  In this paper, we introduce a Homogeneous Second-Order Descent Method (HSODM)
using the homogenized quadratic approximation to the original function. The
merit of homogenization is that only the leftmost eigenvector of a
gradient-Hessian integrated matrix is computed at each iteration. Therefore,
the algorithm is a single-loop method that does not need to switch to other
sophisticated algorithms and is easy to implement. We show that HSODM has a
global convergence rate of $O(\epsilon^{-3/2})$ to find an
$\epsilon$-approximate second-order stationary point, and has a local quadratic
convergence rate under the standard assumptions. The numerical results
demonstrate the advantage of the proposed method over other second-order
methods.
