  The method of alternating projections (MAP) is a common method for solving
feasibility problems. While employed traditionally to subspaces or to convex
sets, little was known about the behavior of the MAP in the nonconvex case
until 2009, when Lewis, Luke, and Malick derived local linear convergence
results provided that a condition involving normal cones holds and at least one
of the sets is superregular (a property less restrictive than convexity).
However, their results failed to capture very simple classical convex instances
such as two lines in three-dimensional space.
  In this paper, we extend and develop the Lewis-Luke-Malick framework so that
not only any two linear subspaces but also any two closed convex sets whose
relative interiors meet are covered. We also allow for sets that are more
structured such as unions of convex sets. The key tool required is the
restricted normal cone, which is a generalization of the classical Mordukhovich
normal cone. We thoroughly study restricted normal cones from the viewpoint of
constraint qualifications and regularity. Numerous examples are provided to
illustrate the theory.
