On the Evaluation Complexity of Constrained Nonlinear Least-Squares and General Constrained Nonlinear Optimization Using Second-Order Methods

Abstract

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 $\epsilon$ can be obtained by a second-order method using cubic regularization in at most $O(\epsilon^{-3/2})$ 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 ($O(\epsilon^{-2})$) evaluation-complexity bound for solving general nonconvexly constrained optimization problems.