Abstract
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 proximal point algorithm 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.
Highlights
A central theme in convex optimization is the computation of zeros z ∈ zer A := A−1(0) of monotone set-valued operators A ⊆ H × H in Hilbert spaceDarmstadt, Germany U
In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant Halpern-type proximal point algorithm (HPPA) to this more general class of operators
E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for ρ-comonotone operators and obtain an effective rate of metastability
Summary
A central theme in convex optimization is the computation of zeros z ∈ zer A := A−1(0) of (maximally) monotone set-valued operators A ⊆ H × H in Hilbert space. In the recent papers [12,13], we studied from a quantitative point of view the PPA as well as a strongly convergent so-called Halpern-type variant HPPA (in Banach spaces) making use essentially only of the fact that all firmly nonexpansive mappings have a common so-called modulus for being strongly nonexpansive (see [10]). This holds true for the class of averaged mappings if we have some control on the averaging constant (see [21]). For the HPPA, to the best of our knowledge, our note provides the first results in the absence of monotonicity
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