Given two nonempty and disjoint intersections of closed and convex subsets,
we look for a best approximation pair relative to them, i.e., a pair of points,
one in each intersection, attaining the minimum distance between the disjoint intersections.
We propose an iterative process based on projections onto the subsets
which generate the intersections. The process is inspired by the Halpern-Lions-
Wittmann-Bauschke algorithm and the classical alternating process of Cheney and
Goldstein, and its advantage is that there is no need to project onto the intersections
themselves, a task which can be rather demanding. We prove that under certain
conditions the two interlaced subsequences converge to a best approximation pair.
These conditions hold, in particular, when the space is Euclidean and the subsets
which generate the intersections are compact and strictly convex. Our result extends
the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of
points for two polyhedra", Computational Optimization and Applications 71 (2018),
509-523] which considered the case of finite-dimensional polyhedra.
Citation
Preprint, April 13, 2023.