OM/hw1/ex3.m

%% Homework 1 - Optimization Methods
% Author: Claudio Maggioni
%
% Sources:
% - https://www.youtube.com/watch?v=91RZYO1cv_o

clear
clc
close all
format short

%% Exercise 3.1

% f(x1, x2) = x1^2 + u * x2^2;

% 1/2 * [x1 x2] [2  0] [x1] + [0][x1]
%               [0 2u] [x2] + [0][x2]

% A = [1 0; 0 u]; b = [0; 0]

%% Exercise 3.2
xaxis = -10:1:10;
yaxis = xaxis;
Zn = zeros(size(xaxis, 2), size(yaxis, 2));
Zs = {Zn,Zn,Zn,Zn,Zn,Zn,Zn,Zn,Zn,Zn};

for u = 1:10
    A = [1 0; 0 u];

    for i = 1:size(xaxis, 2)
        for j = 1:size(yaxis, 2)
            vec = [xaxis(i); yaxis(j)];
            Zs{u}(i, j) = vec' * A * vec;
        end
    end
end

for u = 1:10
    subplot(2, 5, u);
    h = surf(xaxis, yaxis, Zs{u});
    set(h,'LineStyle','none');
    title(sprintf("u=%d", u));
end
sgtitle("Surf plots");

% comment these lines on submission
addpath /home/claudio/git/matlab2tikz/src
matlab2tikz('showInfo', false, './surf.tex')

figure

yi = zeros(30, 25);
ni = zeros(30, 25);
its = zeros(30, 1);

for u = 1:10
    subplot(2, 5, u);
    contour(xaxis, yaxis, Zs{u}, 10);
    title(sprintf("u=%d", u));


%% Exercise 3.3

    A = [2 0; 0 2*u];
    b = [0; 0];
    xs = [[0; 10] [10; 0] [10; 10]];

    syms sx sy
    f = 1/2 * [sx sy] * A * [sx; sy];
        
    g = gradient(f, [sx; sy]);
    
    hold on
    j = 1;
    for x0 = xs
        ri = u * 3 - 3 + j;
        x = x0;
        i = 1;
        
        xi = zeros(2, 25);
        xi(:, 1) = x0;
        
        yi(ri, 1) = subs(f, [sx sy], x0');
        
        while true
            p = -1 * double(subs(g, [sx sy], x'));
            
            ni(ri, i) = log10(norm(p, 2));
            if norm(p, 2) == 0 || ni(ri, i) <= -8
                break
            end
            
            alpha = dot(b - A * x, p) / dot(A * p, p);

            x = x + alpha * p;
            i = i + 1;
            
            xi(:, i) = x;
            yi(ri, i) = subs(f, [sx sy], x');
        end
        
        xi = xi(:, 1:i);
        
        plot(xi(1, :), xi(2, :), '-');
        
        fprintf("u=%2d x0=[%2d,%2d] it=%2d x=[%d,%d]\n", u, ...
            x0(1), x0(2), i, x(1), x(2));
        
        its(ri) = i;
        
        j = j + 1;
    end
    hold off
end
sgtitle("Contour plots and iteration steps");

% comment these lines on submission
addpath /home/claudio/git/matlab2tikz/src
matlab2tikz('showInfo', false, './contour.tex')

figure

for u = 1:10
    subplot(2, 5, u);
    title(sprintf("u=%d", u));
    hold on
    for j = 1:3
        ri = u * 3 - 3 + j;
        vec = yi(ri, :);
        vec = vec(1:its(ri));
        
        plot(1:its(ri), vec);
    end
    hold off
end
sgtitle("Iterations over values of objective function");

% comment these lines on submission
addpath /home/claudio/git/matlab2tikz/src
matlab2tikz('showInfo', false, './yseries.tex')

figure

for u = 1:10
    subplot(2, 5, u);
    hold on
    for j = 1:3
        ri = u * 3 - 3 + j;
        vec = ni(ri, :);
        vec = vec(1:its(ri));
        
        plot(1:its(ri), vec);
    end
    hold off
    title(sprintf("u=%d", u));
end
sgtitle("Iterations over log10 of gradient norms");

% comment these lines on submission
addpath /home/claudio/git/matlab2tikz/src
matlab2tikz('showInfo', false, './norms.tex')