Performance Comparison of Numerical Optimization Algorithms for RSS-TOA-Based Target Localization

In the literature, a hyper-enhanced local positioning system (HELPS) was developed to locate a target mobile device in an emergency. HELPS finds the target mobile device (i.e., emergency caller) using multiple receivers (i.e., signal measurement equipment of first responders) that measure the received signal strength (RSS) and time of arrival (TOA) of the long-term evolution (LTE) uplink signal from the target mobile device. The maximum likelihood (ML) estimator can be applied to localize a target mobile device using the RSS and TOA. However, the ML estimator for the RSS-TOA-based target localization problem is nonconvex and nonlinear, having no analytical solution. Therefore, the ML estimator should be solved numerically, unless it is relaxed into a convex or linear form. This study investigates the target localization performance and computational complexity of numerical methods for solving an ML estimator. The three widely used numerical methods are: grid search, gradient descent, and particle swarm optimization. In the experimental evaluation, the grid search yielded the lowest target localization root-mean-squared error; however, the 95th percentile error of the grid search was larger than those of the other two algorithms. The average code computation time of the grid search was extremely large compared with those of the other two algorithms, and gradient descent exhibited the lowest computation time. HELPS can select numerical algorithms by considering their constraints (e.g., the computational resources of the localization server or target accuracy).