On the solution of the gps localization and circle fitting problems

Amir Beck*, Dror Pan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider the problem of locating a user's position from a set of noisy pseudoranges to a group of satellites. We consider both the nonlinear least squares formulation of the problem, which is nonconvex and nonsmooth, and the nonlinear squared least squares variant, in which the objective function is smooth, but still nonconvex. We show that the squared least squares problem can be reformulated as a generalized trust region subproblem and as such can be solved efficiently. Conditions for attainment of the optimal solutions of both problems are derived. The nonlinear least squares problem is shown to have tight connections to the well-known geometric circle fitting and orthogonal regression problems. Finally, a fixed point method for the nonlinear least squares formulation is derived and analyzed.

Original languageEnglish
Pages (from-to)108-134
Number of pages27
JournalSIAM Journal on Optimization
Issue number1
StatePublished - 2012
Externally publishedYes


  • Existence of optimal solutions
  • GPS localization
  • Generalized trust region subproblem
  • Nonconvex optimization
  • Nonlinear least squares


Dive into the research topics of 'On the solution of the gps localization and circle fitting problems'. Together they form a unique fingerprint.

Cite this