From ca2a294fc74a6b18474488f8a35ff5ac827d9781 Mon Sep 17 00:00:00 2001 From: "Claudio Maggioni (maggicl)" Date: Tue, 20 Apr 2021 15:04:00 +0200 Subject: [PATCH] hw3: done MATLAB --- Claudio_Maggioni_3/BGFS.m | 39 + Claudio_Maggioni_3/GD.m | 30 +- Claudio_Maggioni_3/Newton.m | 38 +- Claudio_Maggioni_3/backtracking.m | 16 +- Claudio_Maggioni_3/ex1-3-gd.tex | 31 + Claudio_Maggioni_3/ex1-3-large.tex | 34 + Claudio_Maggioni_3/ex1-3.tex | 21191 +++++++++++++++++++++ Claudio_Maggioni_3/ex1-4-grad-large.tex | 21165 +++++++++++++++++++++ Claudio_Maggioni_3/ex1-4-grad.tex | 21193 ++++++++++++++++++++++ Claudio_Maggioni_3/ex1-4-ys-large.tex | 36 + Claudio_Maggioni_3/ex1-4-ys.tex | 21193 ++++++++++++++++++++++ Claudio_Maggioni_3/ex2-3.tex | 49 + Claudio_Maggioni_3/ex2-4-grad.tex | 51 + Claudio_Maggioni_3/ex2-4-ys.tex | 51 + Claudio_Maggioni_3/main.m | 143 + 15 files changed, 85216 insertions(+), 44 deletions(-) create mode 100644 Claudio_Maggioni_3/BGFS.m create mode 100644 Claudio_Maggioni_3/ex1-3-gd.tex create mode 100644 Claudio_Maggioni_3/ex1-3-large.tex create mode 100644 Claudio_Maggioni_3/ex1-3.tex create mode 100644 Claudio_Maggioni_3/ex1-4-grad-large.tex create mode 100644 Claudio_Maggioni_3/ex1-4-grad.tex create mode 100644 Claudio_Maggioni_3/ex1-4-ys-large.tex create mode 100644 Claudio_Maggioni_3/ex1-4-ys.tex create mode 100644 Claudio_Maggioni_3/ex2-3.tex create mode 100644 Claudio_Maggioni_3/ex2-4-grad.tex create mode 100644 Claudio_Maggioni_3/ex2-4-ys.tex create mode 100644 Claudio_Maggioni_3/main.m diff --git a/Claudio_Maggioni_3/BGFS.m b/Claudio_Maggioni_3/BGFS.m new file mode 100644 index 0000000..5ff83a9 --- /dev/null +++ b/Claudio_Maggioni_3/BGFS.m @@ -0,0 +1,39 @@ +function [xk, xs, gs] = BGFS(f, x0, H, max_itr, tol) + syms x y; + xk = x0; + gf = gradient(f); + + fl = matlabFunction(f); + gfl = matlabFunction(gf); + + tnorm = norm(at(gf, xk), 2); + xs = zeros(size(x0, 1), max_itr + 1); + xs(:, 1) = x0; + gs = zeros(1, max_itr + 1); + gs(1) = tnorm; + k = 1; + + I = eye(2); + + while tnorm > tol && k <= max_itr + grad = gfl(xk(1), xk(2)); + p = -H * grad; + + alpha = backtracking(fl, gfl, p, xk, 1, 0.9, 1e-4); + xk = xk + alpha * p; + s = alpha * p; + y = gfl(xk(1), xk(2)) - grad; + + rho = 1 / (y' * s); + + H = (I - (rho * s * y')) * H * (I - (rho * y * s')) ... + + (1 / (y' * s) * (s * s')); + + k = k + 1; + xs(:, k) = xk; + tnorm = norm(gfl(xk(1), xk(2)), 2); + gs(k) = tnorm; + end + xs = xs(:, 1:k); + gs = gs(:, 1:k); +end \ No newline at end of file diff --git a/Claudio_Maggioni_3/GD.m b/Claudio_Maggioni_3/GD.m index 6d8a937..55a43af 100644 --- a/Claudio_Maggioni_3/GD.m +++ b/Claudio_Maggioni_3/GD.m @@ -1,17 +1,11 @@ -syms x y; -f = (1 - x)^2 + 100 * (y - x^2)^2; - -[x, xs, gs] = gd(f, [0;0], 500, 1e-6, true); -display(x); -display(xs); -display(gs); -plot(xs(1, :), xs(2, :), '-o'); - -function [xk, xs, gs] = gd(f, x0, max_itr, tol, bt) +function [xk, xs, gs] = GD(f, x0, max_itr, tol, bt) syms x y; xk = x0; gf = gradient(f); - hf = hessian(f, [x, y]); + + fl = matlabFunction(f); + gfl = matlabFunction(gf); + tnorm = norm(at(gf, xk), 2); xs = zeros(size(x0, 1), max_itr + 1); xs(:, 1) = x0; @@ -20,18 +14,20 @@ function [xk, xs, gs] = gd(f, x0, max_itr, tol, bt) k = 1; while tnorm > tol && k <= max_itr - p = -at(gf, xk); - H = at(hf, xk); + p = -gfl(xk(1), xk(2)); + if bt - alpha = backtracking(f, xk, 1, p, 0.01, 1e-4) + alpha = backtracking(fl, gfl, p, xk, 1, 0.9, 1e-4); else - alpha = dot(-H * xk, p) / dot(H * p, p) + alpha = 1; end - xk = xk + alpha * p + + xk = xk + alpha * p; k = k + 1; xs(:, k) = xk; + tnorm = norm(gfl(xk(1), xk(2)), 2); gs(k) = tnorm; end xs = xs(:, 1:k); gs = gs(:, 1:k); -end \ No newline at end of file +end diff --git a/Claudio_Maggioni_3/Newton.m b/Claudio_Maggioni_3/Newton.m index 2e59bd9..1874c71 100644 --- a/Claudio_Maggioni_3/Newton.m +++ b/Claudio_Maggioni_3/Newton.m @@ -1,24 +1,13 @@ - - -syms x y; -f = (1 - x)^2 + 100 * (y - x^2)^2; - -[x, xs, gs] = newton(f, [0;0], 50000, 1e-6); -display(x); -display(xs); -display(gs); -plot(xs(1, :), xs(2, :), '-o'); - -function y = at(f, xk) - syms x y; - y = double(subs(f, [x, y], [xk(1), xk(2)])); -end - -function [xk, xs, gs] = newton(f, x0, max_itr, tol) +function [xk, xs, gs] = Newton(f, x0, max_itr, tol, bt) syms x y; xk = x0; + gf = gradient(f); hf = hessian(f, [x, y]); + + fl = matlabFunction(f); + gfl = matlabFunction(gf); + tnorm = norm(at(gf, xk), 2); xs = zeros(size(x0, 1), max_itr + 1); xs(:, 1) = x0; @@ -27,7 +16,13 @@ function [xk, xs, gs] = newton(f, x0, max_itr, tol) k = 1; while tnorm > tol && k <= max_itr + 1 - xk = xk - (at(hf, xk) \ at(gf, xk)); + pk = - (at(hf, xk) \ at(gf, xk)); + if bt + alpha = backtracking(fl, gfl, pk, xk, 1, 0.9, 1e-4); + else + alpha = 1; + end + xk = xk + alpha * pk; tnorm = norm(at(gf, xk), 2); k = k + 1; xs(:, k) = xk; @@ -35,4 +30,9 @@ function [xk, xs, gs] = newton(f, x0, max_itr, tol) end xs = xs(:, 1:k); gs = gs(:, 1:k); -end \ No newline at end of file +end + +function y = at(f, xk) + syms x y; + y = double(subs(f, [x, y], [xk(1), xk(2)])); +end diff --git a/Claudio_Maggioni_3/backtracking.m b/Claudio_Maggioni_3/backtracking.m index adecdf7..81f4d6a 100644 --- a/Claudio_Maggioni_3/backtracking.m +++ b/Claudio_Maggioni_3/backtracking.m @@ -1,8 +1,8 @@ -function alpha = backtracking(f, xk, alpha_bar, pk, rho, c) - alpha = alpha_bar; - gf = gradient(f); - while at(f, xk + alpha * pk) > at(f, xk) - c * alpha * dot(at(gf, xk), pk) - alpha = alpha * rho - end -end - \ No newline at end of file +function alpha = backtracking(f, gf, p, x, alpha, rho, c) + xn = x + alpha * p; + + while f(xn(1), xn(2)) > f(x(1), x(2)) + c * alpha * gf(x(1), x(2))' * p + alpha = rho * alpha; + xn = x + alpha * p; + end +end \ No newline at end of file diff --git a/Claudio_Maggioni_3/ex1-3-gd.tex b/Claudio_Maggioni_3/ex1-3-gd.tex new file mode 100644 index 0000000..d8da26b --- /dev/null +++ b/Claudio_Maggioni_3/ex1-3-gd.tex @@ -0,0 +1,31 @@ +% This file was 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+3 2.95055682555403e+54\\ +4 6.57585025723101e+169\\ +5 inf\\ +6 inf\\ +7 nan\\ +}; +\addlegendentry{GD (alpha=1)} + +\end{axis} +\end{tikzpicture}% \ No newline at end of file diff --git a/Claudio_Maggioni_3/ex2-3.tex b/Claudio_Maggioni_3/ex2-3.tex new file mode 100644 index 0000000..641e32c --- /dev/null +++ b/Claudio_Maggioni_3/ex2-3.tex @@ -0,0 +1,49 @@ +% This file was created by matlab2tikz. +% +\definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% +% +\begin{tikzpicture} + +\begin{axis}[% +width=4.617in, +height=3.641in, +at={(0.774in,0.491in)}, +scale only axis, +xmin=0, +xmax=1.2, +ymin=0, +ymax=1.2, +axis background/.style={fill=white} +] +\addplot [color=mycolor1, mark=*, mark options={solid, mycolor1}, forget plot] + table[row sep=crcr]{% +0 0\\ +0.243153309181139 0\\ +0.385574283516823 0.0768839344022268\\ +0.406269115526804 0.228605903873976\\ +0.454394143039344 0.204136926895131\\ +0.498557614055172 0.23892892868596\\ +0.620979332605692 0.353718595337535\\ 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b/Claudio_Maggioni_3/ex2-4-grad.tex @@ -0,0 +1,51 @@ +% This file was created by matlab2tikz. +% +\definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}% +% +\begin{tikzpicture} + +\begin{axis}[% +width=4.617in, +height=3.641in, +at={(0.774in,0.491in)}, +scale only axis, +xmin=0, +xmax=30, +ymode=log, +ymin=1e-08, +ymax=100, +yminorticks=true, +axis background/.style={fill=white} +] +\addplot [color=mycolor1, mark=*, mark options={solid, mycolor1}, forget plot] + table[row sep=crcr]{% +0 2\\ +1 12.5607978484932\\ +2 17.4065065901439\\ +3 17.1507084737438\\ +4 0.814005132835468\\ +5 2.13360285320214\\ +6 9.59328224461729\\ +7 1.83406220417747\\ +8 0.960051885187564\\ +9 10.7994609810302\\ +10 0.823371092958359\\ +11 0.399989068016143\\ +12 8.26772606877548\\ +13 0.674513531933847\\ +14 0.994977406942212\\ +15 6.50941712025139\\ +16 0.514849723958323\\ +17 0.619141853913074\\ +18 3.17771232534436\\ +19 0.338310420250268\\ +20 0.223907430737766\\ +21 0.573528649173901\\ +22 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enable_m2tikz + addpath /home/claudio/git/matlab2tikz/src +else + matlab2tikz = @(a,b,c) 0; +end + +syms x y; +f = (1 - x)^2 + 100 * (y - x^2)^2; +global fl +fl = matlabFunction(f); + +%% 1.3 - Newton and GD solutions and energy plot + +[x1, xs1, gs1] = Newton(f, [0;0], 50000, 1e-6, true); +plot(xs1(1, :), xs1(2, :), 'Marker', '.'); +fprintf("Newton backtracking: it=%d\n", size(xs1, 2)-1); +xlim([-0.01 1.01]) +ylim([-0.01 1.01]) +hold on; +[x2, xs2, gs2] = Newton(f, [0;0], 50000, 1e-6, false); +plot(xs2(1, :), xs2(2, :), 'Marker', '.'); +fprintf("Newton: it=%d\n", size(xs2, 2)-1); + +[x3, xs3, gs3] = GD(f, [0;0], 50000, 1e-6, true); +plot(xs3(1, :), xs3(2, :), 'Marker', '.'); +fprintf("GD backtracking: it=%d\n", size(xs3, 2)-1); + +[x4, xs4, gs4] = GD(f, [0;0], 50000, 1e-6, false); +plot(xs4(1, :), xs4(2, :), 'Marker', '.'); +fprintf("GD: it=%d\n", size(xs4, 2)-1); +hold off; +legend('Newton + backtracking', 'Newton', 'GD + backtracking', 'GD (alpha=1)') +sgtitle("Iterations of Newton and Gradient descent methods over 2D energy landscape"); +matlab2tikz('showInfo', false, './ex1-3.tex'); + +figure; +plot(xs4(1, :), xs4(2, :), 'Marker', '.'); +legend('GD (alpha=1)') +sgtitle("Iterations of Newton and Gradient descent methods over 2D energy landscape"); +matlab2tikz('showInfo', false, './ex1-3-large.tex'); + +%% 2.3 - BGFS solution and energy plot + +figure; +[x5, xs5, gs5] = BGFS(f, [0;0], eye(2), 50000, 1e-6); +xlim([-0.01 1.01]) +ylim([-0.01 1.01]) +plot(xs5(1, :), xs5(2, :), 'Marker', '.'); +fprintf("BGFS backtracking: it=%d\n", size(xs5, 2)-1); + +sgtitle("Iterations of BGFS method over 2D energy landscape"); +matlab2tikz('showInfo', false, './ex2-3.tex'); + +%% 1.4 - Newton and GD gradient norm log + +figure; +semilogy(0:size(xs1, 2)-1, gs1, 'Marker', '.'); +ylim([5e-10, 1e12]); +xlim([-1, 30]); +hold on +semilogy(0:size(xs2, 2)-1, gs2, 'Marker', '.'); +semilogy(0:size(xs3, 2)-1, gs3, 'Marker', '.'); +semilogy(0:size(xs4, 2)-1, gs4, 'Marker', '.'); +hold off + +legend('Newton + backtracking', 'Newton', 'GD + backtracking', 'GD (alpha=1)') +sgtitle("Gradient norm w.r.t. iteration number for Newton and GD methods"); +matlab2tikz('showInfo', false, './ex1-4-grad.tex'); + +figure; +semilogy(0:size(xs3, 2)-1, gs3, 'Marker', '.'); +ylim([1e-7, 1e10]); +hold on +semilogy(0:size(xs4, 2)-1, gs4, 'Marker', '.'); +hold off + +legend('GD + backtracking', 'GD (alpha=1)') +sgtitle("Gradient norm w.r.t. iteration number for Newton and GD methods"); +matlab2tikz('showInfo', false, './ex1-4-grad-large.tex'); + +figure; +ys1 = funvalues(xs1); +ys2 = funvalues(xs2); +ys3 = funvalues(xs3); +ys4 = funvalues(xs4); +ys5 = funvalues(xs5); +semilogy(0:size(xs1, 2)-1, ys1, 'Marker', '.'); +ylim([5e-19, 1e12]); +xlim([-1, 30]); +hold on +semilogy(0:size(xs2, 2)-1, ys2, 'Marker', '.'); +semilogy(0:size(xs3, 2)-1, ys3, 'Marker', '.'); +semilogy(0:size(xs4, 2)-1, ys4, 'Marker', '.'); +hold off + +legend('Newton + backtracking', 'Newton', 'GD + backtracking', 'GD (alpha=1)') +sgtitle("Objective function value w.r.t. iteration number for Newton and GD methods"); +matlab2tikz('showInfo', false, './ex1-4-ys.tex'); + +figure; + +semilogy(0:size(xs3, 2)-1, ys3, 'Marker', '.'); +semilogy(0:size(xs4, 2)-1, ys4, 'Marker', '.'); + +legend('GD + backtracking', 'GD (alpha=1)') +sgtitle("Objective function value w.r.t. iteration number for Newton and GD methods"); +matlab2tikz('showInfo', false, './ex1-4-ys-large.tex'); + +%% 2.4 - BGFS gradient norms plot + +figure; +semilogy(0:size(xs5, 2)-1, gs5, 'Marker', '.'); + +sgtitle("Gradient norm w.r.t. iteration number for BGFS method"); +matlab2tikz('showInfo', false, './ex2-4-grad.tex'); + +%% 2.4 - BGFS objective values plot + +figure; +semilogy(0:size(xs5, 2)-1, ys5, 'Marker', '.'); + +sgtitle("Objective function value w.r.t. iteration number for BGFS methods"); +matlab2tikz('showInfo', false, './ex2-4-ys.tex'); + +function ys = funvalues(xs) + ys = zeros(1, size(xs, 2)); + global fl + for i = 1:size(xs, 2) + ys(i) = fl(xs(1,i), xs(2,i)); + end +end