On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

Zhang Leihong(longzlhgmail.com)

Abstract: In this paper, we consider efficient methods for maximizing $x^TDx+x^TBx/x^TWx$ 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 finding a global maximizer is closely associated with solving a nonlinear extreme eigenvalue problem. Rather than resorting some general optimization methods, we introduce a self-consistent-field-like (SCF) iteration for directly solving the resulting nonlinear eigenvalue problem. The SCF iteration is a widely used method for solving the nonlinear eigenvalue problems arising in electronic structure calculations. One attractive feature of SCF for our problem is that once it converges, the limit point not only automatically satisfies the necessary local optimality conditions, but also, and most importantly, a global optimality condition, which in general could not be achieved by optimization-based methods. The global convergence and local quadratic convergence rate are proved in some certain situation. For the general case, we then discuss a trust-region SCF (TRSCF) iteration to stabilize 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.

Keywords: Rayleigh quotient, the nonlinear eigenvalue problem, the self-consistent field iteration, trust region

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

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Entry Submitted: 04/19/2012
Entry Accepted: 04/19/2012
Entry Last Modified: 04/19/2012

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