Core Concepts
The authors propose an efficient iterative algorithm based on projections onto the individual sets that generate the two disjoint intersections, without the need to project directly onto the intersections themselves, which can be computationally demanding. The algorithm is proven to converge to the unique best approximation pair under certain conditions.
Abstract
The paper addresses the problem of finding the best approximation pair between two nonempty and disjoint intersections of closed and convex sets in a Euclidean space. The authors propose an iterative algorithm inspired by the Halpern-Lions-Wittmann-Bauschke (HLWB) algorithm and the classical alternating process of Cheney and Goldstein.
The key highlights are:
The algorithm eliminates the need to project directly onto the intersection sets, which can be computationally demanding. Instead, it uses a weighted sum of projections onto the individual sets that generate the intersections.
The algorithm alternates between the two intersections, performing successive weighted sums of projections, with the number of projections increasing from one sweep to the next.
Under certain conditions, such as the sets being strictly convex, the authors prove that the algorithm converges to the unique best approximation pair.
The proof of convergence involves a generalization of Dini's theorem for uniform convergence of operators, as well as properties of fixed points of compositions of averaged nonexpansive operators.
The result extends the work of Aharoni et al. who considered the case of finite-dimensional polyhedra, by allowing for more general convex sets.