  We propose to use the {\L}ojasiewicz inequality as a general tool for
analyzing the convergence rate of gradient descent on a Hilbert manifold,
without resorting to the continuous gradient flow. Using this tool, we show
that a Sobolev gradient descent method with adaptive inner product converges
exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem.
This method can be extended to a class of general high-degree optimizations or
nonlinear eigenproblems under certain conditions. We demonstrate this
generalization by several examples, in particular a nonlinear Schr\"odinger
eigenproblem with an extra high-order interaction term. Numerical experiments
are presented for these problems.
