A new projection method for finding the closest point in the intersection of convex sets

TL;DR

This paper introduces AAMR, a new iterative projection method that effectively finds the closest point in the intersection of convex sets in a Hilbert space, improving upon existing methods with proven convergence.

Contribution

The paper proposes AAMR, a modified Douglas--Rachford method, with strong convergence guarantees under the strong CHIP condition, advancing the best approximation problem solutions.

Findings

AAMR converges strongly under the strong CHIP condition.

Numerical experiments show AAMR outperforms existing projection methods.

AAMR is effective for finite-dimensional subspace intersections.

Abstract

In this paper we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas--Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of the AAMR method for every point in the space. We report some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces.