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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2512.14507 |
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| _version_ | 1866918250357456896 |
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| author | Allaire, Nathan Digabel, Sébastien Le Orban, Dominique |
| author_facet | Allaire, Nathan Digabel, Sébastien Le Orban, Dominique |
| contents | We introduce iR2N, a modified proximal quasi-Newton method for minimizing the sum of a smooth function $f$ and a lower semi-continuous prox-bounded function $h$, allowing inexact evaluations of $f$, its gradient, and the associated proximal operators. Both $f$ and $h$ may be nonconvex. iR2N is particularly suited to settings where proximal operators are computed via iterative procedures that can be stopped early, or where the accuracy of $f$ and $\nabla f$ can be controlled, leading to significant computational savings. At each iteration, the method approximately minimizes the sum of a quadratic model of $f$, a model of $h$, and an adaptive quadratic regularization term ensuring global convergence. Under standard accuracy assumptions, we prove global convergence in the sense that a first-order stationarity measure converges to zero, with worst-case evaluation complexity $O(ε^{-2})$. Numerical experiments with $\ell_p$ norms, $\ell_p$ total variation, and the indicator of the nonconvex pseudo $p$-norm ball illustrate the effectiveness and flexibility of the approach, and show how controlled inexactness can substantially reduce computational effort. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14507 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Inexact Modified Quasi-Newton Method for Nonsmooth Regularized Optimization Allaire, Nathan Digabel, Sébastien Le Orban, Dominique Optimization and Control 90C53 (Primary) 65F22, 90C30 (Secondary) We introduce iR2N, a modified proximal quasi-Newton method for minimizing the sum of a smooth function $f$ and a lower semi-continuous prox-bounded function $h$, allowing inexact evaluations of $f$, its gradient, and the associated proximal operators. Both $f$ and $h$ may be nonconvex. iR2N is particularly suited to settings where proximal operators are computed via iterative procedures that can be stopped early, or where the accuracy of $f$ and $\nabla f$ can be controlled, leading to significant computational savings. At each iteration, the method approximately minimizes the sum of a quadratic model of $f$, a model of $h$, and an adaptive quadratic regularization term ensuring global convergence. Under standard accuracy assumptions, we prove global convergence in the sense that a first-order stationarity measure converges to zero, with worst-case evaluation complexity $O(ε^{-2})$. Numerical experiments with $\ell_p$ norms, $\ell_p$ total variation, and the indicator of the nonconvex pseudo $p$-norm ball illustrate the effectiveness and flexibility of the approach, and show how controlled inexactness can substantially reduce computational effort. |
| title | An Inexact Modified Quasi-Newton Method for Nonsmooth Regularized Optimization |
| topic | Optimization and Control 90C53 (Primary) 65F22, 90C30 (Secondary) |
| url | https://arxiv.org/abs/2512.14507 |