Parallel Polyhedral Projection Method for the Convex Feasibility Problem

TL;DR

This paper introduces the Parallel Polyhedral Projection Method (3PM) and its approximation A3PM for efficiently finding points in the intersection of convex sets, with proven convergence properties and superior performance in numerical tests.

Contribution

It presents a novel parallel projection framework for convex feasibility problems, including both exact and approximate variants, with comprehensive convergence analysis and improved practical performance.

Findings

A3PM often outperforms classical methods in numerical experiments.

Global convergence is established for both methods without regularity assumptions.

Linear and superlinear convergence are proven under specific conditions.

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.