Algorithm 1.

FISTA with line search

Initialize x0 and t0 > 0, set v0 := x0, select 0 < r < 1, s > 1

for k = 1, 2, … do

t := s tk−1

repeat

$\theta :=\{\begin{array}{ll}1\hfill & \text{if}k=1\hfill \\ \text{positive}\text{root}\text{of}{t}_{k-1}{\theta }^2=t{\theta }_{k-1}^2\left(1-\theta \right)\hfill & \text{if}k>1\hfill \end{array}$

$y:\left(1-\theta \right)x_{k-1}+\theta v_{k-1}$

$x:{\text{prox}}_{th}\left(y-t\nabla g\left(y\right)\right)$

break if $g\left(x\right)\le g\left(y\right)+\langle \nabla g\left(y\right),x-y\rangle +\frac{1}{2t}{\left(x-y\right]}_2^2$

t := rt

tk := t

θk := θ

xk := x

$v_k:=x_{k-1}+\frac{1}{{\theta }_k}\left(x_k-x_{k-1}\right)$

end for