Evaluation complexity bounds for smooth constrained nonlinear optimisation using scaled KKT conditions, high-order models and the criticality measure $\chi$

Abstract

Evaluation complexity for convexly constrained optimization is considered and it is shown first that the complexity bound of $O({\epsilon }^{-3/2})$ proved by Cartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an $\epsilon$-approximate first-order critical point can be obtained under significantly weaker assumptions. Moreover, the result is generalized to the case where high-order derivatives are used, resulting in a bound of $O({\epsilon }^{-(p+1)/p})$ evaluations whenever derivatives of order $p$ are available. It is also shown that the bound of $O({\epsilon }_P^{-1/2}{\epsilon }_D^{-3/2})$ evaluations (${\epsilon }_P$ and ${\epsilon }_D$ being primal and dual accuracy thresholds) suggested by Cartis, Gould and Toint (SINUM, 2015) for the general nonconvex case involving both equality and inequality constraints can be generalized to a bound of $O({\epsilon }_P^{-1/p}{\epsilon }_D^{-(p+1)/p})$ evaluations under similarly weakened assumptions. This paper is variant of a companion report (NTR-11-2015, University of Namur, Belgium) which uses a different first-order criticality measure to obtain the same complexity bounds.