  We prove global convergence of classical projection algorithms for
feasibility problems involving union convex sets, which refer to sets
expressible as the union of a finite number of closed convex sets. We present a
unified strategy for analyzing global convergence by means of studying
fixed-point iterations of a set-valued operator that is the union of a finite
number of compact-valued upper semicontinuous maps. Such a generalized
framework permits the analysis of a class of proximal algorithms for minimizing
the sum of a piecewise smooth function and the difference between pointwise
minimum of finitely many weakly convex functions and a piecewise smooth convex
function. When realized on two-set feasibility problems, this algorithm class
recovers alternating projections and averaged projections as special cases, and
thus we obtain global convergence criterion for these projection algorithms.
Using these general results, we derive sufficient conditions to guarantee
global convergence for several projection algorithms for solving the sparse
affine feasibility problem and a feasibility reformulation of the linear
complementarity problem. Notably, we obtain global convergence of both the
alternating and the averaged projection methods to the solution set for linear
complementarity problems involving $P$-matrices. By leveraging the structures
of the classes of problems we consider, we also propose acceleration algorithms
with guaranteed global convergence. Numerical results further exemplify that
the proposed acceleration schemes significantly improve upon their
non-accelerated counterparts in efficiency.
