Upper bound performance of semi-definite programming for localisation in inhomogeneous media

In this paper, we regarded an absorbing inhomogeneous medium as an assembly of thin layers having different propagation properties. We derived a stochastic model for the refractive index and formulated the localisation problem given noisy distance measurements using graph realisation problem. We relaxed the problem using semi-definite programming (SDP) approach in lp realisation domain and derived upper bounds that follow Edmundson-Madansky bound of order 6p (EM6p) on the SDP objective function to provide an estimation of the techniques' localisation accuracy. Our results showed that the inhomogeneity of the media and the choice of lp norm have significant impact on the ratio of the expected value of the localisation error to the upper bound for the expected optimal SDP objective value. The tightest ratio was derived when l∞ norm was used.