Variational characterization of monotone nonlinear eigenvector problems and geometry of self-consistent field iteration

This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial-symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples demonstrate theoretical properties and computational efficiency of the SCF and its acceleration.