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

E. Nadimi, V. Blanes-Vidal
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Abstract

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.
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非均匀介质中半确定规划的上界性能
本文将非均匀吸收介质看作具有不同传播特性的薄层的集合。我们推导了折射率的随机模型,并利用图形实现问题提出了给定噪声距离测量的定位问题。我们在lp实现域使用半确定规划(SDP)方法放宽问题,并在SDP目标函数上推导出遵循6p阶Edmundson-Madansky界(EM6p)的上界,以提供对技术定位精度的估计。我们的研究结果表明,介质的非均匀性和lp范数的选择对定位误差期望值与期望最优SDP目标值上界的比值有显著影响。采用l∞范数时,得到最紧比值。
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