A Convergence Analysis of Dykstra's Algorithm for Polyhedral Sets

Abstract

Let H be a nonempty closed convex set in a Hilbert space X determined by the intersection of a finite number of closed half spaces. It is well known that given an $x_0 \in X$, Dykstra's algorithm applied to this collection of closed half spaces generates a sequence of iterates that converge to PH(x0 ), the orthogonal projection of x0 onto H. The iterates, however, do not necessarily lie in H. We propose a combined Dykstra--conjugate-gradient method such that, given an $\varepsilon > 0$, the algorithm computes an $x \in H$ with $\|x - P_H(x_0)\| < \varepsilon$. Moreover, for each iterate xm of Dykstra's algorithm we calculate a bound for $\|x_m - P_H(x_0)\|$ that approaches zero as m tends to infinity. Applications are made to computing bounds for $\|x_m - P_H(x_0)\|$ where H is a polyhedral cone. Numerical results are presented from a sample isotone regression problem.