In a recent paper, Bauschke et al. study rho-comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its r...
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In a recent paper, Bauschke et al. study rho-comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent J(A). In this note we show that this result makes it possible to adapt many proofs of properties of the proximalpointalgorithm PPA and its strongly convergent halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for rho-comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. zer A we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a halpern-type variant HPPA of the PPA for rho-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.
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