A quadratic Grassmann manifold optimization problem arising from quantum embedding methods

• URL: http://arxiv.org/abs/2603.17080v1
• Date: Tue, 17 Mar 2026 19:07:12 GMT
• Title: A quadratic Grassmann manifold optimization problem arising from quantum embedding methods
• Abstract summary: This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function $J(P) = textTr(BP)- frac12 textTr(A P A P) - over the Grassmann manifold $A,B in mathbb RM times_rm sym$, over the Grassmann manifold $rm Gr(m,mathbb RM)$.
• Score: 1.1954761974478398
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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