Sharp Taylor Polynomial Enclosures in One Dimension

TL;DR

This paper introduces a method for deriving sharp polynomial bounds on one-dimensional functions over finite intervals, significantly improving the tightness of bounds compared to classical methods, with applications in optimization.

Contribution

The authors develop a novel approach for obtaining sharp polynomial enclosures that outperform classical derivative-based bounds, especially as the trust region shrinks.

Findings

Sharp bounds are at least $k+1$ times tighter than classical bounds.

Bounds become asymptotically optimal as trust region width approaches zero.

Method has potential applications in optimization algorithms like majorization-minimization.

Abstract

It is often useful to have polynomial upper or lower bounds on a one-dimensional function that are valid over a finite interval, called a trust region. A classical way to produce polynomial bounds of degree $k$ involves bounding the range of the $k$th derivative over the trust region, but this produces suboptimal bounds. We improve on this by deriving sharp polynomial upper and lower bounds for a wide variety of one-dimensional functions. We further show that sharp bounds of degree $k$ are at least $k+1$ times tighter than those produced by the classical method, asymptotically as the width of the trust region approaches zero. We discuss how these sharp bounds can be used in majorization-minimization optimization, among other applications.