Finitely Convergent Iterative Methods with Overrelaxations Revisited

TL;DR

This paper investigates the finite convergence of iterative methods for convex feasibility problems, focusing on overrelaxation parameters that tend to zero at a subgeometric rate, and extends convergence guarantees to cases with empty interior solutions.

Contribution

It introduces a novel analysis allowing summable overrelaxation series and provides convergence guarantees even when the solution set has empty interior.

Findings

Finite convergence under subgeometric overrelaxation decay

Asymptotic convergence with empty interior solution sets

Extension of quasi-Fejérian analysis to new overrelaxation conditions

Abstract

We study the finite convergence of iterative methods for solving convex feasibility problems. Our key assumptions are that the interior of the solution set is nonempty and that certain overrelaxation parameters converge to zero, but with a rate slower than any geometric sequence. Unlike other works in this area, which require divergent series of overrelaxations, our approach allows us to consider some summable series. By employing quasi-Fej\'{e}rian analysis in the latter case, we obtain additional asymptotic convergence guarantees, even when the interior of the solution set is empty.