  A new Levenberg--Marquardt (LM) method for solving nonlinear least squares
problems with convex constraints is described. Various versions of the LM
method have been proposed, their main differences being in the choice of a
damping parameter. In this paper, we propose a new rule for updating the
parameter so as to achieve both global and local convergence even under the
presence of a convex constraint set. The key to our results is a new
perspective of the LM method from majorization-minimization methods.
Specifically, we show that if the damping parameter is set in a specific way,
the objective function of the standard subproblem in LM methods becomes an
upper bound on the original objective function under certain standard
assumptions.
  Our method solves a sequence of the subproblems approximately using an
(accelerated) projected gradient method. It finds an $\epsilon$-stationary
point after $O(\epsilon^{-2})$ computation and achieves local quadratic
convergence for zero-residual problems under a local error bound condition.
Numerical results on compressed sensing and matrix factorization show that our
method converges faster in many cases than existing methods.
