hw5: done 1.1, 2, 3

Claudio_Maggioni_5/Claudio_Maggioni_5.md

@@ -16,6 +16,54 @@ header-includes:
 ---
 \maketitle
 
+# Excecise 1
+
+## Exercise 1.1
+
+### The Simplex method
+
+The simplex method solves constrained minimization problems with a linear
+cost function and linearly-defined equality and inequality constraints. The main
+approach used by the simplex method is to consider only basic feasible points
+along the feasible region polytope and to iteratively navigate between them
+hopping through neighbours and trying to find the point that minimizes the cost
+function.
+
+Although the Simplex method is relatively efficient for most in-practice
+applications, it has exponential complexity, since it has been proven that
+a carefully crafted $n$-dimensional problem can have up to $2^n$ polytope
+vertices, thus making the method inefficient for complex problems.
+
+### Interior-point method
+
+The interior point method aims to have a better worst-case complexity than the
+simplex method but still retain an in-practice acceptable performance. Instead
+of performing many inexpensive iterations walking along the polytope boundary,
+the interior point takes Newton-like steps travelling along "interior" points in
+the feasible region (hence the name of the method), thus reaching the
+constrained minimizer in fewer iterations. Additionally, the interior-point
+method is easier to be implemented in a parallelized fashion.
+
+### Penalty method
+
+The penalty method allows for a linear constrained minimization problem with
+equality constraints to be converted in an unconstrained minimization problem,
+and to allow the use of conventional unconstrained minimization algorithms to
+solve the problem. Namely, the penalty method builds a new uncostrained
+objective function with is the summation of:
+
+- The original objective function;
+- An additional term for each constraint, which is positive when the current
+  point $x$ violates that constraint and zero otherwise.
+
+With some fine tuning of the coefficients for these new "penalty" terms, it is
+possible to build an equivalend unconstrained minimization problem whose
+minimizer is also constrained minimizer for the original problem.
+
+## Exercise 1.2
+
+## Exercise 1.3
+
 # Exercise 2
 
 ## Exercise 2.1
@@ -55,7 +103,7 @@ index set $1, 2, \ldots, n$ such that:
   indices in $\beta$ are linearly independent from each other.
 
 The geometric interpretation of basic feasible points is that all of them
-are verticies of the polytope that bounds the feasible region. We will use this
+are vertices of the polytope that bounds the feasible region. We will use this
 proven property to manually solve the constrained minimization problem presented
 in this section by aiding us with the graphical plot of the feasible region in
 figure \ref{fig:a}.
@@ -85,7 +133,7 @@ Figure 1 taken from the book.-->
 
 Since the geometrical interpretation of the definition of basic feasible point
 states that these point are non-other than the vertices of the feasible region,
-we first look at the plot above and to these points (i.e. the verticies of the
+we first look at the plot above and to these points (i.e. the vertices of the
 bright green non-trasparent region). Then, we look which constraint boundaries cross these
 edges, and we formulate an algebraic expression to find these points. In
 clockwise order, we have:
@@ -130,7 +178,8 @@ x^*_3 = \frac{1}{13} \cdot \begin{bmatrix}3\\24\end{bmatrix} \;\;\; f(x^*_3) = 4
 \cdot \frac{3}{13} + 3 \cdot \frac{24}{13} = \frac{84}{13}$$$$
 x^*_4 = \frac12 \cdot \begin{bmatrix}3\\2\end{bmatrix} \;\;\; f(x^*_4) = 4 \cdot
 \frac32 + 3 \cdot 1 = 9$$$$
-x^*_5 = \begin{bmatrix}2\\0\end{bmatrix} \;\;\; f(x^*_5) = 4 \cdot 2 + 1 \cdot 0 = 8$$
+x^*_5 = \begin{bmatrix}2\\0\end{bmatrix} \;\;\; f(x^*_5) = 4 \cdot 2 + 1 \cdot 0
+= 8$$
 
 Therefore, $x^* = x^*_1 = \begin{bmatrix}0 & 0\end{bmatrix}^T$ is the global
 constrained minimizer.
@@ -138,6 +187,73 @@ 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):
+
+$$\min_{x} f(x) = 3x^2_1 + 2x_1x_2 + x_1x_3 +
+2.5x^2_2 + 2x_2x_3 + 2x^2_3 - 8x_1 - 3x_2 - 3x_3
+$$$$\text{ subject to } x_1 + x_3 = 3 \;\;\; x_2 + x_3 = 0$$
+
+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:
+
+$$\begin{bmatrix}G & -A^T\\A & 0 \end{bmatrix} \begin{bmatrix}
+x^*\\\lambda^* \end{bmatrix} = \begin{bmatrix} -c\\b \end{bmatrix}.$$
+
+If we then express the minimizer $x^*$ in terms of $x$, an approximation of it,
+and $p$, a search step (i.e. $x^* = x + p$), we obtain the following system.
+
+$$\begin{bmatrix}
+G & A^T\\
+A & 0
+\end{bmatrix}
+\begin{bmatrix}
+-p\\
+\lambda^*
+\end{bmatrix} =
+\begin{bmatrix}
+g\\
+h
+\end{bmatrix}$$
+
+This is the system the _Uzzawa_ method will solve. Therefore, we need to check
+if the matrix:
+
+$$K = \begin{bmatrix}G & A^T \\ A& 0\end{bmatrix} = \begin{bmatrix}
+6  &   2  &  1  &  1  &  0 \\
+2  &   5  &  2  &  0  &  1 \\
+1  &   2  &  4  &  1  &  1 \\
+1  &   0  &  1  &  0  &  0 \\
+0  &   1  &  1  &  0  &  0 \\
+\end{bmatrix}\text{ recalling the computed values of }A\text{ and }G\text{ from the
+previous assignment}$$
+
+Has non-zero positive and negative eigenvalues. We compute the eigenvalues of this
+matrix with MATLAB, and we find:
+
+$$\begin{bmatrix}
+   -0.4818\\
+   -0.2685\\
+    2.6378\\
+    4.3462\\
+    8.7663\end{bmatrix}$$
+
+Therefore, the system is indeed a saddle point system and it can be solved with
+_Uzzawa_'s method.
 
 ## Exercise 3.2
 

Claudio_Maggioni_5/main.m

@@ -58,21 +58,18 @@ legend('2x1 + 3x2 <= 6', '-3x1 + 2x2 <= 3', '2x2 <= 5', ...
     '2x1 + x2 <= 4', 'x1 > 0 and x2 > 0', 'feasible region');
 hold off
 
-%% gsppn
-
-for i=1:5
-    obj = 4 * px(i) + 3 * py(i);
-    fprintf("x1=%g x2=%g y=%g\n", px(i), py(i), obj);
-end
-    
-
-%% Exercise 3.2
+%% Exercise 3.1
 
 G = [6 2 1; 2 5 2; 1 2 4];
 c = [-8; -3; -3];
 A = [1 0 1; 0 1 1];
 b = [3; 0];
 
+K = [G A'; A zeros(2)];
+eig(K)
+
+%% Exercise 3.2
+
 [x, lambda] = uzawa(G, c, A, b, [0;0;0], [0;0], 1e-8, 100);
 display(x);
 display(lambda);

Claudio_Maggioni_5/uzawa.m

@@ -2,7 +2,7 @@ function [x, lambda] = uzawa(G, c, A, b, x, lambda, tol, max_itr)
     w = 1;
     old_x = ones(size(G, 1)) * 124000000;
     i = 0;
-    
+
     while i < max_itr && norm(x - old_x) > tol
         old_x = x;
         x = G \ (c - (A' * lambda));