Proximal-gradient, FISTA, and fixed-period restarted FISTA

Assume $f:{\mathbb{R}}^n\to \mathbb{R}$ is convex and continuously differentiable with $L$-Lipschitz gradient, $L>0$, and $g:{\mathbb{R}}^n\to (-\mathrm{\infty },+\mathrm{\infty }]$ is proper, closed and convex. For a finite iteration budget $N\ge 1$, initial point $x_0\in {\mathbb{R}}^n$, and exact evaluations of $T_L$ from Proximal-gradient mapping and projected-gradient mapping, the proximal-gradient mode returns $x_k=T_L(x_{k-1}),k=1,\dots ,N\mathrm{.}$

The FISTA mode sets $t_1=1$, $y_1=x_0$, and for $k=1,\dots ,N$ computes $x_k=T_L(y_k),t_{k+1}=\frac{1+\sqrt{1+4t_k^2}}{2},y_{k+1}=x_k+\frac{t_k-1}{t_{k+1}}(x_k-x_{k-1})\mathrm{.}$ It returns $x_N$. A fixed-period restart with period $m\in \mathbb{N}$, $m\ge 1$, runs this FISTA recurrence for at most $m$ updates per block and, after each full block, resets $t_1=1$, $y_1$ to the block's last iterate, and the previous iterate to that same point. A final shorter block stops at the budget. Each call uses the exact proximal point; no inexact call is covered by this algorithm definition. The finite budget makes the output a finite iterate even when no convergence or monotonicity of consecutive accelerated objective values is asserted.