Subgradient and subdifferential

Subgradient and subdifferential

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) = \text{Leaky ReLU}(x) = \begin{cases} x & \text{if } x > 0, \\ 0.01x & \text{otherwise}. \end{cases}
4. Find \partial f(x), if f(x) = \|x\|_p при p = 1,2, \infty
5. Find \partial f(x), if f(x) = \|Ax - b\|_1^2
6. Find \partial f(x), if f(x) = e^{\|x\|}. Try do the task for an arbitrary norm. At least, try \|\cdot\| = \|\cdot\|_{\{2,1,\infty\}}.
7. 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.
8. What can we say about subdifferential of a convex function in those points, where the function is differentiable?
9. Does the subgradient coincide with the gradient of a function if the function is differentiable? Under which condition it holds?
10. If the function is convex on S, whether \partial f(x) \neq \emptyset \;\;\; \forall x \in S always holds or not?
11. Find \partial f(x), if f(x) = x^3
12. 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.
13. 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).
14. Find subdifferential of a function f(x) = \sin x on the set X = [0, \frac32 \pi].
15. Find subdifferential of a function f(x) = \vert c^{\top}x\vert, \; x \in \mathbb{R}^n.
16. Find subdifferential of a function f(x) = \|x\|_1, \; x \in \mathbb{R}^n.
17. Suppose, that if f(x) = \|x\|_\infty. Prove that \partial f(0) = \textbf{conv}\{\pm e_1, \ldots , \pm e_n\}, where e_i is i-th canonical basis vector (column of identity matrix).