# Linear Constrained Rayleigh Quotient Optimization: Theory and Algorithms

@article{Zhou2019LinearCR,
title={Linear Constrained Rayleigh Quotient Optimization: Theory and Algorithms},
author={Yunshen Zhou and Zhaojun Bai and Ren-Cang Li},
journal={ArXiv},
year={2019},
volume={abs/1911.02770}
}
• Published 7 November 2019
• Computer Science
• ArXiv
We consider the following constrained Rayleigh quotient optimization problem (CRQopt) $$\min_{x\in \mathbb{R}^n} x^{T}Ax\,\,\mbox{subject to}\,\, x^{T}x=1\,\mbox{and}\,C^{T}x=b,$$ where $A$ is an $n\times n$ real symmetric matrix and $C$ is an $n\times m$ real matrix. Usually, $m\ll n$. The problem is also known as the constrained eigenvalue problem in the literature because it becomes an eigenvalue problem if the linear constraint $C^{T}x=b$ is removed. We start by equivalently transforming…
2 Citations

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## References

SHOWING 1-10 OF 43 REFERENCES

• Mathematics
• 1988
In this paper we consider the following mathematical and computational problem. Given the quantities A: (n + m)-by-(n + m) matrix, symmetric, n > 0 N: (n + m)-by-m matrix with full rank
• Computer Science
AISTATS
• 2017
The proposed theoretical and computational solutions can be applied to eigenproblems of positive semi-definite pencils arising in other machine learning algorithms, such as generalized linear discriminant analysis in dimension reduction and multisurface classification via eigenvectors.
• Computer Science
Math. Program.
• 1997
A dual simplex type method is studied that solves (TRS) as a parametric eigenvalue problem and the essential cost of the algorithm is the matrix-vector multiplication and, thus, sparsity can be exploited.
• Computer Science
Data Mining and Knowledge Discovery
• 2012
This paper presents a more natural and principled formulation of constrained spectral clustering, which explicitly encodes the constraints as part of a constrained optimization problem, and demonstrates an innovative use of encoding large number of constraints: transfer learning via constraints.
• Computer Science
• 1991
X is the vector space which acts in the n-dimensional (complex) vector space R.1.1 and is related to Varepsilon by the following inequality.
A closed formula for the CG residuals for all 1 ≤ k < N- 1 on Meinardus' example is obtained, and in particular it implies that the bound is always within a factor of -√2 of the actual residuals.
• Computer Science
SIAM J. Optim.
• 2017
A priori upper bounds for the convergence to both the optimal objective value as well as the optimal solution are developed and it is argued that these bounds can be efficiently estimated numerically and serve as stopping criteria for better numerical performance.
• Computer Science
IEEE Transactions on Pattern Analysis and Machine Intelligence
• 2004
It is demonstrated not only that it is possible to integrate both image structures and priors in a single grouping process, but also that objects can be segregated from the background without specific object knowledge.