  We focus on spherical spin glasses whose Parisi distribution has support of
the form $[0,q]$. For such models we construct paths from the origin to the
sphere which consistently remain close to the ground-state energy on the sphere
of corresponding radius. The construction uses a greedy strategy, which always
follows a direction corresponding to the most negative eigenvalues of the
Hessian of the Hamiltonian. For finite mixtures $\nu(x)$ it provides an
algorithm of time complexity $O(N^{{\rm deg}(\nu)})$ to find w.h.p. points with
the ground-state energy, up to a small error.
  For the pure spherical models, the same algorithm reaches the energy
$-E_{\infty}$, the conjectural terminal energy for gradient descent. Using the
TAP formula for the free energy, for full-RSB models with support $[0,q]$, we
are able to prove the correct lower bound on the free energy (namely, prove the
lower bound from Parisi's formula), assuming the correctness of the Parisi
formula only in the replica symmetric case.
