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

• A self-consistent-field method for optimizing the sum of the Rayleigh quotients is proposed.
• Local convergence and global convergence are established under some conditions.
• Quadratic convergence is proved for some situations.
• Numerical testing shows efficiency, in comparison with other optimization-based methods.

In this paper, we consider efficient methods for maximizing x⊤Bxx⊤Wx+x⊤Dx over the unit sphere, where B,DB,D are symmetric matrices, and WW 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.