We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ How does the sequence behave when $\|x_0\| < 1$?
We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by
$$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$
How does the sequence behave when $\|x_0\| < 1$?