Abstract:
The feasibility problem associated with nonempty closed
convex sets $A$ and $B$ is to find some $x\in A \cap B$.
Projection algorithms in general aim to compute such a point.
These algorithms play key roles in optimization and have
many applications outside mathematics - for example in medical
imaging.
Until recently convergence results were only available in the setting of linear spaces (more particularly, Hilbert spaces) and where the two sets are closed and convex.
The extension into geodesic metric spaces allows their use in spaces where there is no natural linear
structure, which is the case for instance in tree spaces, state spaces, phylogenomics
and configuration spaces for robotic movements.
After reviewing the pertinent aspects of CAT(0) spaces introduced in Part I, including results for von Neumann's alternating projection method, we will focus on
the Douglas-Rachford algorithm, in CAT(0) spaces. Two situations arise; spaces with constant curvature and those with non-constant curvature. A prototypical space of the later kind will be introduced and the behavior of the Douglas-Rachford algorithm within it examined.