Geometry of sample sets in derivative free optimization. Part II: polynomial regression and underdetermined interpolation

Abstract

In the recent years, there has been a considerable amount of work in the development of numerical methods for derivative free optimization problems. Some of this work relies on the management of the geometry of sets of sampling points for function evaluation and model building. In this paper, we continue the work developed in [7] for complete or determined interpolation models (when the number of interpolation points equals the number of basis elements), considering now the cases where the number of points is higher (regression models) and lower (underdetermined models) than the number of basis components. We show how the notion of A-poisedness introduced in [7] to quantify the quality of the sample sets can be extended to the nondetermined cases, by extending first the underlying notion of bases of Lagrange polynomials. We also show that Apoisedness is equivalent to a bound on the condition number of the matrix arising from the sampling conditions. We derive bounds for the errors between the function and the (regression and underdetermined) models and between their derivatives.