ON THE EVALUATION COMPLEXITY OF CUBIC REGULARIZATION METHODS FOR POTENTIALLY RANK-DEFICIENT NONLINEAR LEAST-SQUARES PROBLEMS AND ITS RELEVANCE TO CONSTRAINED NONLINEAR OPTIMIZATION.

We propose a new termination criterion suitable for potentially singular, zero or nonzero residual, least-squares problems, with which cubic regularization variants take at most O(∈3/2) residual- and Jacobian-evaluations to drive either the Euclidean norm of the residual or its gradient below e; this is the best known bound for potentially rank-deficient 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 ∈3/2 for nonconvexly constrained problems when higher accuracy is required for primal feasibility than for dual first-order criticality. [ABSTRACT FROM AUTHOR]