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On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space

Kohlenbach, Ulrich (2022)
On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space.
In: Optimization Letters, 16 (2)
doi: 10.1007/s11590-021-01738-9
Artikel, Bibliographie

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Kurzbeschreibung (Abstract)

In a recent paper, Bauschke et al. study ρ-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 JA. 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 ρ-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 ρ-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.

Typ des Eintrags: Artikel
Erschienen: 2022
Autor(en): Kohlenbach, Ulrich
Art des Eintrags: Bibliographie
Titel: On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space
Sprache: Englisch
Publikationsjahr: März 2022
Ort: Berlin ; Heidelberg
Verlag: Springer
Titel der Zeitschrift, Zeitung oder Schriftenreihe: Optimization Letters
Jahrgang/Volume einer Zeitschrift: 16
(Heft-)Nummer: 2
DOI: 10.1007/s11590-021-01738-9
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Kurzbeschreibung (Abstract):

In a recent paper, Bauschke et al. study ρ-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 JA. 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 ρ-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 ρ-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.

Freie Schlagworte: Generalized monotone operators, Proximal point algorithm, Halpern-type proximal point algorithm, Rates of convergence, Metastability, Proof mining
Sachgruppe der Dewey Dezimalklassifikatin (DDC): 500 Naturwissenschaften und Mathematik > 510 Mathematik
Fachbereich(e)/-gebiet(e): 04 Fachbereich Mathematik
04 Fachbereich Mathematik > Logik
Hinterlegungsdatum: 03 Apr 2024 05:15
Letzte Änderung: 03 Apr 2024 05:15
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