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 {An} and {Bn}, each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point a0, we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences {an} and {bn} given by bn=PBn(an−1) and an=PAn(bn). Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences {an} and {bn} converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection A∩B reduces to a singleton and when the interior of A∩B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

De Bernardi, C. A., Miglierina, E., A variational approach to the alternating projections method, <<JOURNAL OF GLOBAL OPTIMIZATION>>, 2021; 2021 (81): 323-350. [doi:10.1007/s10898-021-01025-y] [http://hdl.handle.net/10807/177508]

### A variational approach to the alternating projections method

#### 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 {An} and {Bn}, each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point a0, we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences {an} and {bn} given by bn=PBn(an−1) and an=PAn(bn). Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences {an} and {bn} converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection A∩B reduces to a singleton and when the interior of A∩B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.
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2021
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De Bernardi, C. A., Miglierina, E., A variational approach to the alternating projections method, <<JOURNAL OF GLOBAL OPTIMIZATION>>, 2021; 2021 (81): 323-350. [doi:10.1007/s10898-021-01025-y] [http://hdl.handle.net/10807/177508]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10807/177508`
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