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On a result by Meshulam

Heinz H. Bauschke, Tran Thanh Tung

TL;DR

The paper extends Meshulam's boundedness result for sequences generated by relaxed projections from affine subspaces to (i) finite families of polyhedral sets in finite-dimensional and then general Hilbert spaces, and (ii) analyzes the behavior when relaxation parameters approach reflections ($\lambda_n\to 2$) in the two-subspace setting. The key method is to reduce polyhedral projections to a finite collection of affine subspaces via faces, and, in infinite dimensions, to decompose onto a finite-codimensional subspace so the finite-dimensional argument applies. The authors provide counterexamples showing sharpness (inability to extend to general convex sets or to general Hilbert spaces) and a detailed breakdown of possible dynamics under approaching reflections, including bounded, convergent, and unbounded behaviors depending on the parameter sequence. These results broaden convergence guarantees for projection-based methods in convex feasibility and illuminate the nuanced behavior of relaxed projection dynamics in infinite-dimensional spaces.

Abstract

In 1996, Meshulam proved that every sequence generated by applying projections onto affine subspaces, drawn from a finite collection in Euclidean space, must be bounded. In this paper, we extend his result not only from affine subspaces to convex polyhedral subsets, but also from Euclidean to general Hilbert space. Various examples are provided to illustrate the sharpness of the results.

On a result by Meshulam

TL;DR

The paper extends Meshulam's boundedness result for sequences generated by relaxed projections from affine subspaces to (i) finite families of polyhedral sets in finite-dimensional and then general Hilbert spaces, and (ii) analyzes the behavior when relaxation parameters approach reflections () in the two-subspace setting. The key method is to reduce polyhedral projections to a finite collection of affine subspaces via faces, and, in infinite dimensions, to decompose onto a finite-codimensional subspace so the finite-dimensional argument applies. The authors provide counterexamples showing sharpness (inability to extend to general convex sets or to general Hilbert spaces) and a detailed breakdown of possible dynamics under approaching reflections, including bounded, convergent, and unbounded behaviors depending on the parameter sequence. These results broaden convergence guarantees for projection-based methods in convex feasibility and illuminate the nuanced behavior of relaxed projection dynamics in infinite-dimensional spaces.

Abstract

In 1996, Meshulam proved that every sequence generated by applying projections onto affine subspaces, drawn from a finite collection in Euclidean space, must be bounded. In this paper, we extend his result not only from affine subspaces to convex polyhedral subsets, but also from Euclidean to general Hilbert space. Various examples are provided to illustrate the sharpness of the results.

Paper Structure

This paper contains 14 sections, 3 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Extension to polyhedral sets
  3. Extension to infinite-dimensional spaces
  4. Two limiting examples
  5. Impossibility to extend from polyhedral sets to convex sets
  6. Impossibility to extend Meshulam's result to general Hilbert spaces
  7. Approaching reflections
  8. The case $\lim_n \lambda_n = 2$
  9. The subcase $\sum_n\varepsilon_n<+\infty$
  10. The subcase $\sum_n\varepsilon_n=+\infty$
  11. The case $\varliminf_n\lambda_n<\varlimsup_n \lambda_n = 2$
  12. A case where $(x_n)_{n\in\mathbb{N}}$ is bounded
  13. A case where $(x_n)_{n\in\mathbb{N}}$ is unbounded
  14. Summary

Key Result

Corollary 2.3

Let $C$ be a nonempty closed convex subset of $X$, and let $x\in X$. Then there exists a face $F$ of $C$ such that

Theorems & Definitions (9)

  • proof
  • Corollary 2.3
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.2: main result
  • proof
  • Example 4.1
  • Example 4.2