  Nesterov's accelerated forward-backward algorithm (AFBA) is an efficient
algorithm for solving a class of two-term convex optimization models consisting
of a differentiable function with a Lipschitz continuous gradient plus a
nondifferentiable function with a closed form of its proximity operator. It has
been shown that the iterative sequence generated by AFBA with a modified
Nesterov's momentum scheme converges to a minimizer of the objective function
with an $o\left(\frac{1}{k^2}\right)$ convergence rate in terms of the function
value (FV-convergence rate) and an $o\left(\frac{1}{k}\right)$ convergence rate
in terms of the distance between consecutive iterates (DCI-convergence rate).
In this paper, we propose a more general momentum scheme with an introduced
power parameter $\omega\in(0,1]$ and show that AFBA with the proposed momentum
scheme converges to a minimizer of the objective function with an
$o\left(\frac{1}{k^{2\omega}}\right)$ FV-convergence rate and an
$o\left(\frac{1}{k^{\omega}}\right)$ DCI-convergence rate. The generality of
the proposed momentum scheme provides us a variety of parameter selections for
different scenarios, which makes the resulting algorithm more flexible to
achieve better performance. We then employ AFBA with the proposed momentum
scheme to solve the smoothed hinge loss $\ell_1$-support vector machine model.
Numerical results demonstrate that the proposed generalized momentum scheme
outperforms two existing momentum schemes.
