A finitely convergent circumcenter method for the Convex Feasibility Problem

TL;DR

This paper introduces a novel circumcenter method for the Convex Feasibility Problem that guarantees finite convergence by using projections onto separating halfspaces with decreasing perturbations, a first in the field.

Contribution

It proposes the first circumcenter method for CFP with proven finite convergence under Slater's condition, using perturbed projections onto separating halfspaces.

Findings

Ensures finite convergence under Slater's condition.

Uses projections onto separating halfspaces with decreasing perturbations.

Guarantees finite convergence with slowly decreasing perturbation parameters.

Abstract

In this paper, we present a variant of the circumcenter method for the Convex Feasibility Problem (CFP), ensuring finite convergence under a Slater assumption. The method replaces exact projections onto the convex sets with projections onto separating halfspaces, perturbed by positive exogenous parameters that decrease to zero along the iterations. If the perturbation parameters decrease slowly enough, such as the terms of a diverging series, finite convergence is achieved. To the best of our knowledge, this is the first circumcenter method for CFP that guarantees finite convergence.