Geometry of Sample Sets in Derivative Free Optimization. Part II: Polynomial Regression and Underdetermined Interpolation

In the recent years, there has been a considerable amount of work in the development of numerical methods for derivative free optimizatin problems. Some of this work relies on the management of the geometry of sets of sampling pionts for function evaluation an dmodel 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 notio of -poisedness ingtroduced 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 -poisedness 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

By: Andrew R. Conn; Katya Scheinberg; Luis. N. Vicente

Published in: RC23772 in 2005


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