  This paper proposes and analyzes an iterative minimization formulation for
search- ing index-1 saddle points of an energy function. This formulation
differs from other eigenvector-following methods by constructing a new
objective function near the guess at each iteration step. This leads to a
quadratic convergence rate, in comparison to the linear case of the gentlest
ascent dynamics (E and Zhou, nonlinearity, vol 24, p1831, 2011) and many other
existing methods. We also propose the generalization of the new methodology for
saddle points of higher index and for constrained energy functions on manifold.
