On complexity constants of linear and quadratic models for derivative-free trust-region algorithms

TL;DR

This paper systematically analyzes the complexity constants in derivative-free trust-region algorithms with linear and quadratic models, explicitly relating them to problem dimension, sample set quality, and sample size.

Contribution

It organizes and clarifies existing bounds, extends results to inexact interpolation, and offers a clearer proof for the underdetermined case.

Findings

Constants depend explicitly on problem dimension, sample set quality, and number of points.

Extended results to inexact interpolation sets.

Provided a clearer proof for the underdetermined case.

Abstract

Complexity analysis has become an important tool in the convergence analysis of optimization algorithms. For derivative-free optimization algorithms, it is not different. Interestingly, several constants that appear when developing complexity results hide the dimensions of the problem. This work organizes several results in literature about bounds that appear in derivative-free trust-region algorithms based on linear and quadratic models. All the constants are given explicitly by the quality of the sample set, dimension of the problem and number of sample points. We extend some results to allow "inexact" interpolation sets. We also provide a clearer proof than those already existing in literature for the underdetermined case.