Edge-based semidefinite programming relaxation of sensor network localization with lower bound constraints

In this paper, we strengthen the edge-based semidefinite programming relaxation (ESDP) recently proposed by Wang, Zheng, Boyd, and Ye (SIAM J. Optim. 19:655–673, <CitationRef CitationID="CR28">2008</CitationRef>) by adding lower bound constraints. We show that, when distances are exact, zero individual trace is necessary and sufficient for a sensor to be correctly positioned by an interior solution. To extend this characterization of accurately positioned sensors to the noisy case, we propose a noise-aware version of ESDP<Subscript>lb</Subscript> (ρ-ESDP<Subscript>lb</Subscript>) and show that, for small noise, a small individual trace is equivalent to the sensor being accurately positioned by a certain analytic center solution. We then propose a postprocessing heuristic based on ρ-ESDP<Subscript>lb</Subscript> and a distributed algorithm to solve it. Our computational results show that, when applied to a solution obtained by solving ρ-ESDP proposed of Pong and Tseng (Math. Program. doi:<ExternalRef> <RefSource>10.1007/s10107-009-0338-x</RefSource> <RefTarget Address="10.1007/s10107-009-0338-x" TargetType="DOI"/> </ExternalRef>), this heuristics usually improves the RMSD by at least 10%. Furthermore, it provides a certificate for identifying accurately positioned sensors in the refined solution, which is not common for existing refinement heuristics. Copyright Springer Science+Business Media, LLC 2012