Eigenvector-dependent Nonlinear Eigenvalue Problems (NEPv) have long played critical roles in computational physics and chemistry, and they are becoming increasingly important in data science applications. In this talk, we will focus on a special class of NEPv where the coefficient matrices have an affine-linear structure. These NEPv often arise from optimization problems involving Rayleigh quotients, including the trace-ratio optimization for dimension reduction and robust Rayleigh quotient optimization for handling data uncertainties. Our major goal is to establish variational characterizations for affine-linear NEPv and provide geometric interpretation of the self-consistent field (SCF) iteration to solve the NEPv. The geometric interpretation can help us to understand when SCF globally converges and why it might not in some cases. We will also discuss improvements in SCF, such as local acceleration techniques and global verification methods. Numerical experiments will demonstrate the effectiveness of our approaches.

9.27.pdf