hw1: done report for now
This commit is contained in:
Claudio Maggioni (maggicl)
parent
59094974b1
commit
2fd664a38e
4 changed files with 87 additions and 44 deletions
25
hw1/ex3.m
25
hw1/ex3.m
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@ -9,8 +9,8 @@ clc
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close all
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format short
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colw = 3;
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colh = 4;
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colw = 5;
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colh = 2;
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%% Exercise 3.1
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@ -42,7 +42,7 @@ for u = 1:10
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subplot(colh, colw, u);
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h = surf(xaxis, yaxis, Zs{u});
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set(h,'LineStyle','none');
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title(sprintf("\nu=%d", u));
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title(sprintf("u=%d", u));
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end
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sgtitle("Surf plots");
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@ -52,14 +52,17 @@ matlab2tikz('showInfo', false, './surf.tex')
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figure
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yi = zeros(30, 25);
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ni = zeros(30, 25);
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% max iterations
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c = 100;
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yi = zeros(30, c);
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ni = zeros(30, c);
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its = zeros(30, 1);
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for u = 1:10
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subplot(colh, colw, u);
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contour(xaxis, yaxis, Zs{u}, 10);
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title(sprintf("\nu=%d", u));
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title(sprintf("u=%d", u));
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%% Exercise 3.3
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@ -80,12 +83,12 @@ for u = 1:10
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x = x0;
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i = 1;
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xi = zeros(2, 25);
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xi = zeros(2, c);
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xi(:, 1) = x0;
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yi(ri, 1) = subs(f, [sx sy], x0');
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while true
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while i <= c
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p = -1 * double(subs(g, [sx sy], x'));
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ni(ri, i) = log10(norm(p, 2));
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@ -125,7 +128,7 @@ figure
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for u = 1:10
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subplot(colh, colw, u);
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title(sprintf("\nu=%d", u));
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title(sprintf("u=%d", u));
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hold on
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for j = 1:3
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ri = u * 3 - 3 + j;
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@ -152,10 +155,10 @@ for u = 1:10
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vec = ni(ri, :);
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vec = vec(1:its(ri));
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plot(1:its(ri), vec);
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plot(1:its(ri), vec, '-o');
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end
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hold off
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title(sprintf("\nu=%d", u));
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title(sprintf("u=%d", u));
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end
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sgtitle("Iterations over log10 of gradient norms");
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BIN
hw1/main.pdf
BIN
hw1/main.pdf
Binary file not shown.
46
hw1/main.tex
46
hw1/main.tex
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@ -101,19 +101,59 @@ where:
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\[v = \begin{bmatrix}x\\y\end{bmatrix}\]
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\subsection{Matlab implementation with \texttt{surf} and \texttt{contour}}
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\subsection{Finding the optimal step length $\alpha$}
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Considering $p$, our search direction, as the negative of the gradient (as dictated by the gradient method), we can rewrite the problem of finding an optimal step size $\alpha$ as the problem of minimizing the objective function along the line where $p$ belongs. This can be written as minimizing a function $l(\alpha)$, where:
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\[l(\alpha) = \frac12 \langle A(x + \alpha p), x + \alpha p\rangle\]
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To minimize we compute the gradient of $l(\alpha)$ and fix it to zero to find a stationary point, finding a value for $\alpha$ in function of $A$, $x$ and $p$.
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\[l'(\alpha) = 2 \cdot \frac12 \langle A (x + \alpha p), p \rangle = \langle Ax, p \rangle + \alpha \langle Ap, p \rangle\]
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\[l'(\alpha) = 0 \Leftrightarrow \alpha = \frac{\langle Ax, p \rangle}{\langle Ap, p \rangle}\]
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Since $A$ is s.p.d. by definition the hessian of function $l(\alpha)$ will always be positive, the stationary point found above is a minimizer of $l(\alpha)$ and thus the definition of $\alpha$ given above gives the optimal search step for the gradient method.
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\subsection{Matlab implementation with \texttt{surf} and \texttt{contour} plots}
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The graphs generated by MATLAB are shown below:
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\begin{figure}[h]
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\resizebox{\textwidth}{!}{\input{surf.tex}}
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\caption{Surf plots for different values of $\mu$}
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\end{figure}
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\begin{figure}[h]
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\resizebox{\textwidth}{!}{\input{contour.tex}}
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\caption{Contour plots and iteration steps. Red has $x_0 = \begin{bmatrix}10&0\end{bmatrix}^T$,
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yellow has $x_0 = \begin{bmatrix}10&10\end{bmatrix}^T$, and blue has $x_0 = \begin{bmatrix}0&10\end{bmatrix}^T$}
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\end{figure}
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\begin{figure}[h]
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\resizebox{\textwidth}{!}{\input{yseries.tex}}
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\caption{Iterations over values of the objective function. Red has $x_0 = \begin{bmatrix}10&0\end{bmatrix}^T$,
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yellow has $x_0 = \begin{bmatrix}10&10\end{bmatrix}^T$, and blue has $x_0 = \begin{bmatrix}0&10\end{bmatrix}^T.$}
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\end{figure}
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\begin{figure}[h]
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\resizebox{\textwidth}{!}{\input{norms.tex}}
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\caption{Iterations over base 10 logarithm of gradient norms. Note that for $\mu=1$ the search immediately converges to the
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exact minimizer no matter the value of $x_0$, so no gradient norm other than the very first one is recorded. Again,
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Red has $x_0 = \begin{bmatrix}10&0\end{bmatrix}^T$,
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yellow has $x_0 = \begin{bmatrix}10&10\end{bmatrix}^T$, and blue has $x_0 = \begin{bmatrix}0&10\end{bmatrix}^T.$}
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\end{figure}
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Isolines get stretched along the y axis as $\mu$ increases. For $\mu \neq 1$, points well far away from the axes are a
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problem since picking search directions and steps using the gradient method iterations will zig-zag
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to the minimizer reaching it slowly.
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Additionally, from the \texttt{surf} plots, we can see that the behaviour of isolines is justified by a "stretching" of sorts
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of the function that causes the y axis to be steeper as $\mu$ increases.
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What has been said before about the convergence of the gradient method is additionally showed in the last two sets of plots.
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From the objective function plot we can see that iterations starting from $\begin{bmatrix}10&10\end{bmatrix}^T$ (depicted in yellow) take the highest number of iterations to reach the minimizer (or an acceptable approximation of it). The zig-zag behaviour described before can be also observed in the contour plots, showing the iteration steps taken for each $\mu$ and starting from each $x_0$.
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Finally, in the gradient norm plots a phenomena that creates increasingly flatter plateaus as $\mu$ increases can be observed.
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Isolines get stretched along the y axis as $\mu$ increases. For a large $\mu$, points well far away from the axes could be a
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problem since picking search directions and steps using a naive gradient based method iterations will zig-zag to the minimizer reaching it slowly.
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\end{document}
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@ -22,17 +22,17 @@ title={u=1},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 1.45154499349597\\
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2 -inf\\
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@ -55,17 +55,17 @@ title={u=2},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 1.60205999132796\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 1.65051499783199\\
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2 0.997302484056647\\
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@ -104,17 +104,17 @@ title={u=3},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 1.77815125038364\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 1.80102999566398\\
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2 1.13202321470541\\
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@ -157,17 +157,17 @@ title={u=4},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 1.90308998699194\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 1.91625445635312\\
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2 1.18252234575789\\
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@ -210,17 +210,17 @@ title={u=5},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 2\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 2.00851666964939\\
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2 1.20917612019581\\
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@ -263,17 +263,17 @@ title={u=6},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 2.07918124604762\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 2.08513085769748\\
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2 1.22579237856861\\
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@ -316,17 +316,17 @@ title={u=7},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 2.14612803567824\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 2.15051499783199\\
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2 1.23720584565836\\
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@ -367,17 +367,17 @@ title={u=8},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 2.20411998265593\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 2.20748667398541\\
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2 1.24555733587979\\
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@ -418,17 +418,17 @@ title={u=9},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 2.25527250510331\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 2.25793692185584\\
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2 1.25194655816665\\
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@ -469,17 +469,17 @@ title={u=10},
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axis x line*=bottom,
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axis y line*=left
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]
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\addplot [color=mycolor1, forget plot]
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\addplot [color=mycolor1, mark=o, mark options={solid, mycolor1}, forget plot]
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table[row sep=crcr]{%
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1 2.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor2, forget plot]
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\addplot [color=mycolor2, mark=o, mark options={solid, mycolor2}, forget plot]
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table[row sep=crcr]{%
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1 1.30102999566398\\
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2 -inf\\
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};
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\addplot [color=mycolor3, forget plot]
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\addplot [color=mycolor3, mark=o, mark options={solid, mycolor3}, forget plot]
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table[row sep=crcr]{%
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1 2.3031906825553\\
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2 1.25699911451531\\
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