grandes-ecoles 2025 Q8

grandes-ecoles · France · x-ens-maths__psi Applied differentiation Inequality proof via function study
We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, $f$ is $\alpha$-convex with $\alpha > 0$, and $x_*$ denotes a minimizer of $f$. We suppose $0 < \tau < 2/L$. Show that $\left|x_n - x_*\right| \leq \rho^n \left|x_0 - x_*\right|$, where $\rho$ is a constant that we will specify, and such that $0 \leq \rho < 1$.
We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, $f$ is $\alpha$-convex with $\alpha > 0$, and $x_*$ denotes a minimizer of $f$.\\
We suppose $0 < \tau < 2/L$. Show that $\left|x_n - x_*\right| \leq \rho^n \left|x_0 - x_*\right|$, where $\rho$ is a constant that we will specify, and such that $0 \leq \rho < 1$.