August & September  2019, 12(4&5): 1027-1033. doi: 10.3934/dcdss.2019070

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, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070
References:
[1]

P. BiswasT. C. Lian and T. C. Wang, Semidefinite programming based algorithms for sensor network localization, J.ACM Transactions on Sensor Networks, 2 (2006), 188-220.   Google Scholar

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R. Faragher and R. Harle, Location fingerprinting with bluetooth low energy beacons, J.IEEE Journal on Selected Areas in Communications, 33 (2015), 2418-2428.   Google Scholar

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J. B. Hiriart-Urruty and C. Lemar chal, Convex analysis and minimization algorithms, Springer-Verlag, (1993), 150-159.   Google Scholar

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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.  Google Scholar

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K. LiuG. Motta and T. Ma, Multi-floor indoor navigation with geomagnetic field positioning and ant colony optimization algorithm, C.IEEE, (2016), 314-323.   Google Scholar

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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.   Google Scholar

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V. MorenoM. 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.   Google Scholar

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N. M. NamN. 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.  Google Scholar

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S. PaganoS. 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.   Google Scholar

[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.  Google Scholar

[11]

J. SuhS. You and S. Choi, Vision-based coordinated localization for mobile sensor networks, J.IEEE Transactions on Automation Science and Engineering, 13 (2016), 611-620.   Google Scholar

[12]

Z. SunA. Purohit and K. Chen, PANDAA: physical arrangement detection of networked devices through ambient-sound awareness, C.ACM, 13 (2011), 425-434.   Google Scholar

[13]

R. XuW. Chen and Y. Xu, A new indoor positioning system architecture using GPS signals, J.Sensors, 15 (2015), 10074-10087.   Google Scholar

[14]

L. YangY. 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.   Google Scholar

[15]

Y. ZhuangZ. 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.   Google Scholar

show all references

References:
[1]

P. BiswasT. C. Lian and T. C. Wang, Semidefinite programming based algorithms for sensor network localization, J.ACM Transactions on Sensor Networks, 2 (2006), 188-220.   Google Scholar

[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.   Google Scholar

[3]

J. B. Hiriart-Urruty and C. Lemar chal, Convex analysis and minimization algorithms, Springer-Verlag, (1993), 150-159.   Google Scholar

[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.  Google Scholar

[5]

K. LiuG. Motta and T. Ma, Multi-floor indoor navigation with geomagnetic field positioning and ant colony optimization algorithm, C.IEEE, (2016), 314-323.   Google Scholar

[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.   Google Scholar

[7]

V. MorenoM. 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.   Google Scholar

[8]

N. M. NamN. 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.  Google Scholar

[9]

S. PaganoS. 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.   Google Scholar

[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.  Google Scholar

[11]

J. SuhS. You and S. Choi, Vision-based coordinated localization for mobile sensor networks, J.IEEE Transactions on Automation Science and Engineering, 13 (2016), 611-620.   Google Scholar

[12]

Z. SunA. Purohit and K. Chen, PANDAA: physical arrangement detection of networked devices through ambient-sound awareness, C.ACM, 13 (2011), 425-434.   Google Scholar

[13]

R. XuW. Chen and Y. Xu, A new indoor positioning system architecture using GPS signals, J.Sensors, 15 (2015), 10074-10087.   Google Scholar

[14]

L. YangY. 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.   Google Scholar

[15]

Y. ZhuangZ. 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.   Google Scholar

Figure 1.  Simulation scenario and sensor nodes distribution
Figure 2.  PDF of positioning error for different methods
Figure 3.  RMSE of target location estimate versus $\sigma_{\theta}$ and $\sigma_{t}$
Table 1.  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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