Parallel Polyhedral Projection Method for the Convex Feasibility Problem

Abstract: In this paper, we introduce and study the Parallel Polyhedral Projection Method (3PM) and the Approximate Parallel Polyhedral Projection Method (A3PM) for finding a point in the intersection of finitely many closed convex sets. Each iteration has two phases: parallel projections onto the target sets (exact in 3PM, approximate in A3PM), followed by an exact or approximate projection onto a polyhedron defined by supporting half-spaces. These strategies appear novel, as existing methods largely focus on parallel schemes like Cimmino's method. Numerical experiments demonstrate that A3PM often outperforms both classical and recent projection-based methods when the number of sets is greater than two. Theoretically, we establish global convergence for both 3PM and A3PM without regularity assumptions. Under a Slater condition or error bound, we prove linear convergence, even with inexact projections. Additionally, we show that 3PM achieves superlinear convergence under suitable geometric assumptions.