A quadratic Grassmann manifold optimization problem arising from quantum embedding methods
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866914404606410752 |
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| author | Ayral, Thomas Cancès, Eric Faulstich, Fabian M. Lin, Lin Negre, Alicia |
| author_facet | Ayral, Thomas Cancès, Eric Faulstich, Fabian M. Lin, Lin Negre, Alicia |
| contents | This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function $J(P) = \text{Tr}(BP)- \frac{1}{2} \text{Tr}(A P A P)$, where $A,B \in \mathbb R^{M \times M}_{\rm sym}$, with $A \succeq 0$, over the Grassmann manifold ${\rm Gr}(m,\mathbb R^M)$. While this problem is non-convex and typically admits non-global local minima - posing challenges for Riemannian optimization and self-consistent field (SCF) algorithms - we identify cases where the global minimizer can be obtained by solving an auxiliary convex problem. When this approach is not directly applicable, the solution to the auxiliary problem still serves as an effective initialization for Riemannian optimization methods and SCF algorithms, significantly improving their performance. This work is motivated by applications in quantum embedding methods, particularly in the construction of bath orbitals, where such optimization problems naturally arise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17080 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A quadratic Grassmann manifold optimization problem arising from quantum embedding methods Ayral, Thomas Cancès, Eric Faulstich, Fabian M. Lin, Lin Negre, Alicia Optimization and Control Quantum Physics This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function $J(P) = \text{Tr}(BP)- \frac{1}{2} \text{Tr}(A P A P)$, where $A,B \in \mathbb R^{M \times M}_{\rm sym}$, with $A \succeq 0$, over the Grassmann manifold ${\rm Gr}(m,\mathbb R^M)$. While this problem is non-convex and typically admits non-global local minima - posing challenges for Riemannian optimization and self-consistent field (SCF) algorithms - we identify cases where the global minimizer can be obtained by solving an auxiliary convex problem. When this approach is not directly applicable, the solution to the auxiliary problem still serves as an effective initialization for Riemannian optimization methods and SCF algorithms, significantly improving their performance. This work is motivated by applications in quantum embedding methods, particularly in the construction of bath orbitals, where such optimization problems naturally arise. |
| title | A quadratic Grassmann manifold optimization problem arising from quantum embedding methods |
| topic | Optimization and Control Quantum Physics |
| url | https://arxiv.org/abs/2603.17080 |