Set intersection problems: Supporting hyperplanes and quadratic programming

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.