# American Institute of Mathematical Sciences

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:

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##### References:
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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