  In this paper we consider a nonconvex unconstrained optimization problem
minimizing a twice differentiable objective function with H\"older continuous
Hessian. Specifically, we first propose a Newton-conjugate gradient (Newton-CG)
method for finding an approximate first-order stationary point (FOSP) of this
problem, assuming the associated the H\"older parameters are explicitly known.
Then we develop a parameter-free Newton-CG method without requiring any prior
knowledge of these parameters. To the best of our knowledge, this method is the
first parameter-free second-order method achieving the best-known iteration and
operation complexity for finding an approximate FOSP of this problem.
Furthermore, we propose a Newton-CG method for finding an approximate
second-order stationary point (SOSP) of the considered problem with high
probability and establish its iteration and operation complexity. Finally, we
present preliminary numerical results to demonstrate the superior practical
performance of our parameter-free Newton-CG method over a well-known
regularized Newton method.
