Projection

Projection

  1. Let us have two different points a,b∈Rna, b \in \mathbb{R}^n. Prove that the set of points which in the Euclidean norm are closer to the point aa than to bb make up a half-space. Is this true for another norm?

  2. Find Ο€S(y)=Ο€\pi_S (y) = \pi if S={x∈Rn∣βˆ₯xβˆ’xcβˆ₯≀R}S = \{x \in \mathbb{R}^n \mid \|x - x_c\| \le R \}, yβˆ‰Sy \notin S

  3. Find Ο€S(y)=Ο€\pi_S (y) = \pi if S={x∈Rn∣cTx=b}S = \{x \in \mathbb{R}^n \mid c^T x = b \}, yβˆ‰Sy \notin S

  4. Find Ο€S(y)=Ο€\pi_S (y) = \pi if S={x∈Rn∣Ax=b,A∈RmΓ—n,b∈Rm}S = \{x \in \mathbb{R}^n \mid Ax = b, A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^{m} \}, yβˆ‰Sy \notin S

  5. Illustrate the geometric inequality that connects Ο€S(y),yβˆ‰S,x∈S\pi_S(y), y \notin S, x \in S, from which it follows that Ο€S(y)\pi_S(y) is a projection of the yy point onto a convex set of SS.

  6. For which sets does the projection of the point outside this set exist? Unique?

  7. Find Ο€S(y)=Ο€\pi_S (y) = \pi if S={x∈Rn∣cTxβ‰₯b}S = \{x \in \mathbb{R}^n \mid c^T x \ge b \}

  8. Find Ο€S(y)=Ο€\pi_S (y) = \pi if S={x∈Rn∣x=x0+XΞ±,X∈RnΓ—m,α∈Rm}S = \{x \in \mathbb{R}^n \mid x = x_0 + X \alpha, X \in \mathbb{R}^{n \times m}, \alpha \in \mathbb{R}^{m}\}, y∈Sy \in S

  9. Let SβŠ†RnS \subseteq \mathbb{R}^n be a closed set, and x∈Rnx \in \mathbb{R}^n be a point not lying in it. Show that the projection in l2l_2 norm will be unique, while in l∞l_\infty norm this statement is not valid.

  10. Find the projection of the matrix XX on a set of matrices of rank k,β€…β€Šβ€…β€Šβ€…β€ŠX∈RmΓ—n,k≀n≀mk, \;\;\; X \in \mathbb{R}^{m \times n}, k \leq n \leq m. In Frobenius norm and spectral norm.

  11. Find a projection of the XX matrix on a set of symmetrical positive semi-definite matrices of X∈RnΓ—nX \in \mathbb{R}^{n \times n}. In Frobenius norm and the scalar product associated with it.

  12. Find the projection Ο€S(y)\pi_S(y) of point yy onto the set S={x1,x2∈R2∣∣∣x1∣+∣x2∣=1}S = \{x_1, x_2 \in \mathbb{R}^2 \mid \mid \vert x_1\vert + \vert x_2\vert = 1 \} in βˆ₯β‹…βˆ₯1\| \cdot \|_1 norm. Consider the different positions of yy.

  13. Find Ο€S(y)=Ο€\pi_S (y) = \pi, if S={x∈Rn∣αi≀xi≀βi,i=1,…,n}S = \{x \in \mathbb{R}^n \mid \alpha_i \le x_i \le \beta_i, i = 1, \ldots, n \}.

  14. Prove that projection is a nonexpansive operator, i.e. prove, that if S∈RnS \in \mathbb{R}^{n} is nonempty, closed and convex set, then for any (x1,x2)∈RnΓ—Rn(x_{1}, x_{2}) \in \mathbb{R}^{n} \times \mathbb{R}^{n}

    βˆ₯Ο€S(x2)βˆ’Ο€S(x1)βˆ₯2≀βˆ₯x2βˆ’x1βˆ₯2 \lVert \pi_{S}(x_{2}) - \pi_{S}(x_{1}) \rVert_{2} \leq \lVert x_{2} - x_{1} \rVert_{2}