Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/334875 
Year of Publication: 
2025
Citation: 
[Journal:] Computational Optimization and Applications [ISSN:] 1573-2894 [Volume:] 93 [Issue:] 1 [Publisher:] Springer US [Place:] New York, NY [Year:] 2025 [Pages:] 1-55
Publisher: 
Springer US, New York, NY
Abstract: 
We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure q -superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and q -superlinear convergence of the developed solution algorithm.
Subjects: 
Semismooth Newton method
Quasi-variational inequality
Thermoforming
Nonsmooth analysis
Obstacle problem
Newton differentiability
Semismoothness
Superlinear convergence
Persistent Identifier of the first edition: 
Creative Commons License: 
cc-by Logo
Document Type: 
Article
Document Version: 
Published Version

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