BUP-TR: Bayesian Underdetermined Projection Trust-Region Methods for Derivative-Free Optimization
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866916064406798336 |
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| author | Hu, Wei Xie, Pengcheng Yuan, Ya-Xiang Zhang, Li |
| author_facet | Hu, Wei Xie, Pengcheng Yuan, Ya-Xiang Zhang, Li |
| contents | Underdetermined quadratic interpolation is a standard model-construction tool in model-based derivative-free trust-region methods: it keeps the number of function evaluations manageable, but leaves many quadratic models consistent with the sampled values. Classical solvers remove this ambiguity by prescribing a norm or a model-change measure, such as the least-Frobenius Hessian update in Powell-type methods. We propose BUP-TR (Bayesian Underdetermined Projection Trust-Region), which replaces this prescribed completion rule by a prior-regularized one. At each iteration, the interpolation equations are imposed as exact constraints, and the model is chosen as the maximum-a-posteriori (MAP) element of the affine family of interpolating quadratics, equivalently as the projection of a prior model onto this family in the norm defined by the precision matrix. The precision matrix used in this completion also defines a geometry test for the interpolation set, leading to MAP-poisedness and a corresponding repair procedure. Under standard smoothness, geometry, and prior-accuracy assumptions, the hard-MAP models are fully linear, and the resulting trust-region method attains global first-order convergence with O(epsilon^{-2}) evaluation complexity, including repair evaluations. A NEWUOA-style implementation, BUP-NEWUOA, improves performance under fixed evaluation budgets over NEWUOA and other standard derivative-free solvers on the test set considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30841 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | BUP-TR: Bayesian Underdetermined Projection Trust-Region Methods for Derivative-Free Optimization Hu, Wei Xie, Pengcheng Yuan, Ya-Xiang Zhang, Li Optimization and Control 90C56, 65K05, 90C30 Underdetermined quadratic interpolation is a standard model-construction tool in model-based derivative-free trust-region methods: it keeps the number of function evaluations manageable, but leaves many quadratic models consistent with the sampled values. Classical solvers remove this ambiguity by prescribing a norm or a model-change measure, such as the least-Frobenius Hessian update in Powell-type methods. We propose BUP-TR (Bayesian Underdetermined Projection Trust-Region), which replaces this prescribed completion rule by a prior-regularized one. At each iteration, the interpolation equations are imposed as exact constraints, and the model is chosen as the maximum-a-posteriori (MAP) element of the affine family of interpolating quadratics, equivalently as the projection of a prior model onto this family in the norm defined by the precision matrix. The precision matrix used in this completion also defines a geometry test for the interpolation set, leading to MAP-poisedness and a corresponding repair procedure. Under standard smoothness, geometry, and prior-accuracy assumptions, the hard-MAP models are fully linear, and the resulting trust-region method attains global first-order convergence with O(epsilon^{-2}) evaluation complexity, including repair evaluations. A NEWUOA-style implementation, BUP-NEWUOA, improves performance under fixed evaluation budgets over NEWUOA and other standard derivative-free solvers on the test set considered. |
| title | BUP-TR: Bayesian Underdetermined Projection Trust-Region Methods for Derivative-Free Optimization |
| topic | Optimization and Control 90C56, 65K05, 90C30 |
| url | https://arxiv.org/abs/2605.30841 |