The Number of Global Solutions for GPS Source Localization in Two-Dimension

Abstract

Source localization is widely used in many areas including GPS, but the influence of possible noises cannot be overlooked. Many optimization methods have been attempted to mitigate different kinds of noises. However, the stability of the solution, even the number of global solutions, is not fully known. Only local convergence or stability for the optimization problem is known in simple L1 or L2 settings. In this paper, we prove that the number of possible two-dimensional source locations with three measurements in L2 setting is at most 5. We also showed the sufficient and necessary condition for the number of the solutions being 1, 2, 3, 4, and 5, where the measurement triangle is isosceles and the measurement distance for the two isosceles triangles is the same.