### Technical Report ERGO 13-005

#### On the evaluation complexity of constrained nonlinear least-squares and general constrained nonlinear optimization using second-order methods

*Coralia Cartis, Nicholas I. M. Gould and Philippe L. Toint*

##### Abstract:

When solving the general smooth nonlinear 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}) 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(ε^{-2}))
evaluation-complexity bound for solving general nonconvexly constrained
optimization problems.

##### Download:

ERGO-13-005.pdf

##### History:

Written: 3 April 2013

##### Status:

Submitted for publication