  Intertwining analysis, algebra, numerical analysis and optimization,
computing conjugate co-gradients of real-valued quotients gives rise to
eigenvalue problems. In the linear Hermitian case, by inspecting optimal
quotients in terms of taking the conjugate co-gradient for their critical
points, a generalized folded spectrum eigenvalue problem arises. Replacing the
Euclidean norm in optimal quotients with the $p$-norm, a matrix version of the
so-called $p$-Laplacian eigenvalue problem arises. Such nonlinear eigenvalue
problems seem to be naturally classified as being a special case of homogeneous
problems. Being a quite general class, tools are developed for recovering
whether a given homogeneous eigenvalue problem is a gradient eigenvalue
problem. It turns out to be a delicate issue to come up with a valid quotient.
A notion of nonlinear Hermitian eigenvalue problem is suggested. Cauchy-Schwarz
quotients are introduced.
