On the evaluation complexity of constrained nonlinear least-squares and general constrained nonlinear optimization using second-order methods - Cartis, C; Gould, N; Toint, P (Society for Industrial and Applied Mathematics, 2015)
When solving the general smooth nonlinear and possibly nonconvex optimization problem involving equality and/or inequality constraints, an approximate first-order critical point of accuracy $ε$ can be obtained by a second-order method using cubic regularization in at most $O(ε^{-3/2})$ evaluations of problem functions, 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 nonconvex) 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 ($O(ε^{-2})$) evaluation-complexity bound for solving general nonconvexly constrained optimization problems.