  This paper focuses on target localization in a widely distributed
multiple-input-multiple-output (MIMO) radar system. In this system, range
measurements, which include the sum of distances between transmitter and target
and the distances from the target to receivers, are used. We can obtain an
accurate estimated position of the target by minimizing the measurement errors.
In order to make our model come closer to reality, we introduce two kinds of
noises, namely, Gaussian noise and outliers. When we evaluate a target
localization algorithm, its localization accuracy and computational complexity
are two main criteria. To improve the positioning accuracy, the original
problem is formulated as solving a non-smooth constrained optimization problem
in which the objective function is either l1-norm or l0-norm term. To achieve a
real-time solution, the Lagrange programming neural network (LPNN) is utilized
to solve this problem. However, it is well known that LPNN requires
twice-differentiable objective function and constraints. Obviously, the l1-norm
or l0-norm term in the objective function does not satisfy this requirement. To
address this non-smooth optimization problem, this paper proposes two
modifications based on the LPNN framework. In the first method, a
differentiable proximate l1-norm function is introduced. While in the second
method, locally competitive algorithm is utilized. Simulation and experimental
results demonstrate that the performance of the proposed algorithms outperforms
several existing schemes.
