We propose a new termination criteria suitable for potentially singular, zero or non-zero residual, least-squares problems, with which cubic regularization variants take at most $\mathcal{O}(\epsilon^{-3/2})$ residual- and Jacobian-evaluations to drive either the Euclidean norm of the residual or its gradient below $\epsilon$; this is the best-known bound for potentially singular nonlinear least-squares problems. We then apply the new optimality measure and cubic regularization steps to a family of least-squares merit functions in the context of a target-following algorithm for nonlinear equality-constrained problems; this approach yields the first evaluation complexity bound of order $\epsilon^{-3/2}$ for nonconvexly constrained problems when higher accuracy is required for primal feasibility than for dual first-order criticality.
Citation
ERGO Technical Report 12-001, School of Mathematics, University of Edinburgh, Scotland, UK, 2012.