How good are projection methods for convex feasibility problems?
Computational Optimization and Applications
Tunneling descent level set segmentation of ultrasound images
IPMI'05 Proceedings of the 19th international conference on Information Processing in Medical Imaging
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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)\| 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.