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

We study in this paper the function approximation error of multivariate linear extrapolation. While the sharp error bound of linear interpolation already exists in the literature, linear extrapolation is used far more often in applications such as derivative-free optimization, and its error is not well-studied. A method to numerically compute the sharp error bound is introduced, and several analytical bounds are presented along with the conditions under which they are sharp. The approximation error achievable by quadratic functions and the error bound for the bivariate case are analyzed in depth. Additionally, we provide the convergence theories regarding the simplex derivative-free optimization method as a demonstration of the utility of the derived bounds. All results are under the assumptions that the function being interpolated has Lipschitz continuous gradient and is interpolated on an affinely independent sample set.