When solving the general smooth nonlinear optimization problem involving equality and/or inequality constraints, an approximate first-order critical point of accuracy $\epsilon$ can be obtained by a second-order method using cubic regularization in at most $O(\epsilon^{-3/2})$ problem-functions evaluations, the same order bound as in the unconstrained case. This result is obtained by first showing that the same result holds for inequality constrained nonlinear least-squares. As a consequence, the presence of (possibly nonlinear) equality/inequality constraints does not affect the complexity of finding approximate first-order critical points in nonconvex optimization. This result improves on the best known ($O(\epsilon^{-2})$) evaluation-complexity bound for solving general nonconvexly constrained optimization problems.

## Citation

@techreport{CartGoulToin13a, author={C. Cartis and N. I. M. Gould and Ph. L. Toint}, title = {On the evaluation complexity of constrained nonlinear least-squares and general constrained nonlinear optimization using second-order methods}, institution = {Namur Center for Complex Systems (NAXYS), University of Namur}, address = {Namur, Belgium}, number = {naXys-01-2013}, year = 2013}