Set intersection problems: Supporting hyperplanes and quadratic programming

TL;DR

This paper introduces new algorithms utilizing supporting hyperplanes and quadratic programming to improve convergence in convex set intersection problems, including cases with cones and empty intersections.

Contribution

It proposes algorithms that enhance the method of alternating projections with strong, finite, and superlinear convergence guarantees under various conditions.

Findings

Converges strongly in Hilbert space

Finite convergence for convex cones under alignment

Superlinear convergence under certain conditions

Abstract

We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the intersection of closed convex sets, we propose an algorithm that, like Dykstra's algorithm, converges strongly in a Hilbert space. Moreover, this algorithm converges in finitely many iterations when the closed convex sets are cones in $\mathbb{R}^{n}$ satisfying an alignment condition. Next, we propose modifications of the alternating projection algorithm, and prove its convergence. The algorithm converges superlinearly in $\mathbb{R}^{n}$ under some nice conditions. Under a conical condition, the convergence can be finite. Lastly, we discuss the case where the intersection of the sets is empty.