Technical Report ERGO 12-001

On the evaluation complexity of cubic regularization methods for potentially rank-deficient nonlinear least-squares problems and its relevance to constrained optimization
Coralia Cartis, Nicholas I. M. Gould and Philippe L. Toint


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 O(ε-3/2) residual- and Jacobian-evaluations to drive either the Euclidean norm of the residualior its gradient below ε; 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 ε-3/2 for nonconvexly constrained problems when higher accuracy is required for primal feasibility than for dual first-orde criticality.


Evaluation complexity, worst-case analysis, least-squares, constrained nonlinear optimization, cubic regularization methods




Written: 21 March 2012


To appear in SIAM Journal of Optimization.