  It is well known that finding a global optimum is extremely challenging for
nonconvex optimization. There are some recent efforts
\cite{anandkumar2016efficient, cartis2018second, cartis2020sharp, chen2019high}
regarding the optimization methods for computing higher-order critical points,
which can exclude the so-called degenerate saddle points and reach a solution
with better quality. Desipte theoretical development in
\cite{anandkumar2016efficient, cartis2018second, cartis2020sharp,
chen2019high}, the corresponding numerical experiments are missing. In this
paper, we propose an implementable higher-order method, named adaptive high
order method (AHOM), that aims to find the third-order critical points. This is
achieved by solving an ``easier'' subproblem and incorporating the adaptive
strategy of parameter-tuning in each iteration of the algorithm. The iteration
complexity of the proposed method is established. Some preliminary numerical
results are provided to show AHOM is able to escape the degenerate saddle
points, where the second-order method could possibly get stuck.
