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Exercise 31 We denote by $ (X, Y) $ the scalar product of two vectors $ X= $ $ \left(x_{i}\right)_{1 \leq i \leq n} $ and $ Y=\left(y_{i}\right)_{1 \leq i \leq n} $. The notation $ X \geq Y $ means that for all $ i $ between 1 and $ n, x_{i} \geq y_{i} $. We assume that for all $ X $ in $ \mathbb{R}^{n}, R $ satisfies $ (X, R X) \geq $ $ \alpha(X, X) $ with $ \alpha>0 $. We want to to study the system \[ \left\{\begin{array}{l} R X \geq G \\ X \geq F \\ (R X-G, X-F)=0 . \end{array}\right. \] 1. Show that this is equivalent to finding $ X \geq F $ such that \[ \forall V \geq F \quad(R X-G, V-X) \geq 0 \text {. } \] 2. Prove the uniqueness of a solution of (5.17). 3. Show that if $ R $ is the identity matrix, there exists a unique solution to (5.17). 4. Let $ \rho $ be positive; we denote by $ S_{\rho}(X) $ the unique vector $ Y \geq F $ such that \[ \forall V \geq F \quad(Y-X+\rho(R X-G), V-Y) \geq 0 . \] Show that for sufficiently small $ \rho, S_{\rho} $ is a contraction. 5. Derive the existence of a solution to (5.17).

          Exercise 31 We denote by \( (X, Y) \) the scalar product of two vectors \( X= \) \( \left(x_{i}\right)_{1 \leq i \leq n} \) and \( Y=\left(y_{i}\right)_{1 \leq i \leq n} \). The notation \( X \geq Y \) means that for all \( i \) between 1 and \( n, x_{i} \geq y_{i} \). We assume that for all \( X \) in \( \mathbb{R}^{n}, R \) satisfies \( (X, R X) \geq \) \( \alpha(X, X) \) with \( \alpha>0 \). We want to to study the system
\[
\left\{\begin{array}{l}
R X \geq G \\
X \geq F \\
(R X-G, X-F)=0 .
\end{array}\right.
\]
1. Show that this is equivalent to finding \( X \geq F \) such that
\[
\forall V \geq F \quad(R X-G, V-X) \geq 0 \text {. }
\]
2. Prove the uniqueness of a solution of (5.17).
3. Show that if \( R \) is the identity matrix, there exists a unique solution to (5.17).
4. Let \( \rho \) be positive; we denote by \( S_{\rho}(X) \) the unique vector \( Y \geq F \) such that
\[
\forall V \geq F \quad(Y-X+\rho(R X-G), V-Y) \geq 0 .
\]

Show that for sufficiently small \( \rho, S_{\rho} \) is a contraction.
5. Derive the existence of a solution to (5.17).

$Exercise 31 We denote by (X, Y) the scalar product of two vectors X= (xi)1 ≤ i ≤ n and Y=(yi)1 ≤ i ≤ n. The notation X ≥ Y means that for all i between 1 and n, xi≥ yi. We assume that for all X in ℝ^n, R satisfies (X, R X) ≥ α(X, X) with α>0. We want to to study the system { R X ≥ G X ≥ F (R X-G, X-F)=0 . . 1. Show that this is equivalent to finding X ≥ F such that ∀ V ≥ F (R X-G, V-X) ≥ 0 . 2. Prove the uniqueness of a solution of (5.17). 3. Show that if R is the identity matrix, there exists a unique solution to (5.17). 4. Let ρ be positive; we denote by Sρ(X) the unique vector Y ≥ F such that ∀ V ≥ F (Y-X+ρ(R X-G), V-Y) ≥ 0 . Show that for sufficiently small ρ, Sρ is a contraction. 5. Derive the existence of a solution to (5.17).$

Added by Gregory L.

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