On the Convex Feasibility Problem
This provides a partial solution to a known open problem in convex optimization, relevant to researchers studying projection algorithms and set-theoretic methods.
The paper addresses the convex feasibility problem, specifically the unsolved case where the intersection of closed convex sets is bounded but has empty interior. It proves that for two closed convex sets in R3, the regularity property holds, ensuring convergence of the projection algorithm.
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.