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First work3 code version and report initialization

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hoo2
2024-12-01 22:08:21 +02:00
parent 9565ac2538
commit d5c8ed5e68
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Work 3/scripts/BelongsTo.m
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function [flag] = BelongsTo(x, SetLimmits)
%Checks if the x vector belongs to Set
%
% x: A vector to project
% SetLimmits: The set to project. Each line/dimension off the set has to contain the limits
% of the set to that particular dimension
% flag: True if belongs
%
flag = true; % Have faith
for i = 1:size(SetLimmits, 2)
if x(i) < SetLimmits(i,1)
flag = false;
elseif x(i) > SetLimmits(i,2)
flag = false;
end
end
end
Work 3/scripts/GivenEnv.asv
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% Given environment
clear;
% Setup the function under test
syms x [2 1] real;
fexpr = (1/3)*x(1)^2 +3*x(2)^2;
title_fun = "$f(x) = frac{1}{3}{x_1}^2 + 3{x_2}^2$";
% Calculate the gradient and Hessian
grad_fexpr = gradient(fexpr, x); % Gradient of f
hessian_fexpr = hessian(fexpr, x); % Hessian of f
% Convert symbolic expressions to MATLAB functions
fun = matlabFunction(fexpr, 'Vars', {x}); % Function
grad_fun = matlabFunction(grad_fexpr, 'Vars', {x}); % Gradient
hessian_fun = matlabFunction(hessian_fexpr, 'Vars', {x}); % Hessian
% Minimum reference
%Freference = @(x) x(1).^5 .* exp(-x(1).^2 - x(2).^2);
%[Xmin, Fmin] = fminsearch(Freference, [-1, -1]);
% Amijo globals
global amijo_beta; % Step reduction factor in [0.1, 0.5] (typical range: [0.1, 0.8])
global amijo_sigma; % Sufficient decrease constant in [1e-5, 0.1] (typical range: [0.01, 0.3])
%fixed step size globals
global gamma_fixed_step
global image_width,
global image_height;
image_width = 960;
image_height = 640;
Work 3/scripts/GivenEnv.m
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% Given environment
clear;
% Setup the function under test
syms x [2 1] real;
fexpr = (1/3)*x(1)^2 +3*x(2)^2;
title_fun = "$f(x) = \frac{1}{3}{x_1}^2 + 3{x_2}^2$";
XSetLimmits = [-10, 5 ; -8, 12];
% Calculate the gradient and Hessian
grad_fexpr = gradient(fexpr, x); % Gradient of f
hessian_fexpr = hessian(fexpr, x); % Hessian of f
% Convert symbolic expressions to MATLAB functions
fun = matlabFunction(fexpr, 'Vars', {x}); % Function
grad_fun = matlabFunction(grad_fexpr, 'Vars', {x}); % Gradient
hessian_fun = matlabFunction(hessian_fexpr, 'Vars', {x}); % Hessian
% Minimum reference
[Xmin, Fmin] = fminsearch(fun, [-1, -1]');
Xmin = round(Xmin, 3);
Fmin = round(Fmin, 3);
% Amijo globals
global amijo_beta; % Step reduction factor in [0.1, 0.5] (typical range: [0.1, 0.8])
global amijo_sigma; % Sufficient decrease constant in [1e-5, 0.1] (typical range: [0.01, 0.3])
%fixed step size globals
global gamma_fixed_step
global image_width,
global image_height;
image_width = 960;
image_height = 640;
Work 3/scripts/ProjectionPoint.m
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function [x_p] = ProjectionPoint(x, SetLimmits)
%Project the x vector to Set, returns a point of Set close(st) to x
%
% x: A vector to project
% SetLimmits: The set to project. Each line/dimension off the set has to contain the limits
% of the set to that particular dimension
%x_p: The projected point
%
x_p = x;
for i = 1:size(SetLimmits, 2)
if x(i) < SetLimmits(i,1)
x_p(i) = SetLimmits(i,1);
elseif x(i) > SetLimmits(i,2)
x_p(i) = SetLimmits(i,2);
end
end
end
Work 3/scripts/Script_0_Plots.m
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% Define environment (functions, gradients etc...)
GivenEnv
%
% We plot the function in the domain of x,y in [-3, 3].
% We also plot the contour in order to get a sense of the min and maximum
% points in the x-y plane
%
% 3d plot the function
plot3dFun(fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, title_fun, "figures/Plot_Function.png");
% Plot isobaric lines
plotContour(fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, title_fun, "figures/Plot_Contour.png");
Work 3/scripts/Script_1_SteepDesc.m
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% Define environment (functions, gradients etc...)
GivenEnv
% Define parameters
max_iter = 1000; % Maximum iterations
tol = 1e-4; % Tolerance
% Point x0 = (1, 1)
% =========================================================================
point = 1;
x0 = [5, -5]';
point_str = "[" + x0(1) + ", " + x0(2) + "]";
f = fun(x0);
gf = grad_fun(x0);
hf = hessian_fun(x0);
fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]=> Method applicable\n', x0, f, gf, hf);
for g=[0.1, 0.3, 0.5, 3, 5]
gamma_fixed_step = g;
[x_fixed, f_fixed, kk] = method_SteepDesc(fun, grad_fun, x0, tol, max_iter, 'fixed');
fprintf('Fixed step g=%f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', g, x0, kk, x_fixed(:, end), f_fixed(end));
if g <= 1
plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, "");
else
plotPointsOverContour(x_fixed, fun, [-500, 500], [-500, 500], 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, "");
end
end
Work 3/scripts/Script_2_SteepDesc_Proj.m
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% Define environment (functions, gradients etc...)
GivenEnv
% Settings
max_iter = 1000; % Maximum iterations
% Define parameters
% =========================================================================
x0 = [5, -5]';
gamma = 0.5;
sk_step = 5;
tol = 0.01; % Tolerance
point_str = "[" + x0(1) + ", " + x0(2) + "]";
f = fun(x0);
gf = grad_fun(x0);
hf = hessian_fun(x0);
fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]=> Method applicable\n', x0, f, gf, hf);
gamma_fixed_step = gamma;
[x_fixed, f_fixed, kk] = method_SteepDesc_Proj(fun, grad_fun, x0, sk_step, XSetLimmits, tol, max_iter, 'fixed');
fprintf('Fixed step g=%f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', gamma_fixed_step, x0, kk, x_fixed(:, end), f_fixed(end));
plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, "");
%[x_minimized, f_minimized, kk] = method_SteepDesc_Proj(fun, grad_fun, x0, sk_step, XSetLimmits, tol, max_iter, 'minimized');
%fprintf('Minimized f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', x0, kk, x_fixed(:, end), f_fixed(end));
%plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent minimized", "");
Work 3/scripts/Script_3_SteepDesc_Proj.m
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% Define environment (functions, gradients etc...)
GivenEnv
% Settings
max_iter = 1000; % Maximum iterations
% Define parameters
% =========================================================================
x0 = [-5, 10]';
gamma = 0.1;
sk_step = 55;
tol = 0.01; % Tolerance
point_str = "[" + x0(1) + ", " + x0(2) + "]";
f = fun(x0);
gf = grad_fun(x0);
hf = hessian_fun(x0);
fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]=> Method applicable\n', x0, f, gf, hf);
gamma_fixed_step = gamma;
[x_fixed, f_fixed, kk] = method_SteepDesc_Proj(fun, grad_fun, x0, sk_step, XSetLimmits, tol, max_iter, 'fixed');
fprintf('Fixed step g=%f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', gamma_fixed_step, x0, kk, x_fixed(:, end), f_fixed(end));
plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, "");
Work 3/scripts/Script_4_SteepDesc_Proj.m
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% Define environment (functions, gradients etc...)
GivenEnv
% Settings
max_iter = 1000; % Maximum iterations
% Define parameters
% =========================================================================
x0 = [8, -10]';
gamma = 0.2;
sk_step = 0.1;
tol = 0.01; % Tolerance
point_str = "[" + x0(1) + ", " + x0(2) + "]";
f = fun(x0);
gf = grad_fun(x0);
hf = hessian_fun(x0);
fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]=> Method applicable\n', x0, f, gf, hf);
gamma_fixed_step = gamma;
[x_fixed, f_fixed, kk] = method_SteepDesc_Proj(fun, grad_fun, x0, sk_step, XSetLimmits, tol, max_iter, 'fixed');
fprintf('Fixed step g=%f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', gamma_fixed_step, x0, kk, x_fixed(:, end), f_fixed(end));
plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, "");
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Work 3/scripts/fmin_bisection.m
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function [a, b, k, n] = fmin_bisection(fun, alpha, beta, epsilon, lambda)
% Bisection method for finding the local minimum of a function.
%
% fun: The objective function
% alpha: (number) The starting point of the interval in which we seek
% for minimum
% beta: (number) The ending point of the interval in which we seek
% for minimum
% epsilon: (number) The epsilon value (distance from midpoint)
% lambda: (number) The lambda value (accuracy)
%
% return:
% a: (vector) Starting points of the interval for each iteration
% b: (vector) Ending points of the interval for each iteration
% k: (number) The number of iterations
% n: (number) The calls of objective function fun_expr
%
% Error checking
if alpha > beta || 2*epsilon >= lambda || lambda <= 0
error ('Input criteria not met')
end
% Init
a = alpha;
b = beta;
n = 0;
k=1;
while b(k) - a(k) > lambda
% bisect [a,b]
mid = (a(k) + b(k)) / 2;
x_1 = mid - epsilon;
x_2 = mid + epsilon;
% set new search interval
k = k + 1;
if fun(x_1) < fun(x_2)
a(k) = a(k-1);
b(k) = x_2;
else
a(k) = x_1;
b(k) = b(k-1);
end
end
end
Work 3/scripts/gamma_armijo.m
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function [gamma] = gamma_armijo(f, grad_f, dk, xk)
% Calculates the best step based on amijo method
%
% f(xk​+ γk*dk) ≤ f(xk) + σ * γk * dk^T * ∇f(xk)
% γk = β*γk_0
%
% f: Objective function
% grad_fun: Gradient function of f
% dk: Current value of selected direction -∇f or -inv{H}*∇f or -inv{H + lI}*∇f
% xk: Current point (x,y)
% beta: beta factor in [0.1, 0.5]
% signam: sigma factor in (0, 0.1]
global amijo_beta
global amijo_sigma
gf = grad_f(xk);
gamma = 1; % Start with a step size of 1
% Perform Armijo line search
while f(xk + gamma * dk) > f(xk) + amijo_sigma * gamma * dk * gf
%while f(xk(1) + gamma * dk(1), xk(2) + gamma * dk(2)) > ...
% f(xk(1), xk(2)) + amijo_sigma * gamma * norm(dk)^2
gamma = amijo_beta * gamma; % Reduce step size
if gamma < 1e-12 % Safeguard to prevent infinite reduction
warning('Armijo step size became too small.');
break;
end
end
end
Work 3/scripts/gamma_fixed.m
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function [gamma] = gamma_fixed(~, ~, ~, ~)
% Return a fixed step
%
% This is for completion and code symmetry.
%
global gamma_fixed_step
% Perform line search
gamma = gamma_fixed_step;
end
Work 3/scripts/gamma_minimized.m
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function [gamma] = gamma_minimized(f, ~, dk, xk)
% Calculates the step based on minimizing f(xk​− γk*dk)
%
%
% f: Objective function
% ~: Gradient function of f - Not used
% dk: Current value of selected direction -∇f or -inv{H}*∇f or -inv{H + lI}*∇f
% xk: Current point (x,y)
% Define the line search function fmin(g) = f(xk - g * dk)
fmin = @(g) f(xk) + g * dk;
% find g that minimizes fmin
e = 0.0001;
l = 0.001;
[a,b,k,~] = fmin_bisection(fmin, 0.0001, 1, e, l); % g in (0, 1]
gamma = 0.5*(a(k) + b(k));
% Define the line search function fmin(g) = f(xk - g * dk)
%fmin = @(g) f(xk(1) - gamma * dk(1), xk(2) - gamma * dk(2));
% find g that minimizes fmin
%gamma = fminbnd(g, 0, 1);
end
Work 3/scripts/method_SteepDesc.m
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function [x_vals, f_vals, k] = method_SteepDesc(f, grad_f, xk, tol, max_iter, mode)
% f: Objective function
% grad_f: Gradient of the function
% xk: Initial point [x0; y0]
% tol: Tolerance for stopping criterion
% max_iter: Maximum number of iterations
% x_vals: Vector with the (x,y) values until minimum
% f_vals: Vector with f(x,y) values until minimum
% k: Number of iterations
if strcmp(mode, 'armijo') == 1
gamma_f = @(f, grad_f, dk, xk) gamma_armijo(f, grad_f, dk, xk);
elseif strcmp(mode, 'minimized') == 1
gamma_f = @(f, grad_f, dk, xk) gamma_minimized(f, grad_f, dk, xk);
else % mode == 'fixed'
gamma_f = @(f, grad_f, dk, xk) gamma_fixed(f, grad_f, dk, xk);
end
% Storage for iterations, begin with the first point
x_vals = xk;
f_vals = f(xk);
for k = 1:max_iter
% Check for convergence
if norm(grad_f(xk)) < tol
break;
end
dk = - grad_f(xk); % Steepset descent direction
gk = gamma_f(f, grad_f, dk, xk); % Calculate gamma
x_next = xk + gk * dk; % Update step
f_next = f(x_next);
xk = x_next; % Update point
x_vals = [x_vals, x_next]; % Store values
f_vals = [f_vals, f_next]; % Store function values
end
end
Work 3/scripts/method_SteepDesc_Proj.m
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function [x_vals, f_vals, k] = method_SteepDesc_Proj(f, grad_f, xk, sk, limmits, tol, max_iter, mode)
% f: Objective function
% grad_f: Gradient of the function
% xk: Initial point [x0; y0]
% sk: Step size (fixed positive scalar)
% limits: Bounds of the feasible set for each dimension
% tol: Tolerance for stopping criterion
% max_iter: Maximum number of iterations
% x_vals: Vector with the (x,y) values until minimum
% f_vals: Vector with f(x,y) values until minimum
% k: Number of iterations
if strcmp(mode, 'armijo') == 1
gamma_f = @(f, grad_f, dk, xk) gamma_armijo(f, grad_f, dk, xk);
elseif strcmp(mode, 'minimized') == 1
gamma_f = @(f, grad_f, dk, xk) gamma_minimized(f, grad_f, dk, xk);
else % mode == 'fixed'
gamma_f = @(f, grad_f, dk, xk) gamma_fixed(f, grad_f, dk, xk);
end
% Project the first point if needed
xk = ProjectionPoint(xk, limmits);
% Storage for iterations, begin with the first point
x_vals = xk;
f_vals = f(xk);
for k = 1:max_iter
% Check for convergence
if norm(grad_f(xk)) < tol
break;
end
dk = - grad_f(xk); % Steepset descent direction
% First calculate xk-bar and project it if nessesary
xkbar = xk + sk * dk;
xkbar = ProjectionPoint(xkbar, limmits);
dk = (xkbar - xk); % Steepest descent projection direction
gk = gamma_f(f, grad_f, dk, xk); % Calculate gamma
x_next = xk + gk * dk; % Update step
f_next = f(x_next);
xk = x_next; % Update point
x_vals = [x_vals, x_next]; % Store values
f_vals = [f_vals, f_next]; % Store function values
end
end
Work 3/scripts/plot3dFun.m
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function plot3dFun(fun, x_lim, y_lim, size, plot_title, filename)
% 3D plots a function
% fun: The function to plot
% x_lim: The range for x axis. ex: [-2, 2]
% y_lim: The range for y axis. ex: [0, 2]
% size: The number of points for each axis
% plot_title: The latex title for the plot
%
global image_width,
global image_height;
% Generate a grid for x and y
x_space = linspace(x_lim(1), x_lim(2), size);
y_space = linspace(y_lim(1), y_lim(2), size);
[X, Y] = meshgrid(x_space, y_space);
% Evaluate the function on the grid
for i = 1:size
for j = 1:size
% Pass each [x1; x2] as input to fun
Z(i, j) = fun([X(i, j); Y(i, j)]);
end
end
% 3D plot
figure('Name', 'f(x,y)', 'NumberTitle', 'off');
set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size
surf(X, Y, Z);
% Customize the plot
xlabel('x1'); % Label for x-axis
ylabel('x2'); % Label for y-axis
zlabel('f(x, y)'); % Label for z-axis
title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot
colorbar;
% save the figure
if strcmp(filename, '') == 0
print(gcf, filename, '-dpng', '-r300');
end
end
Work 3/scripts/plotContour.m
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function plotContour(fun, x_lim, y_lim, size, plot_title, filename)
% plot the contour of a function
% fun: The function to plot
% x_lim: The range for x axis. ex: [-2, 2]
% y_lim: The range for y axis. ex: [0, 2]
% size: The number of points for each axis
% plot_title: The latex title for the plot
%
global image_width,
global image_height;
% Generate a grid for x and y
x_space = linspace(x_lim(1), x_lim(2), size);
y_space = linspace(y_lim(1), y_lim(2), size);
[X, Y] = meshgrid(x_space, y_space);
% Evaluate the function on the grid
for i = 1:size
for j = 1:size
% Pass each [x1; x2] as input to fun
Z(i, j) = fun([X(i, j); Y(i, j)]);
end
end
% Contour
figure('Name', 'Contours of f(x1,x2)', 'NumberTitle', 'off');
set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size
contour(X, Y, Z);
% Customize the plot
xlabel('x1'); % Label for x-axis
ylabel('x2'); % Label for y-axis
title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot
colorbar;
% save the figure
if strcmp(filename, '') == 0
print(gcf, filename, '-dpng', '-r300');
end
end
Work 3/scripts/plotConvCompare.m
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function plotConvCompare(points_1, title_1, points_2, title_2, points_3, title_3, Min_point, plot_title, filename)
% 3D plots a function
% points: The points to plot
% contur_fun: The function for contour plot
% x_lim: The range for x axis. ex: [-2, 2]
% y_lim: The range for y axis. ex: [0, 2]
% size: The number of points for each axis
% plot_title: The latex title for the plot
% filename: The filename to save the plot (if exists)
%
global image_width,
global image_height;
distances_1 = sqrt((points_1(1, :) - Min_point(1)).^2 + (points_1(2, :) - Min_point(2)).^2);
distances_2 = sqrt((points_2(1, :) - Min_point(1)).^2 + (points_2(2, :) - Min_point(2)).^2);
distances_3 = sqrt((points_3(1, :) - Min_point(1)).^2 + (points_3(2, :) - Min_point(2)).^2);
% 2D plot
figure('Name', 'Convergence compare', 'NumberTitle', 'off');
set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size
title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot
% One
subplot(3, 1, 1);
plot(distances_1, '-o');
% Customize the plot
ylabel(title_1, 'Interpreter', 'none');
xlabel('Step');
grid on
% One
subplot(3, 1, 2);
plot(distances_2, '-o');
% Customize the plot
ylabel(title_2, 'Interpreter', 'none');
xlabel('Step');
grid on
% One
subplot(3, 1, 3);
plot(distances_3, '-o');
% Customize the plot
ylabel(title_3, 'Interpreter', 'none');
xlabel('Step');
grid on
% save the figure
if strcmp(filename, '') == 0
print(gcf, filename, '-dpng', '-r300');
end
end
Work 3/scripts/plotItersOverGamma.m
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function plotItersOverGamma(gamma, iterations, plot_title, filename)
% 3D plots a function
% fun: The points to plot
% contur_fun: The function for contour plot
% x_lim: The range for x axis. ex: [-2, 2]
% y_lim: The range for y axis. ex: [0, 2]
% size: The number of points for each axis
% plot_title: The latex title for the plot
% filename: The filename to save the plot (if exists)
%
global image_width,
global image_height;
figure('Name', 'Iterations_over_gamma', 'NumberTitle', 'off');
set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size
plot(gamma, iterations, '*r', 'LineWidth', 2);
% Customize the plot
title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot
xlabel('\gamma') ;
ylabel('Iterations');
% save the figure
if strcmp(filename, '') == 0
print(gcf, filename, '-dpng', '-r300');
end
end
Work 3/scripts/plotPointsOverContour.m
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function plotPointsOverContour(points, contour_fun, x_lim, y_lim, size, plot_title, filename)
% 3D plots a function
% points: The points to plot
% contur_fun: The function for contour plot
% x_lim: The range for x axis. ex: [-2, 2]
% y_lim: The range for y axis. ex: [0, 2]
% size: The number of points for each axis
% plot_title: The latex title for the plot
% filename: The filename to save the plot (if exists)
%
global image_width,
global image_height;
% Generate a grid for x and y
x_space = linspace(x_lim(1), x_lim(2), size);
y_space = linspace(y_lim(1), y_lim(2), size);
[X, Y] = meshgrid(x_space, y_space);
% Evaluate the function on the grid
for i = 1:size
for j = 1:size
% Pass each [x1; x2] as input to fun
Z(i, j) = contour_fun([X(i, j); Y(i, j)]);
end
end
% 2D plot
figure('Name', '(x1,x2) convergence', 'NumberTitle', 'off');
set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size
plot(points(1, :), points(2, :), '-or');
hold on
contour(X, Y, Z);
% Customize the plot
xlim(x_lim);
ylim(y_lim);
xlabel('x1'); % Label for x-axis
ylabel('x2'); % Label for y-axis
grid on
title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot
colorbar;
% save the figure
if strcmp(filename, '') == 0
print(gcf, filename, '-dpng', '-r300');
end
end