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The rate of linear convergence of the Douglas-Rachford algorithm for subspaces is the cosine of the Friedrichs angle | | Heinz H. Bauschke
; J.Y. Bello Cruz
; Hung M. Phan
; Xianfu Wang
; | | Date: |
18 Sep 2013 | | Abstract: | The Douglas-Rachford splitting algorithm is a classical optimization method
that has found many applications. When specialized to two normal cone
operators, it gives rise to an algorithm for finding a point in the
intersection of two convex sets. This method for solving feasibility problems
has attracted a lot of attention due to its good performance even in nonconvex
settings.
In this paper, we consider the Douglas-Rachford algorithm for finding a point
in the intersection of two subspaces. We prove that the method converges
strongly to the projection of the starting point onto the intersection.
Moreover, if the sum of the two subspaces is closed, then the convergence is
linear with the rate being the cosine of the Friedrichs angle between the
subspaces. Our results improve upon existing results in three ways: First, we
identify the location of the limit and thus reveal the method as a best
approximation algorithm; second, we quantify the rate of convergence, and
third, we carry out our analysis in general (possibly infinite-dimensional)
Hilbert space. We also provide various examples as well as a comparison with
the classical method of alternating projections. | | Source: | arXiv, 1309.4709 | | Services: | Forum | Review | PDF | Favorites | |
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