# A variational approach to the alternating projections method

## TL;DR

This paper investigates the stability of the alternating projections method for convex feasibility problems in Hilbert spaces, analyzing convergence under perturbations of the sets and various intersection conditions.

## Contribution

It introduces a stability analysis framework for the method of alternating projections with perturbed sets converging to the original sets.

## Key findings

- Sequences converge in norm to a point in the intersection under certain conditions.
- Convergence is established for both singleton and nonempty interior intersections.
- Results include cases where limit sets are subspaces.

## 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 $X$. 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 sets, each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to $A$ and $B$. Given a starting point $a_0$, we consider the sequences of points obtained by projecting on the "perturbed" sets, i.e., the sequences $\{a_n\}$ and $\{b_n\}$ given by $b_n=P_{B_n}(a_{n-1})$ and $a_n=P_{A_n}(b_n)$. Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $\{a_n\}$ and $\{b_n\}$ converge in norm to a point in the intersection of $A$ and $B$. In particular, we consider both when the intersection $A\cap B$ reduces to a singleton and when the interior of $A \cap B$ is nonempty. Finally we consider the case in which the limit sets $A$ and $B$ are subspaces.