Deflation Techniques for an Implicitly Restarted Arnoldi Iteration
SIAM Journal on Matrix Analysis and Applications
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
Matrix algorithms
Trust-Region Methods on Riemannian Manifolds
Foundations of Computational Mathematics
A Trust Region Direct Constrained Minimization Algorithm for the Kohn-Sham Equation
SIAM Journal on Scientific Computing
Optimization Algorithms on Matrix Manifolds
Optimization Algorithms on Matrix Manifolds
SIAM Journal on Matrix Analysis and Applications
Fast Algorithms for the Generalized Foley-Sammon Discriminant Analysis
SIAM Journal on Matrix Analysis and Applications
Computational Optimization and Applications
Hi-index | 7.29 |
In this paper, we consider efficient methods for maximizing x^@?Bxx^@?Wx+x^@?Dx over the unit sphere, where B,D are symmetric matrices, and W is symmetric and positive definite. This problem can arise in the downlink of a multi-user MIMO system and in the sparse Fisher discriminant analysis in pattern recognition. It is already known that the problem of finding a global maximizer is closely associated with a nonlinear extreme eigenvalue problem. Rather than resorting to some general optimization methods, we introduce a self-consistent-field-like (SCF-like) iteration for directly solving the resulting nonlinear eigenvalue problem. The SCF iteration is widely used for solving the nonlinear eigenvalue problems arising in electronic structure calculations. One attractive feature of the SCF for our problem is that once it converges, the limit point not only satisfies the necessary local optimality conditions automatically, but also, and most importantly, satisfies a global optimality condition, which generally is not achievable in some optimization-based methods. The global convergence and local quadratic convergence rate are proved for certain situations. For the general case, we then discuss a trust-region SCF iteration for stabilizing the SCF iteration, which is of good global convergence behavior. Our preliminary numerical experiments show that these algorithms are more efficient than some optimization-based methods.