An Abstract Stochastic Haugazeau Method for Best Approximation

TL;DR

This paper introduces a stochastic version of the Haugazeau method for best approximation in Hilbert spaces, enabling convergence with randomly generated outer approximations and applications to Chebyshev centers.

Contribution

It develops an abstract stochastic Haugazeau method with proven strong convergence, extending the classical approach to stochastic outer approximations and block-operator activation.

Findings

Strong convergence in mean square and almost sure modes.

Effective stochastic algorithms for intersections of fixed point sets.

Numerical demonstration on Chebyshev center computation.

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.