Algorithms for Nonlinear Mixed-Integer Location Estimation

For three decades, carrier-phase observations have been used to obtain the most accurate location estimates using global navigation satellite systems (GNSS). These estimates are computed by minimizing a nonlinear mixed-integer least-squares problem. Existing algorithms linearize the problem, orthogonally project it to eliminate real variables, and then solve the integer least-square problem. There is now considerable interest in developing similar localization techniques for terrestrial and indoor settings. We show that algorithms that linearize first fail in these settings and we propose several algorithms for computing the estimates. Some of our algorithms are elimination algorithms that start by eliminating the non-linear terms in the constraints; others construct a geometric arrangement that allows us to efficiently enumerate integer solutions (in polynomial time). We focus on simplified localization problems in which the measurements are range (distance) measurements and carrier phase range measurements, with no nuisance parameters. The simplified problem allows us to focus on the core question of untangling the nonlinearity and the integer nature of some parameters. We show using simulations that the new algorithms are effective at close ranges at which the linearize-first approach fails.