{"ID":2822908,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2601.01575","arxiv_id":"2601.01575","title":"A MINRES-based Linesearch Algorithm for Nonconvex Optimization with Non-positive Curvature Detection","abstract":"We propose a MINRES-based Newton-type algorithm for solving unconstrained nonconvex optimization problems. Our approach uses the minimal residual method (MINRES), a well-known solver for indefinite symmetric linear systems, to compute descent directions that leverage second-order and non-positive curvature (NPC) information. Comprehensive asymptotic convergence properties are derived under standard assumptions. In particular, under the Kurdyka-Łojasiewicz inequality and a mild NPC-detectability condition, we prove that our algorithm can avoid strict saddle points and converge to second-order critical points. This is primarily achieved by integrating proper regularization techniques and forward linesearch mechanisms along NPC directions. Furthermore, fast local superlinear convergence to potentially non-isolated minima is established, when the local Polyak-Łojasiewicz condition is satisfied. Numerical experiments on the CUTEst test collection and on a deep auto-encoder problem illustrate the efficiency of the proposed method.","short_abstract":"We propose a MINRES-based Newton-type algorithm for solving unconstrained nonconvex optimization problems. Our approach uses the minimal residual method (MINRES), a well-known solver for indefinite symmetric linear systems, to compute descent directions that leverage second-order and non-positive curvature (NPC) inform...","url_abs":"https://arxiv.org/abs/2601.01575","url_pdf":"https://arxiv.org/pdf/2601.01575v1","authors":"[\"Hanfeng Zeng\",\"Yang Liu\",\"Wenqing Ouyang\",\"Andre Milzarek\"]","published":"2026-01-04T15:45:17Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}