# American Institute of Mathematical Sciences

## Error minimization with global optimization for difference of convex functions

 No.10 Xitucheng Road, Haidian District, Beijing, Beijing University of Posts & Telecommunications, Beijing, China

* Corresponding author: Enwen Hu

Received  June 2017 Revised  November 2017 Published  November 2018

Fund Project: The first author is supported by The National Key Research and Development Program of China grant 2016YFB0502001

In this paper, a hybrid positioning method based on global optimization for difference of convex functions (D.C.) with time of arrival (TOA) and angle of arrival (AOA) measurements are proposed. Traditional maximum likelihood (ML) formulation for indoor localization is a nonconvex optimization problem. The relaxation methods can?t provide a global solution. We establish a D.C. model for TOA/AOA fusion positioning model and give a solution with a global optimization. Simulations based on TC-OFDM signal system show that the proposed method is efficient and more robust as compared to the existing ML estimation and convex relaxation.

Citation: Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019070
##### References:
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##### References:
 [1] P. Biswas, T. C. Lian and T. C. Wang, Semidefinite programming based algorithms for sensor network localization, J.ACM Transactions on Sensor Networks, 2 (2006), 188-220. [2] R. Faragher and R. Harle, Location fingerprinting with bluetooth low energy beacons, J.IEEE Journal on Selected Areas in Communications, 33 (2015), 2418-2428. [3] J. B. Hiriart-Urruty and C. Lemar chal, Convex analysis and minimization algorithms, Springer-Verlag, (1993), 150-159. [4] W. L. Lin and Z. L. Deng, Dimensional functional differential convergence for Cramer-Rao lower bound, J. Journal of Difference Equations & Applications, 23 (2017), 249-257. doi: 10.1080/10236198.2016.1216549. [5] K. Liu, G. Motta and T. Ma, Multi-floor indoor navigation with geomagnetic field positioning and ant colony optimization algorithm, C.IEEE, (2016), 314-323. [6] R. Maalek and F. Sadeghpour, Accuracy assessment of ultra-wide band technology in locating dynamic resources in indoor scenarios, J.Automation in Construction, 63 (2016), 12-26. [7] V. Moreno, M. A. Zamora and A. F. Skarmeta, A low-cost indoor lcalization system for energy sustainability in smart buildings, J.IEEE Sensors Journal, 16 (2016), 3246-3262. [8] N. M. Nam, N. T. An and R. B. Rector, Nonsmooth algorithms and nesterov's smoothing technique for generalized fermat--torricelli problems, J. Siam Journal on Optimization, 24 (2013), 1815-1839. doi: 10.1137/130945442. [9] S. Pagano, S. Peirani and M. Valle, Indoor ranging and localisation algorithm based on received signal strength indicator using statistic parameters for wireless sensor networks, J.IET Wireless Sensor Systems, 5 (2015), 243-249. [10] A. S. Strekalovsky, Global optimality conditions in nonconvex Optimization, J.Journal of Optimization Theory & Applications, 173 (2017), 770-792. doi: 10.1007/s10957-016-0998-7. [11] J. Suh, S. You and S. Choi, Vision-based coordinated localization for mobile sensor networks, J.IEEE Transactions on Automation Science and Engineering, 13 (2016), 611-620. [12] Z. Sun, A. Purohit and K. Chen, PANDAA: physical arrangement detection of networked devices through ambient-sound awareness, C.ACM, 13 (2011), 425-434. [13] R. Xu, W. Chen and Y. Xu, A new indoor positioning system architecture using GPS signals, J.Sensors, 15 (2015), 10074-10087. [14] L. Yang, Y. Chen and X. Y. Li, Tagoram: real-time tracking of mobile RFID tags to high precision using COTS devices, C.ACM, 13 (2014), 237-248. [15] Y. Zhuang, Z. Syed and Y. Li, Evaluation of two WiFi positioning systems based on autonomous crowd sourcing on handheld devices for indoor navigation, J.IEEE Transactions on Mobile Computing, 15 (2016), 1982-1995.
Simulation scenario and sensor nodes distribution
PDF of positioning error for different methods
RMSE of target location estimate versus $\sigma_{\theta}$ and $\sigma_{t}$
Global optimization of error minimization
 INPUT: $t_1, \theta_1 \in IR$ For $k = 1, \cdots, N$ do Find $u_k$ approximately by solving the problem $\partial \left(F\left(u_{k}, \aleph \right)+\Lambda \left(u_{k}\right)\right)-\Lambda^{'}\left(u_{k}\right)+N\left(u_{k};S\right)=0$ Find $\Phi_{k+1} \in \Phi\left(u_{k}\right)$ by solving the problem minimize $F\left(u_{k}, \aleph\right)+\Lambda\left(u_k\right)-\Phi_{k+1}$ End for OUTPUT:$u_{N+1}$
 INPUT: $t_1, \theta_1 \in IR$ For $k = 1, \cdots, N$ do Find $u_k$ approximately by solving the problem $\partial \left(F\left(u_{k}, \aleph \right)+\Lambda \left(u_{k}\right)\right)-\Lambda^{'}\left(u_{k}\right)+N\left(u_{k};S\right)=0$ Find $\Phi_{k+1} \in \Phi\left(u_{k}\right)$ by solving the problem minimize $F\left(u_{k}, \aleph\right)+\Lambda\left(u_k\right)-\Phi_{k+1}$ End for OUTPUT:$u_{N+1}$
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