A variational approach to the alternating projections method

Carlo Alberto Bernardi () and Enrico Miglierina ()
Additional contact information
Carlo Alberto Bernardi: Università Cattolica del Sacro Cuore
Enrico Miglierina: Università Cattolica del Sacro Cuore

Journal of Global Optimization, 2021, vol. 81, issue 2, No 3, 323-350

Abstract: Abstract The 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets $$\{A_n\}$$ { A n } and $$\{B_n\}$$ { B n } , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point $$a_0$$ a 0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } given by $$b_n=P_{B_n}(a_{n-1})$$ b n = P B n ( a n - 1 ) and $$a_n=P_{A_n}(b_n)$$ a n = P A n ( b n ) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection $$A\cap B$$ A ∩ B reduces to a singleton and when the interior of $$A \cap B$$ A ∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

Keywords: Convex feasibility problem; Stability; Set-convergence; Alternating projections method; Primary: 47J25; Secondary: 90C25; 90C48 (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10898-021-01025-y Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jglopt:v:81:y:2021:i:2:d:10.1007_s10898-021-01025-y

Ordering information: This journal article can be ordered from
http://www.springer. ... search/journal/10898

DOI: 10.1007/s10898-021-01025-y

Access Statistics for this article

Journal of Global Optimization is currently edited by Sergiy Butenko

More articles in Journal of Global Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().