1. Prove, that $x_0$ - is the minimum point of a convex function $f(x)$ if and only if $0 \in \partial f(x_0)$
2. Find $\partial f(x)$, if $f(x) = \text{ReLU}(x) = \max \{0, x\}$
3. Find $\partial f(x)$, if $f(x) = \|x\|_p$ при $p = 1,2, \infty$
4. Find $\partial f(x)$, if $f(x) = \|Ax - b\|_1^2$
5. Find $\partial f(x)$, if $f(x) = e^{\|x\|}$
6. Describe the connection between subgradient of a scalar function $f: \mathbb{R} \to \mathbb{R}$ and global linear lower bound, which support (tangent) the graph of the function at a point.
9. If the function is convex on $S$, whether $\partial f(x) \neq \emptyset \;\;\; \forall x \in S$ always holds or not?
10. Find $\partial f(x)$, if $f(x) = x^3$
11. Find $f(x) = \lambda_{max} (A(x)) = \sup\limits_{\|y\|_2 = 1} y^T A(x)y$, где $A(x) = A_0 + x_1A_1 + \ldots + x_nA_n$, all the matrices $A_i \in \mathbb{S}^k$ are symmetric and defined.
12. Find subdifferential of a function $f(x,y) = x^2 + xy + y^2 + 3\vert x + y − 2\vert$ at points $(1,0)$ and $(1,1)$.