Second-order optimality and beyond: Characterization and Evaluation Complexity in Convexly Constrained Nonlinear Optimization

High-order optimality conditions for convexly constrained nonlinear optimization problems are analysed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order ϵ-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that if derivatives of the objective function up to order q≥ 1 can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most O(ϵ ^{- ( q + 1 )}) evaluations of f and its derivatives to compute an ϵ-approximate qth-order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed, showing that the obtained evaluation complexity bounds are essentially sharp.