119 lines
5.1 KiB
Matlab
119 lines
5.1 KiB
Matlab
% Define environment (functions, gradients etc...)
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GivenEnv
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% Define parameters
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max_iter = 300; % Maximum iterations
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tol = 1e-4; % Tolerance
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% Point x0 = (0, 0)
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% =========================================================================
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point = 1;
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x0 = [0, 0];
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f = fun(x0(1), x0(2));
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gf = grad_fun(x0(1), x0(2));
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hf = hessian_fun(x0(1), x0(2));
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ev = eig(hf);
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fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Eigenvalues= [%f, %f], Can NOT use method\n', x0, f, gf, hf, ev);
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disp(' ');
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% Point x0 = (-1, 1)
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% =========================================================================
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point = 2;
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x0 = [-1, 1];
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point_str = "[" + x0(1) + ", " + x0(2) + "]";
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f = fun(-1, 1);
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gf = grad_fun(x0(1), x0(2));
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hf = hessian_fun(x0(1), x0(2));
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ev = eig(hf);
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fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Eigenvalues= [%f, %f], Can use method\n', x0, f, gf, hf, ev);
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% Find the best fixed gamma
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k = zeros(100, 1);
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j = 1;
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n = linspace(0.1, 1.5, 100);
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for g = n
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gamma_fixed_step = g;
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[x, f, k(j)] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'fixed');
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if ~(x(end, 1) < -1.57 && x(end, 1) > -1.59 && x(end, 2) < 0.01 && x(end,2) > -0.01 && f(end) < -0.8 && f(end) > -0.82)
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k(j) = 300;
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end
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j = j + 1;
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end
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plotItersOverGamma(n, k, "Iteration for different $\gamma$ values", "figures/LevMar_Iter_o_gamma_" + point + ".png");
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[~, j] = min(k);
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gamma_fixed_step = n(j);
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[x_fixed, f_fixed, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'fixed');
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fprintf('Fixed step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_fixed(end, :), f_fixed(end));
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plotPointsOverContour(x_fixed, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt $\gamma$ = " + gamma_fixed_step, "figures/LevMar_fixed_" + point + ".png");
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[x_minimized, f_minimized, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'minimized');
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fprintf('Minimized f(g): Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_minimized(end, :), f_minimized(end));
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plotPointsOverContour(x_minimized, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt minimized $f(x_k + \gamma_kd_k)$", "figures/LevMar_minimized_" + point + ".png");
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% Armijo Rule
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% Methods tuning
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amijo_beta = 0.4; % typical range: [0.1, 0.8]
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amijo_sigma = 0.1; % typical range: [0.01, 0.3]
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[x_armijo, f_armijo, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'armijo');
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fprintf('Armijo step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_armijo(end, :), f_armijo(end));
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plotPointsOverContour(x_armijo, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/LevMar_armijo_" + point + ".png");
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disp(' ');
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% Compare methods
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plotConvCompare(x_fixed, "Fixed", x_minimized, "Minimized", x_armijo, "Armijo", Xmin, "Convergence compare", "figures/LevMar_compare_" + point + ".png");
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% Point x0 = (1, -1)
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% =========================================================================
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point = 3;
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x0 = [1, -1];
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point_str = "[" + x0(1) + ", " + x0(2) + "]";
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f = fun(-1, 1);
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gf = grad_fun(x0(1), x0(2));
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hf = hessian_fun(x0(1), x0(2));
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ev = eig(hf);
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fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Eigenvalues= [%f, %f], Can use method\n', x0, f, gf, hf, ev);
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% Find the best fixed gamma
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k = zeros(100, 1);
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j = 1;
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n = linspace(0.1, 1.5, 100);
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for g = n
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gamma_fixed_step = g;
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[x, f, k(j)] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'fixed');
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if ~(x(end, 1) < -1.57 && x(end, 1) > -1.59 && x(end, 2) < 0.01 && x(end,2) > -0.01 && f(end) < -0.8 && f(end) > -0.82)
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k(j) = 300;
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end
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j = j + 1;
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end
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[~, j] = min(k);
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gamma_fixed_step = n(j);
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[x_fixed, f_fixed, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'fixed');
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fprintf('Fixed step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_fixed(end, :), f_fixed(end));
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plotPointsOverContour(x_fixed, fun, [-3, 2], [-2, 2], 100, point_str + ": Levenberg-Marquardt $\gamma$ = " + gamma_fixed_step, "figures/LevMar_fixed_" + point + ".png");
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[x_fixed, f_fixed, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'minimized');
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fprintf('Minimized f(g): Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_fixed(end, :), f_fixed(end));
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plotPointsOverContour(x_fixed, fun, [-3, 2], [-2, 2], 100, point_str + ": Levenberg-Marquardt minimized $f(x_k + \gamma_kd_k)$", "figures/LevMar_minimized_" + point + ".png");
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% Armijo Rule
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% Methods tuning
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amijo_beta = 0.4; % typical range: [0.1, 0.8]
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amijo_sigma = 0.1; % typical range: [0.01, 0.3]
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[x_armijo, f_armijo, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'armijo');
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fprintf('Armijo step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_armijo(end, :), f_armijo(end));
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plotPointsOverContour(x_armijo, fun, [-3, 2], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/LevMar_armijo_" + point + ".png");
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disp(' '); |