  This paper investigates a novel gradient algorithm, AGEM, using both energy
and momentum, for addressing general non-convex optimization problems. The
solution properties of the AGEM algorithm, including aspects such as uniformly
boundedness and convergence to critical points, are examined. The dynamic
behavior is studied through a comprehensive analysis of a high-resolution ODE
system. This ODE system, being nonlinear, is derived by taking the limit of the
discrete scheme while preserving the momentum effect through a rescaling of the
momentum parameter. The paper emphasizes the global well-posedness of the ODE
system and the time-asymptotic convergence of solution trajectories.
Furthermore, we establish a linear convergence rate for objective functions
that adhere to the Polyak-{\L}ojasiewicz condition.
