Evaluation complexity bounds for smooth constrained nonlinear optimization using scaled KKT conditions and high-order models

Evaluation complexity for convexly constrained optimization is considered and it is shown first that the complexity bound of O(ǫ) proved by Cartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an ǫ-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(ǫ) evaluations whenever derivatives of order p are available. It is also shown that the bound of O(ǫ −1/2 P ǫ −3/2 D ) evaluations (ǫP and ǫD being primal and dual accuracy thresholds) suggested by Cartis, Gould and Toint (SINUM, 53(2), 2015, pp.836-851) for the general nonconvex case involving both equality and inequality constraints can be generalized to yield a bound of O(ǫ −1/p P ǫ −(p+1)/p D ) evaluations under similarly weakened assumptions.