An Inexact Regularized Proximal Newton Method without Line Search
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| Format: | Preprint |
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2024
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| author | Dahl, Simeon vom Kanzow, Christian |
| author_facet | Dahl, Simeon vom Kanzow, Christian |
| contents | In this paper, we introduce an inexact regularized proximal Newton method (IRPNM) that does not require any line search. The method is designed to minimize the sum of a twice continuously differentiable function $f$ and a convex (possibly non-smooth and extended-valued) function $φ$. Instead of controlling a step size by a line search procedure, we update the regularization parameter in a suitable way, based on the success of the previous iteration. The global convergence of the sequence of iterations and its superlinear convergence rate under a local Hölderian error bound assumption are shown. Notably, these convergence results are obtained without requiring a global Lipschitz property for $ \nabla f $, which, to the best of the authors' knowledge, is a novel contribution for proximal Newton methods. To highlight the efficiency of our approach, we provide numerical comparisons with an IRPNM using a line search globalization and a modern FISTA-type method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_02635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Inexact Regularized Proximal Newton Method without Line Search Dahl, Simeon vom Kanzow, Christian Optimization and Control 49M15, 65K10, 90C26, 90C30, 90C55 In this paper, we introduce an inexact regularized proximal Newton method (IRPNM) that does not require any line search. The method is designed to minimize the sum of a twice continuously differentiable function $f$ and a convex (possibly non-smooth and extended-valued) function $φ$. Instead of controlling a step size by a line search procedure, we update the regularization parameter in a suitable way, based on the success of the previous iteration. The global convergence of the sequence of iterations and its superlinear convergence rate under a local Hölderian error bound assumption are shown. Notably, these convergence results are obtained without requiring a global Lipschitz property for $ \nabla f $, which, to the best of the authors' knowledge, is a novel contribution for proximal Newton methods. To highlight the efficiency of our approach, we provide numerical comparisons with an IRPNM using a line search globalization and a modern FISTA-type method. |
| title | An Inexact Regularized Proximal Newton Method without Line Search |
| topic | Optimization and Control 49M15, 65K10, 90C26, 90C30, 90C55 |
| url | https://arxiv.org/abs/2404.02635 |