177 lines
5.1 KiB
Plaintext
177 lines
5.1 KiB
Plaintext
%% Homework 3 - Optimization Methods
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% Author: Claudio Maggioni
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%
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% Note: exercises are not in the right order due to matlab constraints of
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% functions inside of scripts.
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clc
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clear
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close all
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% Set to non-zero to generate LaTeX for graphs
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enable_m2tikz = 0;
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if enable_m2tikz
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addpath /home/claudio/git/matlab2tikz/src
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else
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matlab2tikz = @(a,b,c) 0;
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end
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syms x y;
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f = (1 - x)^2 + 100 * (y - x^2)^2;
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global fl
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fl = matlabFunction(f);
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%% 1.3 - Newton and GD solutions and energy plot
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[x1, xs1, gs1] = Newton(f, [0;0], 50000, 1e-6, true);
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plot(xs1(1, :), xs1(2, :), 'Marker', '.');
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fprintf("Newton backtracking: it=%d\n", size(xs1, 2)-1);
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xlim([-0.01 1.01])
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ylim([-0.01 1.01])
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hold on;
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[x2, xs2, gs2] = Newton(f, [0;0], 50000, 1e-6, false);
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plot(xs2(1, :), xs2(2, :), 'Marker', '.');
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fprintf("Newton: it=%d\n", size(xs2, 2)-1);
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[x3, xs3, gs3] = GD(f, [0;0], 50000, 1e-6, true);
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plot(xs3(1, :), xs3(2, :), 'Marker', '.');
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fprintf("GD backtracking: it=%d\n", size(xs3, 2)-1);
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[x4, xs4, gs4] = GD(f, [0;0], 50000, 1e-6, false);
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plot(xs4(1, :), xs4(2, :), 'Marker', '.');
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fprintf("GD: it=%d\n", size(xs4, 2)-1);
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hold off;
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legend('Newton + backtracking', 'Newton', 'GD + backtracking', 'GD (alpha=1)')
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sgtitle("Iterations of Newton and Gradient descent methods over 2D energy landscape");
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matlab2tikz('showInfo', false, './ex1-3.tex');
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figure;
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plot(xs4(1, :), xs4(2, :), 'Marker', '.');
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legend('GD (alpha=1)')
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sgtitle("Iterations of Newton and Gradient descent methods over 2D energy landscape");
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matlab2tikz('showInfo', false, './ex1-3-large.tex');
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%% 2.3 - BGFS solution and energy plot
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figure;
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[x5, xs5, gs5] = BGFS(f, [0;0], eye(2), 50000, 1e-6);
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xlim([-0.01 1.01])
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ylim([-0.01 1.01])
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plot(xs5(1, :), xs5(2, :), 'Marker', '.');
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fprintf("BGFS backtracking: it=%d\n", size(xs5, 2)-1);
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sgtitle("Iterations of BGFS method over 2D energy landscape");
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matlab2tikz('showInfo', false, './ex2-3.tex');
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%% 1.4 - Newton and GD gradient norm log
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figure;
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semilogy(0:size(xs1, 2)-1, gs1, 'Marker', '.');
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ylim([5e-10, 1e12]);
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xlim([-1, 30]);
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hold on
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semilogy(0:size(xs2, 2)-1, gs2, 'Marker', '.');
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semilogy(0:size(xs3, 2)-1, gs3, 'Marker', '.');
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semilogy(0:size(xs4, 2)-1, gs4, 'Marker', '.');
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hold off
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legend('Newton + backtracking', 'Newton', 'GD + backtracking', 'GD (alpha=1)')
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sgtitle("Gradient norm w.r.t. iteration number for Newton and GD methods");
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matlab2tikz('showInfo', false, './ex1-4-grad.tex');
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figure;
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plot(0:size(xs1, 2)-1, gs1, 'Marker', '.');
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ylim([5e-10, 25]);
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xlim([-1, 30]);
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hold on
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plot(0:size(xs2, 2)-1, gs2, 'Marker', '.');
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plot(0:size(xs3, 2)-1, gs3, 'Marker', '.');
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plot(0:size(xs4, 2)-1, gs4, 'Marker', '.');
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hold off
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legend('Newton + backtracking', 'Newton', 'GD + backtracking', 'GD (alpha=1)')
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sgtitle("Gradient norm w.r.t. iteration number for Newton and GD methods");
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matlab2tikz('showInfo', false, './ex1-4-grad.tex');
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figure;
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semilogy(0:size(xs3, 2)-1, gs3, 'Marker', '.');
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ylim([1e-7, 1e10]);
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hold on
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semilogy(0:size(xs4, 2)-1, gs4, 'Marker', '.');
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hold off
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legend('GD + backtracking', 'GD (alpha=1)')
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sgtitle("Gradient norm w.r.t. iteration number for Newton and GD methods");
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matlab2tikz('showInfo', false, './ex1-4-grad-large.tex');
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figure;
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ys1 = funvalues(xs1);
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ys2 = funvalues(xs2);
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ys3 = funvalues(xs3);
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ys4 = funvalues(xs4);
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ys5 = funvalues(xs5);
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semilogy(0:size(xs1, 2)-1, ys1, 'Marker', '.');
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ylim([5e-19, 1e12]);
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xlim([-1, 30]);
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hold on
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semilogy(0:size(xs2, 2)-1, ys2, 'Marker', '.');
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semilogy(0:size(xs3, 2)-1, ys3, 'Marker', '.');
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semilogy(0:size(xs4, 2)-1, ys4, 'Marker', '.');
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hold off
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legend('Newton + backtracking', 'Newton', 'GD + backtracking', 'GD (alpha=1)')
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sgtitle("Objective function value w.r.t. iteration number for Newton and GD methods");
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matlab2tikz('showInfo', false, './ex1-4-ys.tex');
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ys1 = funvalues(xs1);
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ys2 = funvalues(xs2);
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ys3 = funvalues(xs3);
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ys4 = funvalues(xs4);
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ys5 = funvalues(xs5);
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semilogy(0:size(xs1, 2)-1, ys1, 'Marker', '.');
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ylim([5e-19, 20]);
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xlim([-1, 30]);
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hold on
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semilogy(0:size(xs2, 2)-1, ys2, 'Marker', '.');
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semilogy(0:size(xs3, 2)-1, ys3, 'Marker', '.');
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semilogy(0:size(xs4, 2)-1, ys4, 'Marker', '.');
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hold off
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legend('Newton + backtracking', 'Newton', 'GD + backtracking', 'GD (alpha=1)')
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sgtitle("Objective function value w.r.t. iteration number for Newton and GD methods");
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matlab2tikz('showInfo', false, './ex1-4-ys.tex');
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figure;
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semilogy(0:size(xs3, 2)-1, ys3, 'Marker', '.');
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semilogy(0:size(xs4, 2)-1, ys4, 'Marker', '.');
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legend('GD + backtracking', 'GD (alpha=1)')
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sgtitle("Objective function value w.r.t. iteration number for Newton and GD methods");
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matlab2tikz('showInfo', false, './ex1-4-ys-large.tex');
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%% 2.4 - BGFS gradient norms plot
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figure;
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semilogy(0:size(xs5, 2)-1, gs5, 'Marker', '.');
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sgtitle("Gradient norm w.r.t. iteration number for BGFS method");
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matlab2tikz('showInfo', false, './ex2-4-grad.tex');
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%% 2.4 - BGFS objective values plot
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figure;
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semilogy(0:size(xs5, 2)-1, ys5, 'Marker', '.');
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sgtitle("Objective function value w.r.t. iteration number for BGFS methods");
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matlab2tikz('showInfo', false, './ex2-4-ys.tex');
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function ys = funvalues(xs)
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ys = zeros(1, size(xs, 2));
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global fl
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for i = 1:size(xs, 2)
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ys(i) = fl(xs(1,i), xs(2,i));
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end
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end
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