# Projection and proximal point methods: convergence results and counterexamples

Bauschke, Heinz H. and Matouskova, Eva and Reich, Simeon (2003) Projection and proximal point methods: convergence results and counterexamples. [Preprint]

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Recently, Hundal has constructed a hyperplane $H$, a cone $K$, and a starting point $y_0$ in $\ell_2$ such that the sequence of alternating projections $\big((P_KP_H)^ny_0\big)_{n \in \NN}$ converges weakly to some point in $H \cap K$, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections; hence, we give an affirmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of firmly nonexpansive maps (\a la Genel and Lindenstrauss) and for the proximal point algorithm (\a la G\"uler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Censor, Elfving, and Herman. Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well.