# 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} }

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…

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## References

SHOWING 1-10 OF 43 REFERENCES

### A Constrained Eigenvalue Problem

- 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…

### LSMR: An Iterative Algorithm for Sparse Least-Squares Problems

- Computer ScienceSIAM J. Sci. Comput.
- 2011

### Robust and Efficient Computation of Eigenvectors in a Generalized Spectral Method for Constrained Clustering

- Computer ScienceAISTATS
- 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.

### A semidefinite framework for trust region subproblems with applications to large scale minimization

- Computer ScienceMath. 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.

### On constrained spectral clustering and its applications

- Computer ScienceData 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.

### Matrix Perturbation Theory

- 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.

### On Meinardus' examples for the conjugate gradient method

- MathematicsMath. Comput.
- 2008

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.

### On the Generalized Lanczos Trust-Region Method

- Computer ScienceSIAM 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.

### Segmentation given partial grouping constraints

- Computer ScienceIEEE 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.