Abstract

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 epsilon -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 ge 1 can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most O(epsilon ^{-(q+1)}) evaluations of f and its derivatives to compute an epsilon -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.

Highlights

  • Recent years have seen a growing interest in the analysis of the worst-case evaluation complexity of nonlinear smooth optimization

  • In the remainder of this section, we show that the example proposed in [18] can be extended to arbitrary order q, and that the complexity bounds (4.9)–(4.10) are essentially sharp for our trust-region algorithm

  • We have analysed the necessary and sufficient optimality conditions of arbitrary order for convexly constrained nonlinear optimization problems, using approximations of the feasible region which generalizes the idea of second-order tangent sets to orders beyond two

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Summary

Introduction

Recent years have seen a growing interest in the analysis of the worst-case evaluation complexity of nonlinear (possibly non-convex) smooth optimization (for the nonconvex case only, see [1,5,6,7,8,9,11,14,15,16,17,19,20,21,22,23,26,27,28,29,31,32,34,35,36,37,41,42,43,44,47,51,53,54,55] among others) In general terms, this analysis aims at giving (sometimes sharp) bounds on the number of evaluations of a minimization problem’s functions (objective and constraints, if relevant) and their derivatives that are, in the worst case, necessary for certain algorithms to find an approximate critical point for the unconstrained, convexly constrained or general nonlinear optimization problem. The analysis of evaluation complexity for orders higher than three is missing both concepts and results

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