We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$, where $p_f(x_0)$ is the unique minimizer of $F_{x_0}(x) := \frac{1}{2}|x - x_0|^2 + \tau f(x)$. In this question only, we set $f(x) := |x|$ for all $x \in \mathbb{R}$. a) Show that $$p_f(x) = \begin{cases} x - \tau & \text{if } x \geq \tau \\ x + \tau & \text{if } x \leq -\tau \\ 0 & \text{otherwise} \end{cases}$$ b) Deduce that $x_n \rightarrow 0$ as $n \rightarrow \infty$, for all $x_0 \in \mathbb{R}$ and $\tau > 0$.
We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$, where $p_f(x_0)$ is the unique minimizer of $F_{x_0}(x) := \frac{1}{2}|x - x_0|^2 + \tau f(x)$.\\
In this question only, we set $f(x) := |x|$ for all $x \in \mathbb{R}$.\\
a) Show that
$$p_f(x) = \begin{cases} x - \tau & \text{if } x \geq \tau \\ x + \tau & \text{if } x \leq -\tau \\ 0 & \text{otherwise} \end{cases}$$
b) Deduce that $x_n \rightarrow 0$ as $n \rightarrow \infty$, for all $x_0 \in \mathbb{R}$ and $\tau > 0$.