Abstract:We consider the optimization problem with a generally quadratic matrix constraint of the form $X^TAX = J$, where $A$ is a given nonsingular, symmetric $n\times n$ matrix and $J$ is a given $k\times k$ symmetric matrix, with $k\leq n$, satisfying $J^2 = I_k$. Since the feasible set constitutes a differentiable manifold, called the indefinite Stiefel manifold, we approach this problem within the framework of Riemannian optimization. Namely, we first equip the manifold with a Riemannian metric and construct the associated geometric structure, then propose a retraction based on the Cayley transform, and finally suggest a Riemannian gradient descent method using the attained materials, whose global convergence is guaranteed. Our results not only cover the known cases, the orthogonal and generalized Stiefel manifolds, but also provide a Riemannian optimization solution for other constrained problems which has not been investigated. As applications, we consider, via trace minimization, several eigenvalue problems of symmetric positive definite matrix pencils, including the linear response eigenvalue problem, and a matrix least square problem, a general framework for the Procrustes problem and constrained matrix equations. The presented numerical results justify the theoretical findings.
| Comments: | 30 pages, 20 figures |
| Subjects: | Optimization and Control (math.OC) |
| MSC classes: | 65K05, 70G45, 90C48 |
| Cite as: | arXiv:2410.22068 [math.OC] |
| (or arXiv:2410.22068v3 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2410.22068 arXiv-issued DOI via DataCite |
|
| Journal reference: | Numerical Linear Algebra with Applications 33 (2), e70077, 2026 |
| Related DOI: | https://doi.org/10.1002/nla.70077
DOI(s) linking to related resources |
From: Nguyen Thanh Son [view email]
[v1]
Tue, 29 Oct 2024 14:27:52 UTC (69 KB)
[v2]
Tue, 19 Aug 2025 07:11:31 UTC (85 KB)
[v3]
Sat, 23 May 2026 07:42:14 UTC (219 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。