{
  "id": "2506.15895",
  "version": "v1",
  "published": "2025-06-18T21:39:35.000Z",
  "updated": "2025-06-18T21:39:35.000Z",
  "title": "Parallel Polyhedral Projection Method for the Convex Feasibility Problem",
  "authors": [
    "Pablo Barros",
    "Roger Behling",
    "Vincent Guigues"
  ],
  "categories": [
    "math.OC"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-18T21:39:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex feasibility problem",
      "approximate parallel polyhedral projection method",
      "3pm achieves superlinear convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}