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

Journal article

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.

Authors: Cartis, C; Gould, N; Toint, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Numerical Analysis
• Volume: 53
• Issue: 2
• Pages: 836–851
• Publication date: 2015-03-26
• Acceptance date: 2015-01-20
• DOI: 10.1137/130915546
• EISSN: 1095-7170
• ISSN: 0036-1429
• Keywords: constrained nonlinear optimization; least-squares problems; cubic regularization methods; evaluation complexity; worst-case analysis