1. We say that the function belongs to the class $f \in C^{k,p}_L (Q)$ if it is $k$ times continuously differentiable on $Q$, and the $p$ derivative has a Lipschitz constant $L$.

The most commonly used $C_L^{1,1}, C_L^{2,2}$ for $\mathbb{R}^n$. Notice that:

• If $q \geq k$, then $C_L^{q,p} \subseteq C_L^{k,p}$. The higher is the order of the derivative, the stronger is the limitation (fewer functions belong to the class).

Prove that the function belongs to the class $C_L^{2,1}. \subseteq C_L^{1,1}$ if and only if $\forall x \in \mathbb{R}^n$:

Prove that the last condition can be rewritten in the form without loss of generality:

2. Show that for gradient descent with the following stepsize selection strategies:

• constant step $h_k = \dfrac{1}{L}$
• Dropping sequence $h_k = \dfrac{\alpha_k}{L}, \quad \alpha_k \to 0$.

you can get the estimation of the function decrease at the iteration of the view:

$\omega > 0$ - some constant, $L$ - Lipschitz constant of the function gradient.