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
Dies ist die neueste Version dieses Eintrags.
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 |
| Zugehörige Links: | |
| 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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| Suche nach Titel in: | TUfind oder in Google |
Verfügbare Versionen dieses Eintrags
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On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space. (deposited 02 Apr 2024 11:22)
- On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space. (deposited 03 Apr 2024 05:15) [Gegenwärtig angezeigt]
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