from enthought.mayavi import mlab
-import surf
+import time
import numpy as np
import matplotlib.pyplot as plt
+import surf
from steepest import steepest
from function import function_g
from newton import newton
b = np.array([-0.3119187, 0.14508516])
c = np.array([ 0.47736331, 1.48834978])
- x0 = np.array([-2.1, 3.2])
+ x0 = np.array([-2.1, 5.2])
function_values = []
- alpha = 0.1
- optimal_points, optimal_residuals = steepest(x0, H, b, c, 0.001, 100, 0)
- constant_points, constant_residuals = steepest(x0, H, b, c, 0.001, 30, alpha)
- newton_points, newton_residuals = newton(x0, H, b, c, 0.001, 100)
- nwst_points, nwst_residuals = newtsteep(x0, H, b, c, 0.1, 0.001)
+ alpha = 0.05
+
+ start = time.clock()
+ optimal_points, optimal_residuals = steepest(x0, H, b, c, 0.000000001, 100000, 0)
+ stop = time.clock()
+ print "optimal steepest brukte %.4f" % (stop - start)
+
+ start = time.clock()
+ constant_points, constant_residuals = steepest(x0, H, b, c, 0.000000001, 100000, alpha)
+ stop = time.clock()
+ print "constant steepest brukte %.4f" % (stop - start)
+
+ start = time.clock()
+ newton_points, newton_residuals = newton(x0, H, b, c, 0.000000001, 100000)
+ stop = time.clock()
+ print "newton brukte %.4f" % (stop - start)
+
+ start = time.clock()
+ nwst_points, nwst_residuals = newtsteep(x0, H, b, c, 0.01, 0.000000001)
+ stop = time.clock()
+ print "newton + steepest brukte %.4f" % (stop - start)
f_optimal = []
f_constant = []
mlab.figure(figure="Surface", bgcolor=(1,1,1), fgcolor=(0,0,0))
- for point in optimal_points:
+ plen = 15
+ for point in optimal_points[:plen]:
z = function_g(point, H, b, c)
f_optimal.append(z)
p = mlab.points3d([point[0]], [point[1]], [z], scale_factor=0.03, color=(0.7,0.2,0.2))
- for point in constant_points:
+ for point in constant_points[:plen]:
z = function_g(point, H, b, c)
f_constant.append(z)
p = mlab.points3d([point[0]], [point[1]], [z], scale_factor=0.02, color=(1,1,1))
- for point in newton_points:
+ for point in newton_points[:plen]:
z = function_g(point, H, b, c)
f_newton.append(z)
- for point in nwst_points:
+ for point in nwst_points[:plen]:
z = function_g(point, H, b, c)
f_nwst.append(z)
plt.title("Relative residual")
plt.xlabel("Iteration")
plt.ylabel("Relative residual")
- plt.plot(optimal_residuals)
- plt.plot(constant_residuals)
- plt.plot(newton_residuals)
- plt.plot(nwst_residuals)
- plt.legend([r'Optimal $\alpha$', r'$\alpha$ = %.1f' % alpha, r'Newtons method', r'Steepest first, then Newton'])
+ plt.plot(optimal_residuals[:plen])
+ plt.plot(constant_residuals[:plen])
+ plt.plot(newton_residuals[:plen])
+ plt.plot(nwst_residuals[:plen])
+ plt.legend([r'Optimal $\alpha$', r'$\alpha$ = %.2f' % alpha, r'Newtons method', r'Steepest first, then Newton'])
ax.set_yscale('log')
ax = plt.subplot(212)
plt.title("Function value")
plt.xlabel("Iteration")
plt.ylabel("g(x)")
- plt.plot(f_optimal)
- plt.plot(f_constant)
- plt.plot(f_newton)
- plt.plot(f_nwst)
+ plt.plot(f_optimal[:plen])
+ plt.plot(f_constant[:plen])
+ plt.plot(f_newton[:plen])
+ plt.plot(f_nwst[:plen])
plt.show()
fig.savefig("steepest.eps", format="eps")
% %\r
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r
\section{Resultater}\r
-We used the following data:\r
-\r
+This project explores how one can approximate the local minima of a function.\r
+The function considered takes in a symmetric positive definite matrix, and two\r
+vectors. For this purpose we used the following randomly generated data:\r
\begin{equation}\r
-H = \r
-\begin{bmatrix}\r
-1.8580682 & -1.01197101 \\\r
--1.01197101 & 0.63456543 \r
-\end{bmatrix}\r
-, \mathbf{b} =\r
-\begin{bmatrix}\r
--0.3119187 \\\r
-0.14508516\r
-\end{bmatrix}\r
-, \mathbf{c} =\r
-\begin{bmatrix}\r
-0.47736331 \\\r
-1.48834978\r
-\end{bmatrix}.\r
-\nonumber\r
+ H = \r
+ \begin{bmatrix}\r
+ 1.8580682 & -1.01197101 \\\r
+ -1.01197101 & 0.63456543 \r
+ \end{bmatrix}\r
+ , \mathbf{b} =\r
+ \begin{bmatrix}\r
+ -0.3119187 \\\r
+ 0.14508516\r
+ \end{bmatrix}\r
+ , \mathbf{c} =\r
+ \begin{bmatrix}\r
+ 0.47736331 \\\r
+ 1.48834978\r
+ \end{bmatrix}.\r
+ \nonumber\r
\end{equation}\r
+The elements of $\mathbf{b}$ are randomly generated numbers from a uniform\r
+distribution over $[-1, 1]$, and the elements of $\mathbf{c}$ are randomly\r
+generated positive numbers from a uniform distribution over $(0, 3]$\r
\r
-Where the elements of $\mathbf{b}$ are randomly generated numbers from the interval INTERVAL, and the elements of $\mathbf{c}$ are randomly generated positive numbers.\r
-\r
-To generate the nxn matrix, $H$, we first generate a upper triangular nxn matrix $U$.\r
-The elements along the diagonal of $U$ are the square roots of randomly generated positive numbers, and the elements above the diagonal are randomly generated numbers from the interval INTERVAL.\r
-$H = U^{T}U$ is then a symmetric positive definite matrix.\r
+To generate the $n\times n$ matrix $H$, we first generate a upper triangular\r
+$n\times n$ matrix $U$. The elements above the diagonal are randomly generated\r
+in the uniform distribution over $[-3, 3]$, while the numbers on the diagonal\r
+are square roots of positive numbers from the same distribution. $H = U^{T}U$ is\r
+then a symmetric positive definite matrix.\r
\r
The function analyzed in this project was\r
-\r
\begin{equation}\r
-\label{eq:func_g}\r
-g(\mathbf{x}) := -\mathbf{b}^{T}\mathbf{x}+\frac{1}{2}\mathbf{x}^{T}H\mathbf{x}+\frac{1}{12}\mathbf{x}^{T}C(\mathbf{x})\mathbf{x}\r
+ \label{eq:func_g}\r
+ g(\mathbf{x}) := -\mathbf{b}^{T}\mathbf{x} +\r
+ \frac{1}{2}\mathbf{x}^{T}H\mathbf{x} +\r
+ \frac{1}{12}\mathbf{x}^{T}C(\mathbf{x})\mathbf{x}.\r
\end{equation}\r
From equation \ref{eq:func_g} we get\r
\begin{eqnarray}\r
-\label{eq:grad_g}\r
-\nabla g(\mathbf{x}) &=& -\mathbf{b}+H\mathbf{x}+\frac{1}{3}C(\mathbf{x})\mathbf{x} \\\r
-\label{eq:hess_g}\r
-\nabla^{2}g(\mathbf{x}) &=& H+C(\mathbf{x}).\r
+ \label{eq:grad_g}\r
+ \nabla g(\mathbf{x}) &=& -\mathbf{b}+H\mathbf{x}+\frac{1}{3}C(\mathbf{x})\mathbf{x} \\\r
+ \label{eq:hess_g}\r
+ \nabla^{2}g(\mathbf{x}) &=& H+C(\mathbf{x}).\r
\end{eqnarray}\r
-The three functions, \ref{eq:func_g}, \ref{eq:grad_g} and \ref{eq:hess_g}, are implemented in the Python code below.\r
+The three functions, \ref{eq:func_g}, \ref{eq:grad_g} and \ref{eq:hess_g}, are\r
+implemented in the Python code below.\r
\r
\lstinputlisting[language=Python]{bj_function.py}\r
\r
-Consider the Hessian $\nabla^{2}g(\mathbf{x})$, and any vector $\mathbf{v} \in \mathbb{R}^{n}_{*}$.\r
-Then we have\r
+Consider the Hessian $\nabla^{2}g(\mathbf{x})$, and any vector $\mathbf{v} \in\r
+\mathbb{R}^{n}_{*}$. Then we have\r
\r
\begin{eqnarray}\r
-\mathbf{v}^{T}(\nabla^{2}g(\mathbf{x}))\mathbf{v} &=& \mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} \nonumber \\\r
-&=& \mathbf{v}^{T}H(\mathbf{x}))\mathbf{v}+\mathbf{v}^{T}C(\mathbf{x})\mathbf{v}.\r
+ \mathbf{v}^{T}(\nabla^{2}g(\mathbf{x}))\mathbf{v} &=& \mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} \nonumber \\\r
+ &=& \mathbf{v}^{T}H(\mathbf{x}))\mathbf{v}+\mathbf{v}^{T}C(\mathbf{x})\mathbf{v}.\r
\end{eqnarray}\r
We already know H is positive definite, so $\mathbf{v}^{T}H(\mathbf{x}))\mathbf{v} > 0$. \r
Since $C(\mathbf{x})$ only has positive entries, we easily see that\r
-\r
\begin{equation}\r
-\mathbf{v}^{T}C(\mathbf{x})\mathbf{v} = \sum\limits_{i=1}^n c_{i}(v_{i}x_{i})^2 \geq 0\r
+ \mathbf{v}^{T}C(\mathbf{x})\mathbf{v} = \sum\limits_{i=1}^n c_{i}(v_{i}x_{i})^2 \geq 0\r
\end{equation}\r
-Hence $\mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} > 0$, and we see by definition that the Hessian, $\nabla^{2}g(\mathbf{x})$, is positive definite.\r
+Hence $\mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} > 0$, and we see by definition\r
+that the Hessian, $\nabla^{2}g(\mathbf{x})$, is positive definite.\r
\r
\begin{figure}[htb]\r
-\centering\r
-\includegraphics[width=0.8\textwidth]{steepest}\r
-\caption{Plot of relative residual and function values for iterations of the steepest descent method with optimal $\alpha$}\r
-\label{fig:fn_resid}\r
+ \centering\r
+ \includegraphics[width=0.9\textwidth]{steepest}\r
+ \caption{Plot of relative residual and function values for iterations of the steepest descent method with optimal $\alpha$}\r
+ \label{fig:fn_resid}\r
\end{figure}\r
\r
\begin{figure}[htb]\r
-\centering\r
-\includegraphics[width=0.9\textwidth]{surface}\r
-\caption{Surface plot of function with points showing iteration of steepest descent method, both with optimal and constant $\alpha$}\r
-\label{fig:surface}\r
+ \centering\r
+ \includegraphics[width=0.9\textwidth]{surface}\r
+ \caption{Surface plot of function with points showing iteration of steepest descent method, both with optimal and constant $\alpha$}\r
+ \label{fig:surface}\r
\end{figure}\r
\r
+Consider the equation\r
+\begin{equation}\r
+ \label{eq:forward}\r
+ \mathbf{r}'(t) = -\nabla g\left(\mathbf{r}(t)\right)\text{, } \mathbf{r}(0) = \mathbf{r}^0.\r
+\end{equation}\r
+One can view the function $\mathbf{r}(t)$ as a parameterisation of the curve\r
+going through the starting point to the minima, always going in the direction of\r
+the steepest descent. By applying the forward Euler method to this differential\r
+vector equation, we get the following iteration:\r
+\begin{equation}\r
+ \label{eq:forward_iter}\r
+ \mathbf{r}^{(k+1)} = \mathbf{r}^{(k)} - \alpha \nabla g\left(\mathbf{r}^{(k)}\right)\r
+\end{equation}\r
+where $t_{n+1} = t_n + \alpha$. Looking at this result, we see that it is equivalent to the steepest descent iteration with constant step size $\alpha$. \r
+\r
To find the optimal step length, $\alpha^*$, we solve the equation\r
\begin{eqnarray}\r
-\frac{\partial}{\partial \mathbf{\alpha}}g\left(\mathbf{x}+\alpha\mathbf{u}\right) &=& 0, \text{where } \mathbf{u} = -\nabla g(\mathbf{x} ) \nonumber \\\r
-&\Updownarrow& \nonumber\\\r
-%HER ER DET FEIL! \r
-A\alpha^3 + B\alpha^2 + C\alpha + D &=& 0,\r
+ \frac{\partial}{\partial \mathbf{\alpha}}g\left(\mathbf{x}+\alpha\mathbf{u}\right) &=& 0, \text{where } \mathbf{u} = -\nabla g(\mathbf{x} ) \nonumber \\\r
+ &\Updownarrow& \nonumber\\\r
+ A\alpha^3 + B\alpha^2 + C\alpha + D &=& 0,\r
+ \end{eqnarray}\r
+where $A$, $B$, $C$, and $D$ is given by\r
+ \begin{eqnarray}\r
+ A &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{u})\mathbf{u} \nonumber \\\r
+ B &=& \mathbf{x}^TC(\mathbf{u}) \mathbf{u} \nonumber \\\r
+ C &=& \mathbf{u}^TH\mathbf{u}+\mathbf{u}^TC(\mathbf{x})\mathbf{u} \nonumber \\\r
+ D &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{x})\mathbf{x}+\mathbf{u}^TH\mathbf{x}-\mathbf{b}^T\mathbf{u}\r
\end{eqnarray}\r
-der $A$, $B$, $C$ og $D$ er gitt ved\r
-\begin{eqnarray}\r
-A &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{u}\mathbf{u} \nonumber \\\r
-B &=& (\mathbf{x})^TC(\mathbf{u}) \mathbf{u} \nonumber \\\r
-C &=& (\mathbf{u})^TH\mathbf{u}+(\mathbf{u})^TC(\mathbf{x})\mathbf{u}) \nonumber \\\r
-D &=& \frac{1}{3}(\mathbf{u})^TC(\mathbf{x})\mathbf{x}+(\mathbf{u})^TH\mathbf{x}-\mathbf{b}^T\mathbf{u}\r
-\end{eqnarray}\r
-\r
-% VI MÅ OGSÅ ha utledning av alt det andre, forklaring \r
-% av figurer, og en slags tabell med sammenlikning av metodene\r
\r
\lstinputlisting[language=Python]{bj_optimal.py}\r
\r
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projects / nummat1 / commitdiff