Superlinear convergence in nonsmooth optimization via higher-order cutting-plane models
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908911636840448 |
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| author | Gebken, Bennet Ulbrich, Michael |
| author_facet | Gebken, Bennet Ulbrich, Michael |
| contents | A cutting-plane model for a nonsmooth function is the maximum of several first-order expansions centered at different points. Using such a model in a bundle method leads to linear convergence (of serious steps) to a minimum. In smooth optimization, superlinear convergence can be achieved by using higher-order models. We show that the same is true for the nonsmooth case, i.e., we show that cutting-plane models involving higher-order expansions can be used to achieve superlinear convergence in nonsmooth optimization. We first formally define higher-order cutting-plane models for lower-$C^2$ functions and derive an error estimate. Afterwards, we construct a trust-region bundle method based on these models that achieves local superlinear convergence of serious steps, and overall superlinear convergence for certain finite max-type functions. Finally, we verify the superlinear convergence in numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23236 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Superlinear convergence in nonsmooth optimization via higher-order cutting-plane models Gebken, Bennet Ulbrich, Michael Optimization and Control A cutting-plane model for a nonsmooth function is the maximum of several first-order expansions centered at different points. Using such a model in a bundle method leads to linear convergence (of serious steps) to a minimum. In smooth optimization, superlinear convergence can be achieved by using higher-order models. We show that the same is true for the nonsmooth case, i.e., we show that cutting-plane models involving higher-order expansions can be used to achieve superlinear convergence in nonsmooth optimization. We first formally define higher-order cutting-plane models for lower-$C^2$ functions and derive an error estimate. Afterwards, we construct a trust-region bundle method based on these models that achieves local superlinear convergence of serious steps, and overall superlinear convergence for certain finite max-type functions. Finally, we verify the superlinear convergence in numerical experiments. |
| title | Superlinear convergence in nonsmooth optimization via higher-order cutting-plane models |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2603.23236 |