  We investigate the problem of high frequency (HF) source localization using
the time-difference-of-arrival (TDOA) observations of ionosphere-refracted
radio rays based on quasi-parabolic (QP) modeling. An unresolved but pertinent
issue in such a field is that the existing gradient-type scheme can easily get
trapped in local optima for practical use. This will lead to the difficulty in
initializing the algorithm and finally degraded positioning performance if the
starting point is inappropriately selected. In this paper, we develop a
collaborative gradient projection (GP) algorithm in order to globally solve the
highly nonconvex QP-based TDOA HF localization problem. The metaheuristic of
particle swarm optimization (PSO) is exploited for information sharing among
multiple GP models, each of which is guaranteed to work out a critical point
solution to the simplified maximum likelihood formulation. Random mutations are
incorporated to avoid the early convergence of PSO. Rather than treating the
geolocation of HF transmitter as a pure optimization problem, we further
provide workarounds for addressing the possible impairments and challenges when
the proposed technique is applied in practice. Numerical results demonstrate
the effectiveness of our PSO-assisted re-initialization strategy in achieving
the global optimality, and the superiority of our method over its competitor in
terms of positioning accuracy.
