  We present a new accelerated gradient-based method for solving smooth
unconstrained optimization problems. The goal is to embed a heavy-ball type of
momentum into the Fast Gradient Method (FGM). For this purpose, we devise a
generalization of the estimating sequences, which allows for encoding any form
of information about the cost function that can aid in further accelerating the
minimization process. In the black box framework, we propose a construction for
the generalized estimating sequences, which is obtained by exploiting the
history of the previously constructed estimating functions. From the viewpoint
of efficiency estimates, we prove that the lower bound on the number of
iterations for the proposed method is $\mathcal{O}
\left(\sqrt{\frac{\kappa}{2}}\right)$. Our theoretical results are further
corroborated by extensive numerical experiments on various types of
optimization problems, often dealt within signal processing. Both synthetic and
real-world datasets are utilized to demonstrate the efficiency of our proposed
method in terms of decreasing the distance to the optimal solution, as well as
in terms of decreasing the norm of the gradient.
