The Error in Multivariate Linear Extrapolation with Applications to Derivative-Free Optimization

TL;DR

This paper analyzes the error bounds of multivariate linear extrapolation, providing numerical and analytical bounds, and applies these results to derivative-free optimization under specific smoothness assumptions.

Contribution

It introduces a method to compute sharp error bounds for linear extrapolation and applies these bounds to improve derivative-free optimization techniques.

Findings

Developed a numerical method for sharp error bound computation.

Presented analytical bounds with conditions for sharpness.

Analyzed the complexity of a simplicial search method in optimization.

Abstract

We study in this paper the function approximation error of multivariate linear extrapolation. The sharp error bound of linear interpolation already exists in the literature. However, linear extrapolation is used far more often in applications such as derivative-free optimization, while its error is not well-studied. We introduce in this paper a method to numerically compute the sharp bound on the error, and then present several analytical bounds along with the conditions under which they are sharp. We also provide a complexity analysis of a basic simplicial search method to illustrate an application of these error bounds in derivative-free optimization. All results are under the assumptions that the function being interpolated has Lipschitz continuous gradient and is interpolated on an affinely independent sample set.