A MINRES-based Linesearch Algorithm for Nonconvex Optimization with Non-positive Curvature Detection
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915707831189504 |
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| author | Zeng, Hanfeng Liu, Yang Ouyang, Wenqing Milzarek, Andre |
| author_facet | Zeng, Hanfeng Liu, Yang Ouyang, Wenqing Milzarek, Andre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01575 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A MINRES-based Linesearch Algorithm for Nonconvex Optimization with Non-positive Curvature Detection Zeng, Hanfeng Liu, Yang Ouyang, Wenqing Milzarek, Andre Optimization and Control 90C30 90C06 65K05 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. |
| title | A MINRES-based Linesearch Algorithm for Nonconvex Optimization with Non-positive Curvature Detection |
| topic | Optimization and Control 90C30 90C06 65K05 |
| url | https://arxiv.org/abs/2601.01575 |