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Evaluation complexity for convexly constrained optimization is considered andit is shown first that the complexity bound of $O(\epsilon^{-3/2})$ proved byCartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an$\epsilon$-approximate first-order critical point can be obtained undersignificantly weaker assumptions. Moreover, the result is generalized to thecase where high-order derivatives are used, resulting in a bound of$O(\epsilon^{-(p+1)/p})$ evaluations whenever derivatives of order $p$ areavailable. 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 bothequality and inequality constraints can be generalized to a bound of$O(\epsilon_P^{-1/p}\epsilon_D^{-(p+1)/p})$ evaluations under similarlyweakened assumptions. This paper is variant of a companion report (NTR-11-2015,University of Namur, Belgium) which uses a different first-order criticalitymeasure to obtain the same complexity bounds.

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