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An open Newton method for piecewise smooth systems.

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Title: An open Newton method for piecewise smooth systems.
Authors: Radons, Manuel1 (AUTHOR) manuel.radons@bdr.de, Lehmann, Lutz1 (AUTHOR), Streubel, Tom1 (AUTHOR), Griewank, Andreas1 (AUTHOR)
Source: Optimization Methods & Software. Apr2026, Vol. 41 Issue 2, p550-576. 27p.
Subjects: Piecewise linear approximation, Topological degree, Jacobian matrices, Applied mathematics, Newton-Raphson method, Smoothness of functions
Abstract: Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed for the solution of PS equation systems. In the present work we relax the criterion of local bijectivity of the linearization to local openness. For this purpose a weak implicit function theorem is proved via local mapping degree theory. It is shown that there exist PS functions $ f:\mathbb {R}^2\rightarrow \mathbb {R}^2 $ f : R 2 → R 2 satisfying the weaker criterion where every neighbourhood of the root of f contains a point x such that all elements of the Clarke Jacobian at x are singular. In such neighbourhoods the steps of classical semismooth Newton are not defined, which establishes the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization, several statements about structure correspondences between the two are proved. Moreover, the influence of the specific representation of the local piecewise linear models on the robustness of our method is studied. An example application from cardiovascular mathematics is given. [ABSTRACT FROM AUTHOR]
Copyright of Optimization Methods & Software is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: An open Newton method for piecewise smooth systems.
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  Data: <searchLink fieldCode="JN" term="%22Optimization+Methods+%26+Software%22">Optimization Methods & Software</searchLink>. Apr2026, Vol. 41 Issue 2, p550-576. 27p.
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  Data: <searchLink fieldCode="DE" term="%22Piecewise+linear+approximation%22">Piecewise linear approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Topological+degree%22">Topological degree</searchLink><br /><searchLink fieldCode="DE" term="%22Jacobian+matrices%22">Jacobian matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Applied+mathematics%22">Applied mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Newton-Raphson+method%22">Newton-Raphson method</searchLink><br /><searchLink fieldCode="DE" term="%22Smoothness+of+functions%22">Smoothness of functions</searchLink>
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  Data: Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed for the solution of PS equation systems. In the present work we relax the criterion of local bijectivity of the linearization to local openness. For this purpose a weak implicit function theorem is proved via local mapping degree theory. It is shown that there exist PS functions $ f:\mathbb {R}^2\rightarrow \mathbb {R}^2 $ f : R 2 → R 2 satisfying the weaker criterion where every neighbourhood of the root of f contains a point x such that all elements of the Clarke Jacobian at x are singular. In such neighbourhoods the steps of classical semismooth Newton are not defined, which establishes the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization, several statements about structure correspondences between the two are proved. Moreover, the influence of the specific representation of the local piecewise linear models on the robustness of our method is studied. An example application from cardiovascular mathematics is given. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Optimization Methods & Software is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1080/10556788.2026.2624457
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      – Code: eng
        Text: English
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        PageCount: 27
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      – SubjectFull: Piecewise linear approximation
        Type: general
      – SubjectFull: Topological degree
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      – SubjectFull: Jacobian matrices
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      – SubjectFull: Applied mathematics
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      – SubjectFull: Newton-Raphson method
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      – SubjectFull: Smoothness of functions
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      – TitleFull: An open Newton method for piecewise smooth systems.
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              M: 04
              Text: Apr2026
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              Y: 2026
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