On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere

Given symmetric matrices B,D∈ℝ n×n and a symmetric positive definite matrix W∈ℝ n×n , maximizing the sum of the Rayleigh quotient x ⊤ D x and the generalized Rayleigh quotient $\frac{\mathbf{x}^{\top}B \mathbf{x}}{\vphantom{\mathrm{I}^{\mathrm{I}}}\mathbf{x}^{\top}W\mathbf{x} }$ on the unit sphere not only is of mathematical interest in its own right, but also finds applications in practice. In this paper, we first present a real world application arising from the sparse Fisher discriminant analysis. To tackle this problem, our first effort is to characterize the local and global maxima by investigating the optimality conditions. Our results reveal that finding the global solution is closely related with a special extreme nonlinear eigenvalue problem, and in the special case D=μW (μ>0), the set of the global solutions is essentially an eigenspace corresponding to the largest eigenvalue of a specially-defined matrix. The characterization of the global solution not only sheds some lights on the maximization problem, but motives a starting point strategy to obtain the global maximizer for any monotonically convergent iteration. Our second part then realizes the Riemannian trust-region method of Absil, Baker and Gallivan (Found. Comput. Math. 7:303–330, 2007 ) into a practical algorithm to solve this problem, which enjoys the nice convergence properties: global convergence and local superlinear convergence. Preliminary numerical tests are carried out and empirical evaluation of its performance is reported. Copyright Springer Science+Business Media, LLC 2013