On the Convex Feasibility Problem

TL;DR

This paper investigates the convergence of projection algorithms for convex feasibility problems, focusing on a specific challenging case where the intersection is bounded but has an empty interior, providing partial solutions in three-dimensional space.

Contribution

It offers a partial solution for the convex feasibility problem in R3 when the intersection's interior is empty but the intersection is bounded, addressing an unsolved case.

Findings

Regularity property holds for two convex sets in R3 under the specified conditions.

Provides partial theoretical understanding of the convergence in complex intersection scenarios.

Abstract

The convergence of the projection algorithm for solving the convex feasibility problem for a family of closed convex sets, is in connection with the regularity properties of the family. In the paper [18] are pointed out four cases of such a family depending of the two characteristics: the emptiness and boudedness of the intersection of the family. The case four (the interior of the intersection is empty and the intersection itself is bounded) is unsolved. In this paper we give a (partial) answer for the case four: in the case of two closed convex sets in R3 the regularity property holds.