong duality and minimal representations for cone optimization
Computational Optimization and Applications
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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 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.