The Method of Alternating Relaxed Projections for Two Nonconvex Sets

Abstract

The Method of Alternating Projections (MAP), a classical algorithm for solving feasibility problems, has recently been intensely studied for nonconvex sets. However, intrinsically available are only local convergence results: convergence occurs if the starting point is not too far away from solutions to avoid getting trapped in certain regions. Instead of taking full projection steps, it can be advantageous to underrelax, i.e., to move only part way towards the constraint set, in order to enlarge the regions of convergence.In this paper, we thus systematically study the Method of Alternating Relaxed Projections (MARP) for two (possibly nonconvex) sets. Complementing our recent work on MAP, we establish local linear convergence results for the MARP. Several examples illustrate our analysis.