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

Convergence Rate Analysis of Accelerated Forward-Backward Algorithm with Generalized Nesterov Momentum Scheme