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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 disjointintersections. We propose an iterative process based on projections onto thesubsets which generate the intersections. The process is inspired by theHalpern-Lions-Wittmann-Bauschke algorithm and the classical alternating processof Cheney and Goldstein, and its advantage is that there is no need to projectonto the intersections themselves, a task which can be rather demanding. Weprove that under certain conditions the two interlaced subsequences converge toa best approximation pair. These conditions hold, in particular, when the spaceis Euclidean and the subsets which generate the intersections are compact andstrictly convex. Our result extends the one of Aharoni, Censor and Jiang["Finding a best approximation pair of points for two polyhedra", ComputationalOptimization and Applications 71 (2018), 509-523] which considered the case offinite-dimensional polyhedra.

The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm  for finding the best approximation pair for two disjoint intersections of  convex sets