Uzzawa

Claudio_Maggioni_5/Claudio_Maggioni_5.md

@@ -121,12 +121,6 @@ central path C converges to anything then $\tau$ approaches zero, therefore
 leading the path to the constrained minimizer while keeping $x$ and $s$
 positive but at the same time minimizing their pairwise products to zero.
 
-In practice, most central path implementation use Newton steps toward points on
-$\mathcal{C}$ for which $\tau > 0$, rather than pure Newton steps for $F$. This
-is usually possible since those newton steps are biased to stay in the
-hyperspace region where $x > 0$ and $s > 0$ even without enforcing these
-constraints.
-
 # Exercise 2
 
 ## Exercise 2.1
@@ -213,27 +207,8 @@ constrained minimizer.
 # Exercise 3
 
 ## Exercise 3.1
-<!--
-I consider the given problem, which is exactly the same as one of the problems
-of the previous assignment (Homework 4):
 
-$$\min_{x} f(x) = 3x^2_1 + 2x_1x_2 + x_1x_3 +
+Yes, the problem can be solved with Uzawa's method since the problem can be
-2.5x^2_2 + 2x_2x_3 + 2x^2_3 - 8x_1 - 3x_2 - 3x_3
-$$$$\text{ subject to } x_1 + x_3 = 3 \;\;\; x_2 + x_3 = 0$$
-
-defining $x$ as $(x_1,\,x_2,\,x_3)^T$, that can be written in the form of a
-quadratic minimization problem:
-
-$$\min_{x} f(x) = \dfrac{1}{2} \langle x,\, Gx\rangle + \langle x,\, c\rangle \\
-\text{ subject to } Ax = b$$
-
-Where $G\in \mathbb{R}^{n\times n}$ is a symmetric positive definite matrix,
-$x$, $c \in \mathbb{R}^n$. The equality constraints are defined in terms of the
-matrix $A\in \mathbb{R}^{m\times n}$, with $m \leq n$ and vector $b \in
-\mathbb{R}^m$. Here, matrix $A$ has full rank.
--->
-
-Yes, the problem can be solved with _Uzzawa_'s method since the problem can be
 reformulated as a saddle point system. The KKT conditions of the problem can be
 reformulated as a matrix-vector to vector equality in the following way:
 
@@ -256,7 +231,7 @@ g\\
 h
 \end{bmatrix}$$
 
-This is the system the _Uzzawa_ method will solve. Therefore, we need to check
+This is the system the Uzawa method will solve. Therefore, we need to check
 if the matrix:
 
 $$K = \begin{bmatrix}G & A^T \\ A& 0\end{bmatrix} = \begin{bmatrix}
@@ -279,7 +254,7 @@ $$\begin{bmatrix}
     8.7663\end{bmatrix}$$
 
 Therefore, the system is indeed a saddle point system and it can be solved with
-_Uzzawa_'s method.
+Uzawa's method.
 
 ## Exercise 3.2