Separation

Separation

  1. Let S_1, S_2 be closed convex sets such that: S_1 \cap S_2 = \varnothing. Is it true that \exists p: (p,x) < (p,y) \;\; \forall x \in S_1, \forall y \in S_2

  2. Find a separating hyperplane between S_1 and S_2:

    S_1 = \left\{ x \in \mathbb{R}^2 \mid x_1 x_2 \ge 1, x_1 > 0\right\},\quad S_2 = \left\{ x \in \mathbb{R}^2 \mid x_2 \le \frac{4}{x_1 - 1} +9\right\}

  3. Find a supporting hyperplane for a set of S = \left\{ x \in \mathbb{R}^2 \mid e^{x_1} \le x_2\right\} at the boundary point x_0 = (0, 1)

  4. Find a supporting hyperplane for the set of S = \left\{ x \in \mathbb{R}^3 \mid x_3 \ge x_1^2 + x_2^2\right\} such that separates it from the point x_0 = \left(-\frac{5}{4}, \frac{5}{16}, \frac{15}{16}\right)

  5. Приведите пример двух строго, но не сильно отделимых множеств. Двух отделимых, но не собственно отделимых множеств.

  6. Illustrate the difference between the tangent hyperplane and the separating hyperplane by considering a convex set with a non-smooth boundary.

  7. Derive the equation of the supporting hyperplane of the set of \{ x \in \mathbb{R}^3 \mid \dfrac{x_1^2}{4} + \dfrac{x_2^2}{9} + \dfrac{x_3^2}{25} \le 1 \} at (-6/5, 12/5, 0), (0, 9/5, 4), (6/5, 0, -4) (any choice)

  8. Derive the equation of the supporting hyperplane of the set of \{ x \in \mathbb{R}^3 \mid x_3 \ge x_1^2 + x_2^2 \}, which separates it from the points (5/4, 5/16, 15/16), (4/3, 2/3, 13/12), (11/9, 11/27, 1) (any choice)

  9. Find a separating hyperplane between S_1 and S_2:

    S_1 = \left\{ x \in \mathbb{R}^n \mid x_1^2 + x_2^2 + \ldots + x_n^2 \le 1\right\}, \;\;\; S_2 = \left\{ x \in \mathbb{R}^n \mid x_1^2 + x_2^2 + \ldots + x_{n-1}^2 + 1 \le x_n \right\}

  10. Find a supporting hyperplane for a set S = \left\{ x \in \mathbb{R}^3 \mid \frac{x_1^2}{4}+\frac{x_2^2}{8}+\frac{x_3^2}{25} \le 1 \right\} at the border point x_0 = \left(-1, \frac{12}{5}, \frac{\sqrt{3}}{2}\right)