Abstract
In this paper we propose a modification of the von Neumann method of alternating projection x k+1=P A P B x k where A,B are closed and convex subsets of a real Hilbert space ℋ. If Fix P A P B ≠∅ then any sequence generated by the classical method converges weakly to a fixed point of the operator T=P A P B . If the distance δ=inf x∈A,y∈B ‖ x−y ‖ is known then one can efficiently apply a modification of the von Neumann method, which has the form x k+1=P A (x k +λ k (P A P B x k −x k )) for λ k >0 depending on x k (for details see: Cegielski and Suchocka, SIAM J. Optim. 19:1093–1106, 2008). Our paper contains a generalization of this modification, where we do not suppose that we know the value δ. Instead of δ we apply its approximation which is updated in each iteration.
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Cegielski, A., Dylewski, R. Variable target value relaxed alternating projection method. Comput Optim Appl 47, 455–476 (2010). https://doi.org/10.1007/s10589-009-9233-x
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DOI: https://doi.org/10.1007/s10589-009-9233-x