Sharp Taylor Polynomial Enclosures in One Dimension
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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 kth 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.
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