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This paper concerns a class of monotone eigenvalue problems with eigenvectornonlinearities (mNEPv). The mNEPv is encountered in applications such as thecomputation of joint numerical radius of matrices, best rank-one approximationof third-order partial symmetric tensors, and distance to singularity fordissipative Hamiltonian differential-algebraic equations. We first present avariational characterization of the mNEPv. Based on the variationalcharacterization, we provide a geometric interpretation of theself-consistent-field (SCF) iterations for solving the mNEPv, prove the globalconvergence of the SCF, and devise an accelerated SCF. Numerical examples froma variety of applications demonstrate the theoretical properties andcomputational efficiency of the SCF and its acceleration.

Variational Characterization of Monotone Nonlinear Eigenvector Problems and Geometry of Self-Consistent-Field Iteration