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\usepackage[ruled,vlined]{algorithm2e} +- \usepackage{float} +- \floatplacement{figure}{H} + --- -\title{Midterm -- Optimization Methods} -\author{Claudio Maggioni} \maketitle # Exercise 1 @@ -139,3 +143,157 @@ https://en.wikipedia.org/wiki/Definite_matrix#Multiplication) Thanks to this we have indeed proven that the delta $\|e_k\|_A - \|e_{k+1}\|_A$ is indeed positive and thus as $i$ increases the energy norm of the error monotonically decreases. + +# Question 2 + +## Point 1 + +TBD + +## Point 2 + +The trust region algorithm is the following: + +\begin{algorithm}[H] +\SetAlgoLined +Given $\hat{\Delta} > 0, \Delta_0 \in (0,\hat{\Delta})$, +and $\eta \in [0, \frac14)$\; + + \For{$k = 0, 1, 2, \ldots$}{% + Obtain $p_k$ by using Cauchy or Dogleg method\; + $\rho_k \gets \frac{f(x_k) - f(x_k + p_k)}{m_k(0) - m_k(p_k)}$\; + \uIf{$\rho_k < \frac14$}{% + $\Delta_{k+1} \gets \frac14 \Delta_k$\; + }\Else{% + \uIf{$\rho_k > \frac34$ and $\|\rho_k\| = \Delta_k$}{% + $\Delta_{k+1} \gets \min(2\Delta_k, \hat{\Delta})$\; + } + \Else{% + $\Delta_{k+1} \gets \Delta_k$\; + }} + \uIf{$\rho_k > \eta$}{% + $x_{k+1} \gets x_k + p_k$\; + } + \Else{ + $x_{k+1} \gets x_k$\; + } + } + \caption{Trust region method} +\end{algorithm} + +The Cauchy point algorithm is the following: + +\begin{algorithm}[H] +\SetAlgoLined +Input $B$ (quadratic term), $g$ (linear term), $\Delta_k$\; + \uIf{$g^T B g \geq 0$}{% + $\tau \gets 1$\; + }\Else{% + $\tau \gets \min(\frac{\|g\|^3}{\Delta_k \cdot g^T B g}, 1)$\; + } + + $p_k \gets -\tau \cdot \frac{\Delta_k}{\|g\|^2 \cdot g}$\; + \Return{$p_k$} + \caption{Cauchy point} +\end{algorithm} + +Finally, the Dogleg method algorithm is the following: + +\begin{algorithm}[H] +\SetAlgoLined +Input $B$ (quadratic term), $g$ (linear term), $\Delta_k$\; + $p_N \gets - B^{-1} g$\; + + \uIf{$\|p_N\| < \Delta_k$}{% + $p_k \gets p_N$\; + }\Else{% + $p_u = - \frac{g^T g}{g^T B g} g$\; + + \uIf{$\|p_u\| > \Delta_k$}{% + compute $p_k$ with Cauchy point algorithm\; + }\Else{% + solve for $\tau$ the equality $\|p_u + \tau * (p_N - p_u)\|^2 = + \Delta_k^2$\; + $p_k \gets p_u + \tau \cdot (p_N - p_u)$\; + } + } + \caption{Dogleg method} +\end{algorithm} + +## Point 3 + +The trust region, dogleg and Cauchy point algorithms were implemented +respectively in the files `trust_region.m`, `dogleg.m`, and `cauchy.m`. + +## Point 4 + +### Taylor expansion + +The Taylor expansion up the second order of the function is the following: + +$$f(x_0, w) = f(x_0) + \langle\begin{bmatrix}48x^3 - 16xy + 2x - 2\\2y - 8x^2 +\end{bmatrix}, w\rangle + \frac12 \langle\begin{bmatrix}144x^2 -16y + 2 - 16 & +-16 \\ -16 & 2 \end{bmatrix}w, w\rangle$$ + +### Minimization + +The code used to minimize the function can be found in the MATLAB script +`main.m` under section 2.4. The resulting minimizer (found in 10 iterations) is: + +$$x_m = \begin{bmatrix}1\\4\end{bmatrix}$$ + +### Energy landscape + +The following figure shows a `surf` plot of the objective function overlayed +with the iterates used to reach the minimizer: + +![Energy landscape of the function overlayed with iterates and steps (the white +dot is $x_0$ while the black dot is $x_m$)](./2-4-energy.png) + +The code used to generate such plot can be found in the MATLAB script `main.m` +under section 2.4c. + +## Point 5 + +### Minimization + +The code used to minimize the function can be found in the MATLAB script +`main.m` under section 2.5. The resulting minimizer (found in 25 iterations) is: + +$$x_m = \begin{bmatrix}1\\5\end{bmatrix}$$ + +### Energy landscape + +The following figure shows a `surf` plot of the objective function overlayed +with the iterates used to reach the minimizer: + +![Energy landscape of the Rosenbrock function overlayed with iterates and steps +(the white dot is $x_0$ while the black dot is $x_m$)](./2-5-energy.png) + +The code used to generate such plot can be found in the MATLAB script `main.m` +under section 2.5b. + +### Gradient norms + +The following figure shows the logarithm of the norm of the gradient w.r.t. +iterations: + +![Gradient norms (y-axis, log-scale) w.r.t. iteration number +(x-axis)](./2-5-gnorms.png) + +The code used to generate such plot can be found in the MATLAB script `main.m` +under section 2.5c. + +Comparing the behaviour shown above with the figures obtained in the previous +assignment for the Newton method with backtracking and the gradient descent with +backtracking, we notice that the trust-region method really behaves like a +compromise between the two methods. First of all, we notice that TR converges in +25 iterations, almost double of the number of iterations of regular NM + +backtracking. The actual behaviour of the curve is somewhat similar to the +Netwon gradient norms curve w.r.t. to the presence of spikes, which however are +less evident in the Trust region curve (probably due to Trust region method +alternating quadratic steps with linear or almost linear steps while iterating). +Finally, we notice that TR is the only method to have neighbouring iterations +having the exact same norm: this is probably due to some proposed iterations +steps not being validated by the acceptance criteria, which makes the method mot +move for some iterations. diff --git a/Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.pdf b/Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.pdf index 4c894182048984ff247397bf8f1c16a6d9f997f4..e31878eabb1f688821da892ea3a41a23ca255545 100644 GIT binary patch delta 516116 zcmZs?1yodF*EW3U?vhSHLb`^QmhSEr>5vkJltGXb0V$F0lJ1i3ZUO1;68R23&-;G= zTHim5wV0W6pS|yEU)PSg2PU>K$EF!#pTi5%B*oFdGsUS0Lc46li;9YiTg&B*j7aKl 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