  We present two novel methods for approximating minimizers of the abstract
Rayleigh quotient $\Phi(u)/ \|u\|^p$. Here $\Phi$ is a strictly convex
functional on a Banach space with norm $\|\cdot\|$, and $\Phi$ is assumed to be
positively homogeneous of degree $p\in (1,\infty)$. Minimizers are shown to
satisfy $\partial \Phi(u)- \lambda\mathcal{J}_p(u)\ni 0$ for a certain
$\lambda\in \mathbb{R}$, where $\mathcal{J}_p$ is the subdifferential of
$\frac{1}{p}\|\cdot\|^p$. The first approximation scheme is based on inverse
iteration for square matrices and involves sequences that satisfy $$ \partial
\Phi(u_k)- \mathcal{J}_p(u_{k-1})\ni 0 \quad (k\in \mathbb{N}). $$ The second
method is based on the large time behavior of solutions of the doubly nonlinear
evolution $$ \mathcal{J}_p(\dot v(t))+\partial\Phi(v(t))\ni 0 \quad(a.e.\;t>0)
$$ and more generally $p$-curves of maximal slope for $\Phi$. We show that both
schemes have the remarkable property that the Rayleigh quotient is
nonincreasing along solutions and that properly scaled solutions converge to a
minimizer of $\Phi(u)/ \|u\|^p$. These results are new even for Hilbert spaces
and their primary application is in the approximation of optimal constants and
extremal functions for inequalities in Sobolev spaces.
