version 2

Work2/scripts/GivenEnv.m

@@ -15,6 +15,10 @@ fun         = matlabFunction(fexpr, 'Vars', [x, y]);        % Function
 grad_fun    = matlabFunction(grad_fexpr, 'Vars', [x, y]); % Gradient
 hessian_fun = matlabFunction(hessian_fexpr, 'Vars', [x, y]);  % 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])

Work2/scripts/Script_1_Plots.m

@@ -8,7 +8,7 @@ GivenEnv
 %
 
 % 3d plot the function
-plot3dFun(fun, [-3, 3], [-3, 3], 100, title_fun, "figures/FunctionPlot.png");
+plot3dFun(fun, [-3, 3], [-3, 3], 100, title_fun, "figures/Plot_Function.png");
 
 % Plot isobaric lines
-plotContour(fun, [-3, 3], [-3, 3], 100, title_fun, "figures/FunctionContour.png");
+plotContour(fun, [-3, 3], [-3, 3], 100, title_fun, "figures/Plot_Contour.png");

Work2/scripts/Script_2_Steepest_descent.m

@@ -63,6 +63,9 @@ fprintf('Armijo step:    Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f),
 plotPointsOverContour(x_armijo, fun, [-2, 0], [-2, 2], 100, point_str + ": Steepest descent Armijo method", "figures/StDes_armijo_" + point + ".png");  
 disp(' ');
 
+% Compare methods
+plotConvCompare(x_fixed, "Fixed", x_minimized, "Minimized", x_armijo, "Armijo", Xmin, "Convergence compare", "figures/StDes_compare_" + point + ".png");
+
 % Point x0 = (1, -1)
 % =========================================================================
 point = 3;
@@ -111,4 +114,3 @@ fprintf('Armijo step:    Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f),
 plotPointsOverContour(x_armijo, fun, [-1, 2], [-2, 2], 100, point_str + ": Steepest descent Armijo method", "figures/StDes_armijo_" + point + ".png");
 
 
-

Work2/scripts/Script_3_Newton.m

@@ -40,4 +40,3 @@ ev = eig(hf);
 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);
 
 
-

Work2/scripts/Script_4_LevMar.m

@@ -43,6 +43,7 @@ for g = n
     end
     j = j + 1;
 end
+plotItersOverGamma(n, k, "Iteration for different $\gamma$ values", "figures/LevMar_Iter_o_gamma_" + point + ".png");
 
 [~, j] = min(k);
 gamma_fixed_step = n(j);
@@ -51,9 +52,9 @@ gamma_fixed_step = n(j);
 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));
 plotPointsOverContour(x_fixed, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt $\gamma$ = " + gamma_fixed_step, "figures/LevMar_fixed_" + point + ".png");
 
-[x_fixed, f_fixed, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'minimized');
-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));
-plotPointsOverContour(x_fixed, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt minimized $f(x_k + \gamma_kd_k)$", "figures/LevMar_minimized_" + point + ".png");
+[x_minimized, f_minimized, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'minimized');
+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));
+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");
 
 % Armijo Rule
 
@@ -63,9 +64,11 @@ amijo_sigma = 0.1; % typical range: [0.01, 0.3]
 
 [x_armijo, f_armijo, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'armijo');
 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));
-plotPointsOverContour(x_armijo, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/StDes_armijo_" + point + ".png");  
+plotPointsOverContour(x_armijo, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/LevMar_armijo_" + point + ".png");  
 disp(' ');
 
+% Compare methods
+plotConvCompare(x_fixed, "Fixed", x_minimized, "Minimized", x_armijo, "Armijo", Xmin, "Convergence compare", "figures/LevMar_compare_" + point + ".png");
 
 % Point x0 = (1, -1)
 % =========================================================================
@@ -112,5 +115,5 @@ amijo_sigma = 0.1; % typical range: [0.01, 0.3]
 
 [x_armijo, f_armijo, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'armijo');
 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));
-plotPointsOverContour(x_armijo, fun, [-3, 2], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/StDes_armijo_" + point + ".png");  
-disp(' ');
+plotPointsOverContour(x_armijo, fun, [-3, 2], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/LevMar_armijo_" + point + ".png");  
+disp(' ');

Work2/scripts/figures/ReadME.md

@@ -1 +0,0 @@
-File to keep directory

Work2/scripts/fmin_bisection.m

@@ -46,4 +46,4 @@ while b(k) - a(k) > lambda
     end
 end
 
-end
+end

Work2/scripts/gamma_armijo.m

@@ -30,4 +30,4 @@ function [gamma] = gamma_armijo(f, grad_f, dk, xk)
         end
     end
 
-end
+end

Work2/scripts/gamma_minimized.m

@@ -21,4 +21,4 @@ function [gamma] = gamma_minimized(f, ~, dk, xk)
 
     % find g that minimizes fmin
     %gamma = fminbnd(g, 0, 1);
-end
+end

Work2/scripts/method_lev_mar.m

@@ -2,7 +2,9 @@ function [x_vals, f_vals, k] = method_lev_mar(f, grad_f, hessian_f, e, xk, tol,
     % f:         Objective function
     % grad_f:    Gradient of the function
     % hessian_f: Hessian of the function
-    % e:         mu offset for hessian damping H' = H_k + mI
+    % e:         Offset for hessian damping Hk' = Hk + mI
+    %            - when:  Hk not positive defined
+    %            - Where: m = abs(min(eig(Hk))) + e
     % xk:        Initial point [xk, yk]
     % tol:       Tolerance for stopping criterion
     % max_iter:  Maximum number of iterations
@@ -35,10 +37,10 @@ function [x_vals, f_vals, k] = method_lev_mar(f, grad_f, hessian_f, e, xk, tol,
         % Check if hessian is not positive defined
         lmin = min(eig(hess));
         if lmin <= 0
+            % Select m with offset to stear hess to positive eigenvalues
             m = abs(lmin) + e;
             mI = m * eye(size(hess));
-            nev = eig(hess + mI);
-            if min(nev) <= 0
+            if min(eig(hess + mI)) <= 0 % Fail-check
                 warning('Can not normalize hessian matrix.');
             end
         end
@@ -56,4 +58,4 @@ function [x_vals, f_vals, k] = method_lev_mar(f, grad_f, hessian_f, e, xk, tol,
         x_vals = [x_vals; x_next]; % Store values
         f_vals = [f_vals; f_next]; % Store function values
     end
-end
+end

Work2/scripts/method_newton.m

@@ -44,4 +44,4 @@ function [x_vals, f_vals, k] = method_newton(f, grad_f, hessian_f, xk, tol, max_
         x_vals = [x_vals; x_next]; % Store values
         f_vals = [f_vals; f_next]; % Store function values
     end
-end
+end

Work2/scripts/method_steepest_descent.m

@@ -40,4 +40,4 @@ function [x_vals, f_vals, k] = method_steepest_descent(f, grad_f, xk, tol, max_i
         x_vals = [x_vals; x_next]; % Store values
         f_vals = [f_vals; f_next]; % Store function values
     end
-end
+end

Work2/scripts/plotConvCompare.m

@@ -0,0 +1,54 @@
+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