  A Nehari manifold optimization method (NMOM) is introduced for finding
1-saddles, i.e., saddle points with the Morse index equal to one, of a generic
nonlinear functional in Hilbert spaces. Actually, it is based on the
variational characterization that 1-saddles of the generic functional are local
minimizers of the same functional restricted on the associated Nehari manifold.
The framework contains two important ingredients: one is the retraction mapping
to make the iteration points always lie on the Nehari manifold; the other is
the tangential search direction to decrease the generic functional with
suitable step-size search rules. Particularly, the global convergence is
rigorously established by virtue of some crucial analysis techniques (including
a weak convergence method) overcoming difficulties in the infinite-dimensional
setting. In practice, combining with an easy-to-implement Nehari retraction and
the negative Riemannian gradient direction, the NMOM is successfully applied to
compute the unstable ground-state solutions of a class of typical semilinear
elliptic PDEs such as H\'enon equation and the stationary nonlinear
Schr\"odinger equation. In particular, the symmetry-breaking phenomenon of the
ground states of H\'enon equation is explored numerically in 1D and 2D with
interesting numerical findings on the critical value of symmetry-breaking
reported.
