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
Article, Bibliographie
This is the latest version of this item.
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.
| Item Type: | Article |
|---|---|
| Erschienen: | 2022 |
| Creators: | Kohlenbach, Ulrich |
| Type of entry: | Bibliographie |
| Title: | On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space |
| Language: | English |
| Date: | March 2022 |
| Place of Publication: | Berlin ; Heidelberg |
| Publisher: | Springer |
| Journal or Publication Title: | Optimization Letters |
| Volume of the journal: | 16 |
| Issue Number: | 2 |
| DOI: | 10.1007/s11590-021-01738-9 |
| Corresponding Links: | |
| 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. |
| Uncontrolled Keywords: | Generalized monotone operators, Proximal point algorithm, Halpern-type proximal point algorithm, Rates of convergence, Metastability, Proof mining |
| Classification DDC: | 500 Science and mathematics > 510 Mathematics |
| Divisions: | 04 Department of Mathematics 04 Department of Mathematics > Logic |
| Date Deposited: | 03 Apr 2024 05:15 |
| Last Modified: | 03 Apr 2024 05:15 |
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| Suche nach Titel in: | TUfind oder in Google |
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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) [Currently Displayed]
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