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We propose an accelerated forward-backward method with fast convergence ratefor finding a minimizer of a decomposable nonsmooth convex function over aclosed convex set, and name it smoothing accelerated proximal gradient (SAPG)algorithm. The proposed algorithm combines the smoothing method with theproximal gradient algorithm with extrapolation $\frac{k-1}{k+\alpha-1}$ and$\alpha>3$. The updating rule of smoothing parameter $\mu_k$ is a smart schemeand guarantees the global convergence rate of $o(\ln^{\sigma}k/k)$ with$\sigma\in(\frac{1}{2},1]$ on the objective function values. Moreover, we provethat the sequence is convergent to an optimal solution of the problem.Furthermore, we introduce an error term in the SAPG algorithm to get theinexact smoothing accelerated proximal gradient algorithm. And we obtain thesame convergence results as the SAPG algorithm under the summability conditionon the errors. Finally, numerical experiments show the effectiveness andefficiency of the proposed algorithm.

Accelerated forward-backward method with fast convergence rate for nonsmooth convex optimization beyond differentiability