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An Abstract Stochastic Haugazeau Method for Best Approximation

Javier I. Madariaga

TL;DR

An abstract stochastic version of the Haugazeau method to compute the best approximation from a closed convex set by successive projections onto randomly generated stochastic outer approximations of that set is proposed.

Abstract

The Haugazeau method was originally designed to compute the best approximation from an intersection of closed convex sets in Hilbert spaces using the projection operators onto the individual sets iteratively. We propose an abstract stochastic version of it to compute the best approximation from a closed convex set by successive projections onto randomly generated stochastic outer approximations of that set. Strong convergence in the mean square and the almost sure modes is derived under general hypotheses on the outer approximations. The results are applied to the development of stochastic algorithms to construct the best approximation from an arbitrary intersection of fixed point sets by random activation of blocks of operators. A numerical application to the computation of Chebyshev centers is provided.

An Abstract Stochastic Haugazeau Method for Best Approximation

TL;DR

An abstract stochastic version of the Haugazeau method to compute the best approximation from a closed convex set by successive projections onto randomly generated stochastic outer approximations of that set is proposed.

Abstract

The Haugazeau method was originally designed to compute the best approximation from an intersection of closed convex sets in Hilbert spaces using the projection operators onto the individual sets iteratively. We propose an abstract stochastic version of it to compute the best approximation from a closed convex set by successive projections onto randomly generated stochastic outer approximations of that set. Strong convergence in the mean square and the almost sure modes is derived under general hypotheses on the outer approximations. The results are applied to the development of stochastic algorithms to construct the best approximation from an arbitrary intersection of fixed point sets by random activation of blocks of operators. A numerical application to the computation of Chebyshev centers is provided.
Paper Structure (9 sections, 6 theorems, 53 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 6 theorems, 53 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Notation
  4. Preliminary results
  5. Convergence analysis
  6. Application to common fixed point best approximation
  7. Algorithm and convergence
  8. Best approximation from a finite collection of closed convex sets
  9. Application to the computation of a Chebyshev center

Key Result

Lemma 2.1

Set $\upchi=\scal{\mathsf{x-y}}{\mathsf{y-z}}_{\mathsf{H}}$, $\upmu=\|\mathsf{x-y}\|_{\mathsf{H}}^2$, $\upnu=\|\mathsf{y-z}\|_{\mathsf{H}}^2$, and $\uprho=\upmu\upnu-\upchi^2$. Then

Figures (1)

  • Figure 1: Three instances of the experiment of Section \ref{['sec:5']}. Blue: The subset $\mathsf{S}$. Red: The ball with center and radius given by the $\upalpha$-approximation.

Theorems & Definitions (13)

  • Lemma 2.1: Livre1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Moco25
  • Theorem 3.1
  • proof
  • Example 4.2: Mor1
  • Theorem 4.3
  • proof
  • Remark 4.4
  • ...and 3 more