Algorithm 3: FISTA with Backtracking Step for solving (17)

Require: $A,\xi ,w\lambda >0$ Ensure: solution $\tilde{\beta }$ 1: Step 0. Select $L^0>0,\eta >1,{\tilde{\beta }}^0\in {\mathbb{R}}^{2p}$ Let ${\xi }^1={\tilde{\beta }}^0,t^1=1$ 2: Step $k(k\ge 1)$. 3: Determine the smallest non-negative integer $i^k$ which make $\overline{L}={\eta }^{i^k}{L}^{k-1}$ satisfy 4: $$F\left({\mathsf{\Theta }}_{\overline{L}}\left({\xi }^k\right)\right)\le Q_{\overline{L}}\left({\mathsf{\Theta }}_{\overline{L}}\left({\xi }^k\right),{\xi }^k\right).$$ 5: Let $L^k={\eta }^{i^k}{L}^{k-1}$ according to (19), calculate: 6: ${\tilde{\beta }}^k={\mathsf{\Theta }}_{L^k}\left({\xi }^k\right)$ 7: $t^{k+1}=\frac{1}{2}\left[1+\sqrt{1+4{\left(t^k\right)}^2}\right]$ 8: ${\xi }^{k+1}={\tilde{\beta }}^k+\frac{t^k-1}{t^{k+1}}\left({\tilde{\beta }}^k-{\tilde{\beta }}^{k-1}\right)$ 9: Output $\tilde{\beta }:={\tilde{\beta }}^k$.