Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces
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| Format: | Preprint |
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2023
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| _version_ | 1866913334682451968 |
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| author | Schiavo, Lorenzo Dello Maas, Jan Pedrotti, Francesco |
| author_facet | Schiavo, Lorenzo Dello Maas, Jan Pedrotti, Francesco |
| contents | This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka--Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on $\mathbb{R}^n$. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on $\mathbb{R}^n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_05239 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces Schiavo, Lorenzo Dello Maas, Jan Pedrotti, Francesco Optimization and Control 45J05, 49Q20 (Primary) 39B62, 37N40, 49J52, 65K10 (Secondary) This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka--Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on $\mathbb{R}^n$. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on $\mathbb{R}^n$. |
| title | Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces |
| topic | Optimization and Control 45J05, 49Q20 (Primary) 39B62, 37N40, 49J52, 65K10 (Secondary) |
| url | https://arxiv.org/abs/2304.05239 |