AbstractThe 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.
A variational approach to the alternating projections method
