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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2602.10374 |
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| _version_ | 1866914321600086016 |
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| author | Chen, Yiwen |
| author_facet | Chen, Yiwen |
| contents | Quadratic interpolation models and simplex derivatives are fundamental tools in numerical optimization, particularly in derivative-free optimization. When constructed in suitably chosen affine subspaces, these tools have been shown to be especially effective for high-dimensional derivative-free optimization problems, where full-space model construction is often impractical. In this paper, we analyze the relationships between full-space and subspace formulations of these tools. In particular, we derive explicit conversion formulas between full-space and subspace models, including minimum-norm models, minimum Frobenius norm models, least Frobenius norm updating models, as well as models constructed via generalized simplex gradients and Hessians. We show that the full-space and subspace models coincide on the affine subspace and, in general, along directions in the orthogonal complement. Overall, our results provide a theoretical framework for understanding subspace approximation techniques and offer insight into the design and analysis of derivative-free optimization methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10374 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relationships between full-space and subspace quadratic interpolation models and simplex derivatives Chen, Yiwen Optimization and Control Numerical Analysis Quadratic interpolation models and simplex derivatives are fundamental tools in numerical optimization, particularly in derivative-free optimization. When constructed in suitably chosen affine subspaces, these tools have been shown to be especially effective for high-dimensional derivative-free optimization problems, where full-space model construction is often impractical. In this paper, we analyze the relationships between full-space and subspace formulations of these tools. In particular, we derive explicit conversion formulas between full-space and subspace models, including minimum-norm models, minimum Frobenius norm models, least Frobenius norm updating models, as well as models constructed via generalized simplex gradients and Hessians. We show that the full-space and subspace models coincide on the affine subspace and, in general, along directions in the orthogonal complement. Overall, our results provide a theoretical framework for understanding subspace approximation techniques and offer insight into the design and analysis of derivative-free optimization methods. |
| title | Relationships between full-space and subspace quadratic interpolation models and simplex derivatives |
| topic | Optimization and Control Numerical Analysis |
| url | https://arxiv.org/abs/2602.10374 |