A Globalized Semismooth Newton Method for Prox-regular Optimization Problems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916937616850944 |
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| author | Wu, Yuqia Wu, Pengcheng Hu, Yaohua Pan, Shaohua Yang, Xiaoqi |
| author_facet | Wu, Yuqia Wu, Pengcheng Hu, Yaohua Pan, Shaohua Yang, Xiaoqi |
| contents | We are concerned with a class of nonconvex and nonsmooth composite optimization problems, comprising a twice differentiable function and a prox-regular function. We establish a sufficient condition for the proximal mapping of a prox-regular function to be single-valued and locally Lipschitz continuous. By virtue of this property, we propose a hybrid of proximal gradient and semismooth Newton methods for solving these composite optimization problems, which is a globalized semismooth Newton method. The whole sequence is shown to converge to an $L$-stationary point under a Kurdyka-Łojasiewicz exponent assumption. Under an additional error bound condition and some other mild conditions, we prove that the sequence converges to a nonisolated $L$-stationary point at a superlinear convergence rate. Numerical comparison with several existing second order methods reveal that our approach performs comparably well in solving both the $\ell_q(0<q<1)$ quasi-norm regularized problems and the fused zero-norm regularization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05765 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Globalized Semismooth Newton Method for Prox-regular Optimization Problems Wu, Yuqia Wu, Pengcheng Hu, Yaohua Pan, Shaohua Yang, Xiaoqi Optimization and Control We are concerned with a class of nonconvex and nonsmooth composite optimization problems, comprising a twice differentiable function and a prox-regular function. We establish a sufficient condition for the proximal mapping of a prox-regular function to be single-valued and locally Lipschitz continuous. By virtue of this property, we propose a hybrid of proximal gradient and semismooth Newton methods for solving these composite optimization problems, which is a globalized semismooth Newton method. The whole sequence is shown to converge to an $L$-stationary point under a Kurdyka-Łojasiewicz exponent assumption. Under an additional error bound condition and some other mild conditions, we prove that the sequence converges to a nonisolated $L$-stationary point at a superlinear convergence rate. Numerical comparison with several existing second order methods reveal that our approach performs comparably well in solving both the $\ell_q(0<q<1)$ quasi-norm regularized problems and the fused zero-norm regularization problems. |
| title | A Globalized Semismooth Newton Method for Prox-regular Optimization Problems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2509.05765 |