Optimal Spectral Design with Prior Information
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910264300929024 |
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| author | Kleywegt, Anton J. Milz, Johannes Singh, Mohit Xie, Weijun |
| author_facet | Kleywegt, Anton J. Milz, Johannes Singh, Mohit Xie, Weijun |
| contents | We study a class of spectral design problems in which a prior positive semidefinite information matrix is updated by a sum of rank-one matrices constructed from chosen design vectors subject to a bound on their Euclidean norm. The objective of a spectral design problem is any symmetric convex function of the eigenvalues of the updated information matrix. This framework unifies classical optimal experimental design criteria, including A-, D-, and E-optimality. It also arises in model-based derivative-free optimization, where sampling directions determine the conditioning and accuracy of regression models. Although the objective is symmetric and convex in the eigenvalues, the optimization problem with design vectors/matrix as decision variables is nonconvex, and optimal solutions of their convex relaxations may not be feasible for the spectral design problem. We use tight eigenvalue relaxations to obtain a convex reformulation, and we apply the Schur--Horn theorem to construct a simple polynomial-time algorithm for solving the spectral design problem. We illustrate the optimal spectral designs computed by our algorithm. Moreover, a small set of numerical experiments shows the potential of spectral designs for derivative-free optimization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27837 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal Spectral Design with Prior Information Kleywegt, Anton J. Milz, Johannes Singh, Mohit Xie, Weijun Optimization and Control We study a class of spectral design problems in which a prior positive semidefinite information matrix is updated by a sum of rank-one matrices constructed from chosen design vectors subject to a bound on their Euclidean norm. The objective of a spectral design problem is any symmetric convex function of the eigenvalues of the updated information matrix. This framework unifies classical optimal experimental design criteria, including A-, D-, and E-optimality. It also arises in model-based derivative-free optimization, where sampling directions determine the conditioning and accuracy of regression models. Although the objective is symmetric and convex in the eigenvalues, the optimization problem with design vectors/matrix as decision variables is nonconvex, and optimal solutions of their convex relaxations may not be feasible for the spectral design problem. We use tight eigenvalue relaxations to obtain a convex reformulation, and we apply the Schur--Horn theorem to construct a simple polynomial-time algorithm for solving the spectral design problem. We illustrate the optimal spectral designs computed by our algorithm. Moreover, a small set of numerical experiments shows the potential of spectral designs for derivative-free optimization. |
| title | Optimal Spectral Design with Prior Information |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2605.27837 |