Title: On Multivariate Polynomial Interpolation and Its Use in Derivative-Free Optimization

We will introduce a new, comprehensive approach for multivariate polynomial interpolation under which it is possible to derive appropriate bounds for the error between the function being interpolated and its interpolating polynomial. This derivation is based on a new concept of well-poisedness for the interpolation set, directly connecting the accuracy of the error bounds with the geometry of the points in the set. Our approach includes error bounds for function values as well as for derivatives. We show how to design algorithms to build sets of wellpoised interpolation points or to modify existing interpolation sets to ensure wellpoisedness. We will also talk about derivative free optimization techniques, relating geometrical concepts like well-poisedness to the key ingredients for attaining global convergence.
If time permits a new approach to solve derivative-free bilevel optimization problems based on interpolation-based models for both upper and lower level objective functions is also briefly discussed. The underlying algorithms are of trust-region type and the setting is such that both function and gradient values can be inexact.