On Josephy-Halley method for generalized equations
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arXiv
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| Format: | Preprint |
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2025
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| author | Roubal, Tomáš Valdman, Jan |
| author_facet | Roubal, Tomáš Valdman, Jan |
| contents | We extend the classical third-order Halley iteration to the setting of generalized equations of the form \[ 0 \in f(x) + F(x), \] where \(f\colon X\longrightarrow Y\) is twice continuously Fréchet-differentiable on Banach spaces and \(F\colon X\tto Y\) is a set-valued mapping with closed graph. Building on predictor-corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor \(u_{k+1}\), then incorporates second-order information in a Halley-type corrector step to obtain \(x_{k+1}\). Under metric regularity of the linearization at a reference solution and Hölder continuity of \(f''\), we prove that the iterates converge locally with order \(2+p\) (cubically when \(p=1\)). Moreover, by constructing a suitable scalar majorant function we derive semilocal Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments-including one- and two-dimensional test problems confirm the theoretical convergence rates and illustrate the efficiency of the Josephy-Halley method compared to its Josephy-Newton counterpart. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Josephy-Halley method for generalized equations Roubal, Tomáš Valdman, Jan Numerical Analysis Optimization and Control 49J53, 65K15, 90C33 We extend the classical third-order Halley iteration to the setting of generalized equations of the form \[ 0 \in f(x) + F(x), \] where \(f\colon X\longrightarrow Y\) is twice continuously Fréchet-differentiable on Banach spaces and \(F\colon X\tto Y\) is a set-valued mapping with closed graph. Building on predictor-corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor \(u_{k+1}\), then incorporates second-order information in a Halley-type corrector step to obtain \(x_{k+1}\). Under metric regularity of the linearization at a reference solution and Hölder continuity of \(f''\), we prove that the iterates converge locally with order \(2+p\) (cubically when \(p=1\)). Moreover, by constructing a suitable scalar majorant function we derive semilocal Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments-including one- and two-dimensional test problems confirm the theoretical convergence rates and illustrate the efficiency of the Josephy-Halley method compared to its Josephy-Newton counterpart. |
| title | On Josephy-Halley method for generalized equations |
| topic | Numerical Analysis Optimization and Control 49J53, 65K15, 90C33 |
| url | https://arxiv.org/abs/2504.17649 |