Iterative Methods for Fixed Point Problems in Hilbert Spaces by Andrzej Cegielski

Author:Andrzej Cegielski
Language: eng
Format: epub, pdf
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(see Fig. 4.9). The details are left to the reader.

Fig. 4.9Metric projection onto a ball

4.1.8 Metric Projection onto an Ellipsoid

An ellipsoidin has the form

where D is a positive definite matrix, and . An ellipsoid is a closed convex subset as a sublevel set of a convex function

(note that the Hessian is positive definite). We present a method for calculating the metric projection of a point onto the ellipsoid C (cf. [316, Sect. 3.4] and [228]). A different method was also presented in [128].

It follows from the definition of the metric projection that if and only if y is a solution of the following convex minimization problem

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