  We propose an adaptive step size with an energy approach for a suitable class
of preconditioned gradient descent methods. We focus on settings where the
preconditioning is applied to address the constraints in optimization problems,
such as the Hessian-Riemannian and natural gradient descent methods. More
specifically, we incorporate these preconditioned gradient descent algorithms
in the recently introduced Adaptive Energy Gradient Descent (AEGD) framework.
In particular, we discuss theoretical results on the unconditional
energy-stability and convergence rates across three classes of objective
functions. Furthermore, our numerical results demonstrate excellent performance
of the proposed method on several test bed optimization problems.
