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, 2008) 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 ESDPlb (ρ‐ESDPlb) 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‐ESDPlb 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: ‐‐‐ 10.1007/s10107‐009‐0338‐x" TargetType="DOI"/> ), 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.