  Optimization problems constrained by partial differential equations (PDEs)
naturally arise in scientific computing, as those constraints often model
physical systems or the simulation thereof. In an implicitly constrained
approach, the constraints are incorporated into the objective through a reduced
formulation. To this end, a numerical procedure is typically applied to solve
the constraint system, and efficient numerical routines with quantifiable cost
have long been developed for that purpose. Meanwhile, the field of complexity
in optimization, that estimates the cost of an optimization algorithm, has
received significant attention in the literature, with most of the focus being
on unconstrained or explicitly constrained problems.
  In this paper, we analyze an algorithmic framework based on quadratic
regularization for implicitly constrained nonlinear least squares. By
leveraging adjoint formulations, we can quantify the worst-case cost of our
method to reach an approximate stationary point of the optimization problem.
Our definition of such points exploits the least-squares structure of the
objective, and provides new complexity insights even in the unconstrained
setting. Numerical experiments conducted on PDE-constrained optimization
problems demonstrate the efficiency of the proposed framework.
