Concrete convergence rates for common fixed point problems under Karamata regularity
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arXiv
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| Format: | Preprint |
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2024
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| author | Liu, Tianxiang Lourenço, Bruno F. |
| author_facet | Liu, Tianxiang Lourenço, Bruno F. |
| contents | We introduce the notion of Karamata regular operators, which is a notion of regularity that is suitable for obtaining concrete convergence rates for common fixed point problems. This provides a broad framework that includes, but goes beyond, Hölderian error bounds and Hölder regular operators. By concrete, we mean that the rates we obtain are explicitly expressed in terms of a function of the iteration number $k$ instead, of say, a function of the iterate $x^k$. While it is well-known that under Hölderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy Hölderian assumptions, which may happen if a problem involves exponentials and logarithms. Our main innovation is the usage of the theory of regularly varying functions which we showcase by obtaining concrete convergence rates for quasi-cylic algorithms in non-Hölderian settings. This includes certain rates that are neither sublinear nor linear but sit somewhere in-between, including a case where the rate is expressed via the Lambert W function. Finally, we connect our discussion to o-minimal geometry and show that, under mild assumptions, definable operators in any o-minimal structure are always Karamata regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13234 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concrete convergence rates for common fixed point problems under Karamata regularity Liu, Tianxiang Lourenço, Bruno F. Optimization and Control Numerical Analysis Functional Analysis Metric Geometry We introduce the notion of Karamata regular operators, which is a notion of regularity that is suitable for obtaining concrete convergence rates for common fixed point problems. This provides a broad framework that includes, but goes beyond, Hölderian error bounds and Hölder regular operators. By concrete, we mean that the rates we obtain are explicitly expressed in terms of a function of the iteration number $k$ instead, of say, a function of the iterate $x^k$. While it is well-known that under Hölderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy Hölderian assumptions, which may happen if a problem involves exponentials and logarithms. Our main innovation is the usage of the theory of regularly varying functions which we showcase by obtaining concrete convergence rates for quasi-cylic algorithms in non-Hölderian settings. This includes certain rates that are neither sublinear nor linear but sit somewhere in-between, including a case where the rate is expressed via the Lambert W function. Finally, we connect our discussion to o-minimal geometry and show that, under mild assumptions, definable operators in any o-minimal structure are always Karamata regular. |
| title | Concrete convergence rates for common fixed point problems under Karamata regularity |
| topic | Optimization and Control Numerical Analysis Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2407.13234 |