{"ID":2828570,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.14507","arxiv_id":"2512.14507","title":"An Inexact Modified Quasi-Newton Method for Nonsmooth Regularized Optimization","abstract":"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.","short_abstract":"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 se...","url_abs":"https://arxiv.org/abs/2512.14507","url_pdf":"https://arxiv.org/pdf/2512.14507v1","authors":"[\"Nathan Allaire\",\"Sébastien Le Digabel\",\"Dominique Orban\"]","published":"2025-12-16T15:43:37Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}