latex rapport.tex
latex rapport.tex
dvips rapport.dvi
- ps2pdf rapport.ps
+ #ps2pdf rapport.ps
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%%Title: steepest.eps
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(Relative residual) show
--- /dev/null
+# Calculate the next iteration of the Newton method
+def newton_iter(x, grad_g, hess_g):
+ return x - inv(hess_g).dot(grad_g)
+
+# Run the Newton method from point xk
+def newton(xk, H, b, c, tolerance, maxiter, x0_norm=0):
+ if x0_norm == 0:
+ x0_norm = norm(gradient_g(xk, H, b, c))
+
+ relative_residual = norm(gradient_g(xk, H, b, c)) / x0_norm
+ relative_residuals = []
+ points = []
+
+ while relative_residual > tolerance and maxiter > 0:
+ maxiter -= 1
+
+ relative_residuals.append(relative_residual)
+ points.append(xk)
+
+ xk = newton_iter(xk, gradient_g(xk, H, b, c), hessian_g(xk, H, b, c))
+ relative_residual = norm(gradient_g(xk, H, b, c)) / x0_norm
+
+ return (points, relative_residuals)
# Calculate gradient of g in point x
def gradient_g(x, H, b, c):
- print _C(x,c)
return -b + H.dot(x) + (1.0/3.0) * _C(x,c).dot(x)
# Calculate hessian of g in point x
def hessian_g(x, H, b, c):
- return 0.5 * H.T + 0.5 * H + _C(x,c)
+ return H + _C(x,c)
# Small test of the above functions
def main():
b = np.array([-0.3119187, 0.14508516])
c = np.array([ 0.47736331, 1.48834978])
- x0 = np.array([-2.1, 5.2])
+ x0 = np.array([-50, 50])
function_values = []
- alpha = 0.05
+ alpha = 0.001
+ tolerance = 0.000000001
- start = time.clock()
- optimal_points, optimal_residuals = steepest(x0, H, b, c, 0.000000001, 100000, 0)
- stop = time.clock()
+ start = time.time()
+ optimal_points, optimal_residuals = steepest(x0, H, b, c, tolerance, 100000, 0)
+ stop = time.time()
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()
+ start = time.time()
+ constant_points, constant_residuals = steepest(x0, H, b, c, tolerance, 100000, alpha)
+ stop = time.time()
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()
+ start = time.time()
+ newton_points, newton_residuals = newton(x0, H, b, c, tolerance, 100000)
+ stop = time.time()
print "newton brukte %.4f" % (stop - start)
- start = time.clock()
- nwst_points, nwst_residuals = steepestnewton(x0, H, b, c, 0.01, 0.000000001)
- stop = time.clock()
+ start = time.time()
+ nwst_points, nwst_residuals = steepestnewton(x0, H, b, c, 0.00001, 4, tolerance)
+ stop = time.time()
print "newton + steepest brukte %.4f" % (stop - start)
f_optimal = []
plt.plot(f_constant[:plen])
plt.plot(f_newton[:plen])
plt.plot(f_nwst[:plen])
- plt.legend([r'Optimal $\alpha$', r'$\alpha$ = %.2f' % alpha, r'Newtons method', r'Steepest first, then Newton'])
+ plt.legend([r'Optimal $\alpha$', r'$\alpha$ = %.3f' % alpha, r'Newtons method', r'Steepest first, then Newton'])
ax = plt.subplot(212)
plt.title("Relative residual")
--- /dev/null
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\r
\title{Semester Project 1 in TMA4215}\r
\author{\r
- 10012 \\\r
- 10023 \\\r
- 10056\r
+ Candidate numbers: 10012, 10023, 10056\r
}\r
\r
\maketitle\r
% %\r
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r
\section{Resultater}\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
+This project explores how one can approximate the solution to a certain type of\r
+minimization problems. The function considered takes in a symmetric positive\r
+definite matrix, and two vectors. For this purpose we used the following\r
+randomly generated data:\r
\begin{equation}\r
H = \r
\begin{bmatrix}\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
+are square roots of positive numbers from the same distribution. The matrix $H\r
+= U^{T}U$ is then symmetric positive definite.\r
\r
The function analyzed in this project was\r
\begin{equation}\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}H\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
+We already know H is positive definite, so $\mathbf{v}^{T}H\mathbf{v} > 0$. \r
Since $C(\mathbf{x})$ only has positive entries, we easily see that\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\r
that the Hessian, $\nabla^{2}g(\mathbf{x})$, is positive definite.\r
\begin{figure}[htb]\r
\centering\r
\includegraphics[width=0.9\textwidth]{backup}\r
- \caption{Plot of relative residual and function values for iterations of the\r
- steepest descent method with optimal $\alpha$} \label{fig:fn_resid}\r
+ \caption{\sf Plot of relative residual and function values for iterations of the\r
+ 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\r
- descent method, both with optimal and constant $\alpha$} \label{fig:surface}\r
+ \caption{\sf Surface plot of function with points showing iteration of steepest\r
+ 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
+ \mathbf{r}'(t) = -\nabla g\left(\mathbf{r}(t)\right)\text{, } \mathbf{r}(0) =\r
+ \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
\frac{\partial}{\partial\r
\mathbf{\alpha}}g\left(\mathbf{x}+\alpha\mathbf{u}\right) &=& 0, \text{where }\r
\mathbf{u} = -\nabla g(\mathbf{x} ) \nonumber \\\r
- &\Updownarrow& \nonumber \\\r
+ &\Downarrow& \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
+for $\alpha$. Where $A$, $B$, $C$, and $D$ are 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
-Deriving the polynomial results in:\r
+Differentiating the polynomial results in:\r
\begin{eqnarray}\r
\frac{\partial^2}{\partial \alpha^2}\left(g\left(\mathbf{x} +\r
\alpha\mathbf{u}\right)\right) &=& \mathbf{u}^T H \mathbf{u} + \mathbf{u}^T\r
C\left(\mathbf{x}\right)\right]\mathbf{u} + 2\mathbf{x}^T\r
C\left(\mathbf{u}\right) \mathbf{u}\alpha \nonumber \\\r
&=& \mathbf{u}^T H \mathbf{u} + \mathbf{u}^T C\left(\mathbf{x} +\r
- \alpha\mathbf{u}\right) \nonumber \\\r
+ \alpha\mathbf{u}\right)\mathbf{u} \nonumber \\\r
&=& \mathbf{u}^T\nabla^2 g\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u}.\r
\end{eqnarray}\r
From this we see that the polynomial \eqref{eq:poly} is monotonically increasing\r
and therefore only has one real zero. This is the zero we want to use as our\r
-optimal step length.\r
-\r
+optimal step length. This is implemented in the Python code below.\r
\lstinputlisting[language=Python]{bj_optimal.py}\r
\r
+The Newton method was also implemented to solve $\nabla g\left(\mathbf{x}^*\right)\r
+= 0$. The Python code used for this method is as follows.\r
+\lstinputlisting[language=Python]{bj_newton.py}\r
+\r
+\begin{table}[h]\r
+ \caption{\sf Comparison of the execution time of the four different methods.}\r
+ \begin{center}\r
+ \begin{tabular}{l c}\r
+ \hline\r
+ Method & Time ($s$) \\\r
+ \hline\r
+ Steepest descent, constant $\alpha$ & 7.2021 \\\r
+ Steepest descent, optimal $\alpha$ & 0.0514 \\\r
+ Newtons method & 0.0037 \\\r
+ Steepest first, then Newton & 0.0040 \\\r
+ \hline\r
+ \end{tabular}\r
+ \end{center}\r
+ \label{tab:time}\r
+\end{table}\r
+We see from table \ref{tab:time} that newtons method uses the least time to\r
+solve the problem, even though figure \ref{fig:fn_resid} shows that the\r
+combination method results in fewer iterations. This comes from the fact that\r
+computing the optimal step size requires solving a qubic equation and is\r
+computationally expensive.\r
+\r
\end{document}\r
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(Relative residual) show
function_values = []
for point in points:
- print point
+ #print point
z = function_g(point, H, b, c)
function_values.append(z)
p = mlab.points3d([point[0]], [point[1]], [z], scale_factor=0.05)
# Run steepest method until tolerance steep_tol, then continue with Newtons
# method until tolerance newt_tol
-def steepestnewton(x0, H, b, c, steep_tol, newt_tol):
+def steepestnewton(x0, H, b, c, steep_tol, steep_max, newt_tol):
x = []
r = []
# Use the starting norm to calculate relative residuals in both algorithms
x0_norm = norm(gradient_g(x0, H, b, c))
- points, residuals = steepest(x0, H, b, c, steep_tol, 100, 0, x0_norm)
+ points, residuals = steepest(x0, H, b, c, steep_tol, steep_max, 0, x0_norm)
x.extend(points)
r.extend(residuals)
projects / nummat1 / commitdiff