{"ID":2874004,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.05765","arxiv_id":"2509.05765","title":"A Globalized Semismooth Newton Method for Prox-regular Optimization Problems","abstract":"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 $\\ell_q(0\u003cq\u003c1)$ quasi-norm regularized problems and the fused zero-norm regularization problems.","short_abstract":"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 thi...","url_abs":"https://arxiv.org/abs/2509.05765","url_pdf":"https://arxiv.org/pdf/2509.05765v1","authors":"[\"Yuqia Wu\",\"Pengcheng Wu\",\"Yaohua Hu\",\"Shaohua Pan\",\"Xiaoqi Yang\"]","published":"2025-09-06T16:32:26Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}