Finding best approximation pairs for two intersections of closed convex sets

TL;DR

This paper introduces new algorithms using projection and proximity operators to find best approximation pairs for intersections of convex sets, improving upon previous methods with competitive numerical results.

Contribution

The paper proposes alternative algorithms leveraging projection and proximity operators for the best approximation pair problem, extending prior approaches for finite intersections of halfspaces.

Findings

Algorithms are competitive with existing methods.

Numerical experiments show sometimes superior performance.

Methods are applicable to intersections of convex sets.

Abstract

The problem of finding a best approximation pair of two sets, which in turn generalizes the well known convex feasibility problem, has a long history that dates back to work by Cheney and Goldstein in 1959. In 2018, Aharoni, Censor, and Jiang revisited this problem and proposed an algorithm that can be used when the two sets are finite intersections of halfspaces. Motivated by their work, we present alternative algorithms that utilize projection and proximity operators. Numerical experiments indicate that these methods are competitive and sometimes superior to the one proposed by Aharoni et al.