hw2: done 1.1 and 1.2
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Claudio Maggioni (maggicl)
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%% Homework 1 - Optimization Methods
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% Author: Claudio Maggioni
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%
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% Sources:
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% - https://www.youtube.com/watch?v=91RZYO1cv_o
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clear
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clc
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close all
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format short
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colw = 5;
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colh = 2;
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%% Exercise 3.1
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% f(x1, x2) = x1^2 + u * x2^2;
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% 1/2 * [x1 x2] [2 0] [x1] + [0][x1]
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% [0 2u] [x2] + [0][x2]
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% A = [1 0; 0 u]; b = [0; 0]
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%% Exercise 3.2
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xaxis = -10:0.1:10;
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yaxis = xaxis;
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Zn = zeros(size(xaxis, 2), size(yaxis, 2));
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Zs = {Zn,Zn,Zn,Zn,Zn,Zn,Zn,Zn,Zn,Zn};
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for u = 1:10
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A = [1 0; 0 u];
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for i = 1:size(xaxis, 2)
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for j = 1:size(yaxis, 2)
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vec = [xaxis(i); yaxis(j)];
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Zs{u}(i, j) = vec' * A * vec;
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end
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end
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end
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for u = 1:10
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subplot(colh, colw, u);
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h = surf(xaxis, yaxis, Zs{u});
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set(h,'LineStyle','none');
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title(sprintf("u=%d", u));
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end
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sgtitle("Surf plots");
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% comment these lines on submission
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% addpath /home/claudio/git/matlab2tikz/src
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% matlab2tikz('showInfo', false, './surf.tex')
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figure
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% max iterations
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c = 100;
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yi = zeros(30, c);
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ni = zeros(30, c);
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its = zeros(30, 1);
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for u = 1:10
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subplot(colh, colw, u);
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contour(xaxis, yaxis, Zs{u}, 10);
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title(sprintf("u=%d", u));
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%% Exercise 3.3
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A = [2 0; 0 2*u];
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b = [0; 0];
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xs = [[0; 10] [10; 0] [10; 10]];
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syms sx sy
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f = 1/2 * [sx sy] * A * [sx; sy];
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g = gradient(f, [sx; sy]);
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hold on
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j = 1;
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for x0 = xs
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ri = u * 3 - 3 + j;
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x = x0;
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i = 1;
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xi = zeros(2, c);
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xi(:, 1) = x0;
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yi(ri, 1) = subs(f, [sx sy], x0');
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while i <= c
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p = -1 * double(subs(g, [sx sy], x'));
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ni(ri, i) = log10(norm(p, 2));
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if norm(p, 2) == 0 || ni(ri, i) <= -8
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break
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end
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alpha = dot(b - A * x, p) / dot(A * p, p);
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x = x + alpha * p;
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i = i + 1;
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xi(:, i) = x;
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yi(ri, i) = subs(f, [sx sy], x');
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end
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xi = xi(:, 1:i);
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plot(xi(1, :), xi(2, :), '-');
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fprintf("u=%2d x0=[%2d,%2d] it=%2d x=[%d,%d]\n", u, ...
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x0(1), x0(2), i, x(1), x(2));
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its(ri) = i;
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j = j + 1;
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end
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hold off
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end
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sgtitle("Contour plots and iteration steps");
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% comment these lines on submission
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% addpath /home/claudio/git/matlab2tikz/src
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% matlab2tikz('showInfo', false, './contour.tex')
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figure
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for u = 1:10
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subplot(colh, colw, u);
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title(sprintf("u=%d", u));
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hold on
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for j = 1:3
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ri = u * 3 - 3 + j;
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vec = yi(ri, :);
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vec = vec(1:its(ri));
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plot(1:its(ri), vec);
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end
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hold off
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end
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sgtitle("Iterations over values of objective function");
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% comment these lines on submission
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% addpath /home/claudio/git/matlab2tikz/src
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% matlab2tikz('showInfo', false, './yseries.tex')
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figure
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for u = 1:10
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subplot(colh, colw, u);
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hold on
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for j = 1:3
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ri = u * 3 - 3 + j;
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vec = ni(ri, :);
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vec = vec(1:its(ri));
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plot(1:its(ri), vec, '-o');
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end
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hold off
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title(sprintf("u=%d", u));
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end
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sgtitle("Iterations over log10 of gradient norms");
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% comment these lines on submission
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% addpath /home/claudio/git/matlab2tikz/src
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% matlab2tikz('showInfo', false, './norms.tex')
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@ -0,0 +1,50 @@
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%% Homework 2 - Optimization Methods
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% Author: Claudio Maggioni
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%
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% Sources:
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clear
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clc
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close all
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[A, b] = build_poisson(4);
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disp(A)
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disp(b)
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% 1.1
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function [A,b] = build_poisson(n)
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A = diag(2 * ones(1,n));
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A(1,1) = 1;
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A(n,n) = 1;
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for i = 2:n-1
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A(i, i+1) = -1;
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A(i, i-1) = -1;
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end
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h = 1 / (n - 1);
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b = h^2 * ones(n, 1);
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b(1) = 0;
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b(n) = 0;
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end
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%% 1.1 (check this)
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% Answer is a energy function does not exist. Since A is not symmetric
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% (even if it is pd), the minimizer used for the c.g. method
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% (i.e. (1/2)x^TAx - b^x) won't work
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% since x^TAx might be negative and thus the minimizer does not point to
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% the solution of Ax=B necessairly
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%% 1.2
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% we already enforce x(1) = x(n) = 0, since b(1) = b(n) = 0 and thus
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% A(1, :) * x = b(0) = 0 and same for n can be solved only for x(1) = x(n)
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% = 0
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%
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% The objective is therefore \phi(x) = (1/2)x^T\overline{A}x - b^x with a and b
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% defined above, gradient is = \overline{A}x - b, hessian is = \overline{A}
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