  Least squares form one of the most prominent classes of optimization
problems, with numerous applications in scientific computing and data fitting.
When such formulations aim at modeling complex systems, the optimization
process must account for nonlinear dynamics by incorporating constraints. In
addition, these systems often incorporate a large number of variables, which
increases the difficulty of the problem, and motivates the need for efficient
algorithms amenable to large-scale implementations.
  In this paper, we propose and analyze a Levenberg-Marquardt algorithm for
nonlinear least squares subject to nonlinear equality constraints. Our
algorithm is based on inexact solves of linear least-squares problems, that
only require Jacobian-vector products. Global convergence is guaranteed by the
combination of a composite step approach and a nonmonotone step acceptance
rule. We illustrate the performance of our method on several test cases from
data assimilation and inverse problems: our algorithm is able to reach the
vicinity of a solution from an arbitrary starting point, and can outperform the
most natural alternatives for these classes of problems.
