Accuracy and Relationships of Quadratic Models in Derivative-free Optimization
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| Format: | Preprint |
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2026
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| _version_ | 1866909038452670464 |
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| author | Chen, Yiwen Hare, Warren Roberts, Lindon |
| author_facet | Chen, Yiwen Hare, Warren Roberts, Lindon |
| contents | We study three quadratic models in model-based derivative-free optimization: the minimum norm (MN), minimum Frobenius norm (MFN), and quadratic generalized simplex derivative (QS) models. Despite their widespread use, their approximation accuracy and relationships have not been systematically explored. We establish fully linear error bounds for all three models, removing the uniformly bounded model Hessian assumption required in existing MN analyses and deriving the first such results for the QS model. We further analyze Hessian approximation accuracy via directional error bounds, showing that all three models achieve fully quadratic accuracy along sample directions under a mild condition on the sample set. This reveals a form of directional fully quadratic accuracy not captured by existing theory. Finally, we characterize the relationships among these models, identifying conditions under which they coincide and clarifying their structural connections. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12819 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Accuracy and Relationships of Quadratic Models in Derivative-free Optimization Chen, Yiwen Hare, Warren Roberts, Lindon Optimization and Control Numerical Analysis 90C56, 65G99, 65K05 We study three quadratic models in model-based derivative-free optimization: the minimum norm (MN), minimum Frobenius norm (MFN), and quadratic generalized simplex derivative (QS) models. Despite their widespread use, their approximation accuracy and relationships have not been systematically explored. We establish fully linear error bounds for all three models, removing the uniformly bounded model Hessian assumption required in existing MN analyses and deriving the first such results for the QS model. We further analyze Hessian approximation accuracy via directional error bounds, showing that all three models achieve fully quadratic accuracy along sample directions under a mild condition on the sample set. This reveals a form of directional fully quadratic accuracy not captured by existing theory. Finally, we characterize the relationships among these models, identifying conditions under which they coincide and clarifying their structural connections. |
| title | Accuracy and Relationships of Quadratic Models in Derivative-free Optimization |
| topic | Optimization and Control Numerical Analysis 90C56, 65G99, 65K05 |
| url | https://arxiv.org/abs/2605.12819 |