hw3: done MATLAB
This commit is contained in:
Claudio Maggioni (maggicl)
parent
b29cdc34e3
commit
ca2a294fc7
15 changed files with 85216 additions and 44 deletions
39
Claudio_Maggioni_3/BGFS.m
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39
Claudio_Maggioni_3/BGFS.m
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@ -0,0 +1,39 @@
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function [xk, xs, gs] = BGFS(f, x0, H, max_itr, tol)
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syms x y;
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xk = x0;
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gf = gradient(f);
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fl = matlabFunction(f);
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gfl = matlabFunction(gf);
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tnorm = norm(at(gf, xk), 2);
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xs = zeros(size(x0, 1), max_itr + 1);
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xs(:, 1) = x0;
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gs = zeros(1, max_itr + 1);
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gs(1) = tnorm;
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k = 1;
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I = eye(2);
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while tnorm > tol && k <= max_itr
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grad = gfl(xk(1), xk(2));
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p = -H * grad;
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alpha = backtracking(fl, gfl, p, xk, 1, 0.9, 1e-4);
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xk = xk + alpha * p;
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s = alpha * p;
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y = gfl(xk(1), xk(2)) - grad;
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rho = 1 / (y' * s);
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H = (I - (rho * s * y')) * H * (I - (rho * y * s')) ...
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+ (1 / (y' * s) * (s * s'));
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k = k + 1;
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xs(:, k) = xk;
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tnorm = norm(gfl(xk(1), xk(2)), 2);
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gs(k) = tnorm;
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end
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xs = xs(:, 1:k);
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gs = gs(:, 1:k);
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end
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@ -1,17 +1,11 @@
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syms x y;
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f = (1 - x)^2 + 100 * (y - x^2)^2;
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[x, xs, gs] = gd(f, [0;0], 500, 1e-6, true);
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display(x);
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display(xs);
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display(gs);
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plot(xs(1, :), xs(2, :), '-o');
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function [xk, xs, gs] = gd(f, x0, max_itr, tol, bt)
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function [xk, xs, gs] = GD(f, x0, max_itr, tol, bt)
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syms x y;
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xk = x0;
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gf = gradient(f);
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hf = hessian(f, [x, y]);
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fl = matlabFunction(f);
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gfl = matlabFunction(gf);
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tnorm = norm(at(gf, xk), 2);
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xs = zeros(size(x0, 1), max_itr + 1);
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xs(:, 1) = x0;
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@ -20,16 +14,18 @@ function [xk, xs, gs] = gd(f, x0, max_itr, tol, bt)
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k = 1;
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while tnorm > tol && k <= max_itr
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p = -at(gf, xk);
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H = at(hf, xk);
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p = -gfl(xk(1), xk(2));
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if bt
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alpha = backtracking(f, xk, 1, p, 0.01, 1e-4)
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alpha = backtracking(fl, gfl, p, xk, 1, 0.9, 1e-4);
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else
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alpha = dot(-H * xk, p) / dot(H * p, p)
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alpha = 1;
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end
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xk = xk + alpha * p
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xk = xk + alpha * p;
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k = k + 1;
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xs(:, k) = xk;
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tnorm = norm(gfl(xk(1), xk(2)), 2);
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gs(k) = tnorm;
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end
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xs = xs(:, 1:k);
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@ -1,24 +1,13 @@
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syms x y;
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f = (1 - x)^2 + 100 * (y - x^2)^2;
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[x, xs, gs] = newton(f, [0;0], 50000, 1e-6);
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display(x);
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display(xs);
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display(gs);
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plot(xs(1, :), xs(2, :), '-o');
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function y = at(f, xk)
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syms x y;
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y = double(subs(f, [x, y], [xk(1), xk(2)]));
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end
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function [xk, xs, gs] = newton(f, x0, max_itr, tol)
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function [xk, xs, gs] = Newton(f, x0, max_itr, tol, bt)
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syms x y;
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xk = x0;
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gf = gradient(f);
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hf = hessian(f, [x, y]);
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fl = matlabFunction(f);
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gfl = matlabFunction(gf);
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tnorm = norm(at(gf, xk), 2);
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xs = zeros(size(x0, 1), max_itr + 1);
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xs(:, 1) = x0;
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@ -27,7 +16,13 @@ function [xk, xs, gs] = newton(f, x0, max_itr, tol)
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k = 1;
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while tnorm > tol && k <= max_itr + 1
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xk = xk - (at(hf, xk) \ at(gf, xk));
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pk = - (at(hf, xk) \ at(gf, xk));
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if bt
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alpha = backtracking(fl, gfl, pk, xk, 1, 0.9, 1e-4);
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else
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alpha = 1;
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end
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xk = xk + alpha * pk;
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tnorm = norm(at(gf, xk), 2);
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k = k + 1;
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xs(:, k) = xk;
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@ -36,3 +31,8 @@ function [xk, xs, gs] = newton(f, x0, max_itr, tol)
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xs = xs(:, 1:k);
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gs = gs(:, 1:k);
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end
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function y = at(f, xk)
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syms x y;
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y = double(subs(f, [x, y], [xk(1), xk(2)]));
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end
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@ -1,8 +1,8 @@
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function alpha = backtracking(f, xk, alpha_bar, pk, rho, c)
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alpha = alpha_bar;
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gf = gradient(f);
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while at(f, xk + alpha * pk) > at(f, xk) - c * alpha * dot(at(gf, xk), pk)
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alpha = alpha * rho
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function alpha = backtracking(f, gf, p, x, alpha, rho, c)
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xn = x + alpha * p;
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while f(xn(1), xn(2)) > f(x(1), x(2)) + c * alpha * gf(x(1), x(2))' * p
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alpha = rho * alpha;
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xn = x + alpha * p;
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end
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end
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31
Claudio_Maggioni_3/ex1-3-gd.tex
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31
Claudio_Maggioni_3/ex1-3-gd.tex
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% This file was created by matlab2tikz.
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%
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\definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
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%
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\begin{tikzpicture}
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\begin{axis}[%
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width=4.617in,
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scale only axis,
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unbounded coords=jump,
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xmin=-5e+127,
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xmax=3e+128,
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ymin=0,
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ymax=1.8e+86,
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axis background/.style={fill=white}
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]
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\addplot [color=mycolor1, mark=*, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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13106176003202 2047840800\\
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-9.00508836393827e+41 3.43543698853816e+28\\
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2.92094868655132e+128 1.62183232884673e+86\\
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-inf 1.70638824589317e+259\\
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nan inf\\
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};
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\end{axis}
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\end{tikzpicture}%
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34
Claudio_Maggioni_3/ex1-3-large.tex
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34
Claudio_Maggioni_3/ex1-3-large.tex
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@ -0,0 +1,34 @@
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% This file was created by matlab2tikz.
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%
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\definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
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%
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\begin{axis}[%
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scale only axis,
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unbounded coords=jump,
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xmin=-5e+127,
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xmax=3e+128,
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ymin=0,
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ymax=1.8e+86,
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axis background/.style={fill=white},
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legend style={legend cell align=left, align=left, draw=white!15!black}
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\addplot [color=mycolor1, mark=*, mark options={solid, mycolor1}]
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table[row sep=crcr]{%
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2.92094868655132e+128 1.62183232884673e+86\\
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-inf 1.70638824589317e+259\\
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nan inf\\
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};
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\addlegendentry{GD (alpha=1)}
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\end{axis}
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\end{tikzpicture}%
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21191
Claudio_Maggioni_3/ex1-3.tex
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21191
Claudio_Maggioni_3/ex1-3.tex
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21165
Claudio_Maggioni_3/ex1-4-grad-large.tex
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21165
Claudio_Maggioni_3/ex1-4-grad-large.tex
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21193
Claudio_Maggioni_3/ex1-4-grad.tex
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21193
Claudio_Maggioni_3/ex1-4-grad.tex
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36
Claudio_Maggioni_3/ex1-4-ys-large.tex
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36
Claudio_Maggioni_3/ex1-4-ys-large.tex
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@ -0,0 +1,36 @@
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% This file was created by matlab2tikz.
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%
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\definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
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\begin{axis}[%
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table[row sep=crcr]{%
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6 inf\\
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7 nan\\
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\addlegendentry{GD + backtracking}
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\end{axis}
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\end{tikzpicture}%
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21193
Claudio_Maggioni_3/ex1-4-ys.tex
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21193
Claudio_Maggioni_3/ex1-4-ys.tex
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49
Claudio_Maggioni_3/ex2-3.tex
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49
Claudio_Maggioni_3/ex2-3.tex
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@ -0,0 +1,49 @@
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% This file was created by matlab2tikz.
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%
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51
Claudio_Maggioni_3/ex2-4-grad.tex
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51
Claudio_Maggioni_3/ex2-4-grad.tex
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@ -0,0 +1,51 @@
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% This file was created by matlab2tikz.
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25 1.39324401637967e-05\\
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26 6.48313521241376e-07\\
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};
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\end{axis}
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\end{tikzpicture}%
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51
Claudio_Maggioni_3/ex2-4-ys.tex
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51
Claudio_Maggioni_3/ex2-4-ys.tex
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@ -0,0 +1,51 @@
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% This file was created by matlab2tikz.
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17 0.0111113740774223\\
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18 0.00649240799550402\\
|
||||
19 0.00267559882211434\\
|
||||
20 0.00126486514852597\\
|
||||
21 0.000205881258659308\\
|
||||
22 1.95308875917095e-05\\
|
||||
23 2.42470212441251e-07\\
|
||||
24 4.60765435070989e-10\\
|
||||
25 1.67328301523398e-13\\
|
||||
26 4.18737896779034e-16\\
|
||||
};
|
||||
\end{axis}
|
||||
\end{tikzpicture}%
|
||||
143
Claudio_Maggioni_3/main.m
Normal file
143
Claudio_Maggioni_3/main.m
Normal file
|
|
@ -0,0 +1,143 @@
|
|||
%% Homework 3 - Optimization Methods
|
||||
% Author: Claudio Maggioni
|
||||
%
|
||||
% Note: exercises are not in the right order due to matlab constraints of
|
||||
% functions inside of scripts.
|
||||
|
||||
clc
|
||||
clear
|
||||
close all
|
||||
|
||||
% Set to non-zero to generate LaTeX for graphs
|
||||
enable_m2tikz = 1;
|
||||
if 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
|
||||
Reference in a new issue