hw3: done MATLAB
This commit is contained in:
Claudio Maggioni (maggicl)
parent
b29cdc34e3
commit
ca2a294fc7
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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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@ -20,18 +14,20 @@ 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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gs = gs(:, 1:k);
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end
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end
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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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@ -35,4 +30,9 @@ function [xk, xs, gs] = newton(f, x0, max_itr, tol)
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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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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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end
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end
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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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@ -0,0 +1,31 @@
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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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0 0\\
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2 0\\
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-3200 800\\
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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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@ -0,0 +1,34 @@
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% This file was created by matlab2tikz.
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%
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%
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scale only axis,
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ymin=0,
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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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]
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table[row sep=crcr]{%
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0 0\\
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2 0\\
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-3200 800\\
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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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\addlegendentry{GD (alpha=1)}
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\end{axis}
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\end{tikzpicture}%
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File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
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@ -0,0 +1,36 @@
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% This file was created by matlab2tikz.
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scale only axis,
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ymax=1e+200,
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yminorticks=true,
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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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0 1\\
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1 1601\\
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2 1.04841216742464e+16\\
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3 2.95055682555403e+54\\
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4 6.57585025723101e+169\\
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5 inf\\
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6 inf\\
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7 nan\\
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};
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\addlegendentry{GD + backtracking}
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File diff suppressed because it is too large
Load Diff
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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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scale only axis,
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axis background/.style={fill=white}
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table[row sep=crcr]{%
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0 0\\
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0.243153309181139 0\\
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0.385574283516823 0.0768839344022268\\
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0.406269115526804 0.228605903873976\\
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0.454394143039344 0.204136926895131\\
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0.498557614055172 0.23892892868596\\
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0.620979332605692 0.353718595337535\\
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0.57936460358975 0.327866742545248\\
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0.616121196551721 0.375114354375997\\
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0.776077091062027 0.572038852536037\\
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0.697592763590646 0.487371411589351\\
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0.737499732511574 0.541927499854004\\
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0.851186743684398 0.702934657899626\\
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0.792280441629556 0.62507087775202\\
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0.817946864998747 0.665679934708557\\
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0.92248773523621 0.835150610480633\\
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0.873933239049152 0.761975932983517\\
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0.896369084677219 0.801548786456236\\
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0.968508670106431 0.930592370870331\\
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0.94922990019977 0.90004747521168\\
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0.965038718894672 0.930647241632658\\
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0.994184674394427 0.98709143799204\\
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0.995726402584417 0.991358496483784\\
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0.999560495084773 0.999098978460342\\
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0.999985154690863 0.999968759175614\\
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0.999999722333829 0.999999474706011\\
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1.00000001502439 1.00000002865953\\
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};
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@ -0,0 +1,51 @@
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%
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%
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0 2\\
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1 12.5607978484932\\
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2 17.4065065901439\\
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3 17.1507084737438\\
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4 0.814005132835468\\
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5 2.13360285320214\\
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6 9.59328224461729\\
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7 1.83406220417747\\
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8 0.960051885187564\\
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9 10.7994609810302\\
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11 0.399989068016143\\
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12 8.26772606877548\\
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13 0.674513531933847\\
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14 0.994977406942212\\
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15 6.50941712025139\\
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16 0.514849723958323\\
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17 0.619141853913074\\
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18 3.17771232534436\\
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21 0.573528649173901\\
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22 0.0427061378580603\\
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23 0.00914914389448632\\
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24 0.000666939747400779\\
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25 1.39324401637967e-05\\
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26 6.48313521241376e-07\\
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};
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@ -0,0 +1,51 @@
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% This file was created by matlab2tikz.
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%
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%
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\begin{axis}[%
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0 1\\
|
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1 0.922376114248889\\
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2 0.89280739362912\\
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||||
3 0.756393258911934\\
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||||
4 0.298231959620787\\
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5 0.260719631500909\\
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7 0.18301283597033\\
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8 0.149379820900806\\
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9 0.141688856117852\\
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10 0.0915042691116316\\
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11 0.0692977795197531\\
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12 0.0687332177822504\\
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13 0.0438430135856068\\
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14 0.0342703823600005\\
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15 0.0310765754933112\\
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16 0.0162108702605621\\
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17 0.0111113740774223\\
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18 0.00649240799550402\\
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19 0.00267559882211434\\
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20 0.00126486514852597\\
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21 0.000205881258659308\\
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22 1.95308875917095e-05\\
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23 2.42470212441251e-07\\
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24 4.60765435070989e-10\\
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25 1.67328301523398e-13\\
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26 4.18737896779034e-16\\
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};
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\end{tikzpicture}%
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@ -0,0 +1,143 @@
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%% 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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|
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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
|
||||
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
|
||||
Loading…
Reference in New Issue