# Efficient Distributed Method for NLOS Cooperative Localization in WSNs
## 1 Introduction
### Related Work  
The maximum-likelihood (ML) problem of sensor cooperative localization is a complicated nonconvex problem of high dimensionality. It is quite difficult to obtain an optimal solution. Most existing approaches try to find a feasible solution by applying relaxation methods to the original nonconvex problem. Among them, the most typical ones are listed. Semidefinite programming (SDP) relaxation for cooperative localization is proposed in References [12; 13; 14]. In Reference [13], the authors formulated the cooperative localization problem via graph realization theory, and derived the upper and lower bounds of the SDP objective function. In Reference [14], the authors developed three relaxations: node-based SDP (NSDP), edge-based SDP (ESDP), and sub-SDP (SSDP). NSDP and ESDP are weaker than the original SDP relaxation of Reference [12], but they both remain efficient and accurate. SSDP waves a faster way to efficiently solve a general SDP problem without sacrificing solution quality. In Reference [15], the authors proposed a novel model by combining angle information with range measurements, and transformed the model into an SDP problem. This method has excellent performance, but it also has considerably high computational complexity. The work in References [16; 17] proposes second-order cone programming (SOCP) by relaxing distance constraints. The work in Reference [16] further applies SOCP to alleviate the computational burden of the standard SDP problem at the price of performance degradation. The authors in Reference [17] designed a method suited for both Gaussian and Laplacian noise environments. In addition to the aforementioned convex relaxations, some other methods also contribute to improving localization performance. In Reference [18], the authors formulated the problem as a regression problem over adaptive bases. They utilized the eigenvector of a distance affinity matrix as the initial point and implemented iterations by conjugate gradient descent. In Reference [19], the authors derived a majorization-minimization (MM) algorithm with quadratic objective function. However, all the methods or algorithms above are implemented in a centralized framework. The centralized framework is a classical paradigm that transmits data to the central or fusion node to fulfil entire network tasks. A centralized framework is easy to implement, but the computational burden becomes extremely heavy for large-scale sensor networks. Centralized algorithms are prone to data-traffic bottlenecks among sensors around the central node [20]. As the number of sensor nodes grows, the problem size and the computational complexity in the centralized framework dramatically increase. Scale limits and time delay brought by the centralized paradigm may affect the engineering implementation of WSNs. Due to the advent of large-scale networks, it is urgent to find an effective framework to satisfy the requirements on both processing speed and localization accuracy. Therefore, the distributed paradigm has become a new tendency in this field. Compared with the centralized paradigm, the distributed paradigm is an algorithm with all nodes performing the same type of computation. A number of distributed approaches have been proposed over the years, such as the distributed gradient descent method [21], multidimensional scaling method [22], and the sequential greedy optimization (SGO) algorithm [23]. In Reference [24], the authors developed a parallel distributed algorithm based on SOCP relaxation, but the convergence property of the algorithm was not established. The work in Reference [25] proposes a sequential method: the estimated neighbors are regarded as new anchors, which are then used to estimate other sensors. This process only stops when the whole network is covered. In Reference [23], the authors applied the SGO algorithm to ESDP and SOCP relaxation formulations.