Conjugate functions

Conjugate functions

1. Find f^*(y), if f(x) = ax + b
2. Find f^*(y), if f(x) = -\log x, \;\; x\in \mathbb{R}_{++}
3. Find f^*(y), if f(x) = e^x
4. Find f^*(y), if f(x) = x \log x, x \neq 0, \;\;\; f(0) = 0, \;\;\; x \in \mathbb{R}_+
5. Find f^*(y), if f(x) =\frac{1}{2} x^T A x, \;\;\; A \in \mathbb{S}^n_{++}
6. Find f^*(y), if f(x) =\max\limits_{i} x_i, \;\;\; x \in \mathbb{R}^n
7. Find f^*(y), if f(x) = -\dfrac{1}{x}, \;\; x\in \mathbb{R}_{++}
8. Find f^*(y), if f(x) = -0,5 - \log x, \;\; x>0
9. Find f^*(y), if f(x) = \log \left( \sum\limits_{i=1}^n e^{x_i} \right)
10. Find f^*(y), if f(x) = - (a^2 - x^2)^{1/2}, \;\;\; \vert x\vert \le a, \;\;\; a>0
11. Find f^*(Y), if f(X) = - \ln \det X, X \in \mathbb{S}^n_{++}
12. Find f^*(y), if f(x) = \|x\|
13. Find f^*(y), if f(x) = \dfrac{1}{2}\|x\|^2
14. Name any 3 non-trivial facts about conjugate function.
15. Find conjugate function to the f(x) = \dfrac{1}{x}, \;\; x \in \mathbb{R}_{++}
16. Find conjugate function to the f(x) = x^p, \;\; x \in \mathbb{R}_{++}, \;\; p>1
17. Prove, that if f(x_1, x_2) = g_1(x_1) + g_2(x_2), then f^*(y_1, y_2) = g_1^*(y_1) + g_2^*(y_2)
18. Prove, that if f(x) = g(x-b), then f^*(y) = b^\top y + g^*(y)
19. Prove, that if f(x) = \alpha g(x) and $ > 0 $, then f^*(y) = \alpha g^*(y/\alpha)
20. Prove, that if f(x) = g(Ax), then f^*(y) = g^*(A^{-\top}y)
21. Prove, that if f(x) = \inf\limits_{u+v = x} (g(u) + h(v)), then f^*(y) = g^*(y) + h^*(y)