Technion, IEM faculty - Operations Research seminar
Speaker: Dror Pan
Title: On the Solution of the GPS Localization Problem
Date: 28/02/2011
Time: 12:30
Place: Bloomfield-527
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Abstract: We consider the problem of localization of a user's position using a finite set of its pseudo-ranges from a
set of satellites. The pseudo-ranges are subjected to some unknown clock-bias and other unknown noise. Two
formulations of the problem as optimization problems are being considered: the nonlinear least squares formulation
and the nonlinear squared least squares variant. The least squares problem is nonsmooth and nonconvex, whereas the
squared least squares problem has a smooth objective function, although also nonconvex. Both problems are shown to
have tight connection to the well known circle fitting problem, which is in fact, identical to a private case of
them. We show that a relaxation of the squared least squares problem is equivalent to a generalized trust region
subproblem, and therefore can be solved efficiently. We show a simple algorithm for solving it using a procedure for
finding roots of a univariate equation. Sufficient conditions for both problems to attain a global minimum are
derived. In the circle fitting least squares problem, there is also a tight connection between the attainment
condition to the known orthogonal regression problem. Finally, a fixed point method for solving the least squares
problem is introduced and analyzed. The squared least squares solution is used as an initial guess for it. The
quality of the solutions obtained from both problems are tested numerically, using standard Monte-Carlo simulations.
This talk is based on the M.Sc work done under the supervision of Prof. Amir Beck
Technion Math. Net (TECHMATH)
Editor: Michael Cwikel   <> 
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