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Έτσι οι μέθοδοι αντιγράφηκαν και εδώ για ολότητα, ακόμα και αν για την παρούσα εργασία χρησιμοποιείται μόνο το σταθερό βήμα $\gamma_k$. -\subsection{Symbolic expression functions} -Μία ακόμη προγραμματιστική τεχνική που ακολουθήθηκε είναι η χρήση \textbf{symbolic expression} για την αναπαράσταση της αντικειμενικής συνάρτησης. -Η εξήγηση υπάρχει και στις προηγούμενες εργασίες αλλά την παραθέτουμε εδώ για ολότητα. -Ο λόγος που επιλέχθηκε είναι η \textbf{δυνατότητα εξαγωγής ενός symbolic expression που αναπαριστά την κλίση $\nabla f$ και τον Εσσιανό $\nabla^2f$ μιας συνάρτησης} από την MATLAB, κάνοντας χρήση των εντολών \textit{gradient()} και \textit{hessian()}. -Αν αντίθετα χρησιμοποιούσαμε απλές συναρτήσεις, πολυώνυμα ή lambdas για την αναπαράσταση των αντικειμενικών συναρτήσεων, τότε για τον υπολογισμό της κλίσης και του Εσσιανού θα έπρεπε: -\begin{itemize} - \item Είτε να υπολογίζαμε αριθμητικά τις παραγώγους gradient και hessian μέσα στις μεθόδους, κάτι που θα εισήγαγε \textit{\textbf{αχρείαστο αριθμητικό σφάλμα}}. - \item Είτε να κάναμε χρήση δύο επιπλέων συναρτήσεων (ή πολυωνύμων) για την αναπαράσταση τους, κάτι που ουσιαστικά θα δημιουργούσε \textit{\textbf{πλεονασμό πληροφορίας εισόδου}} και άρα μεγαλύτερη πιθανότητα να κάνουμε λάθος. -\end{itemize} -Η αναπαράσταση όμως με χρήση symbolic expression είναι πιο “βαριά” όταν χρειάζεται να υπολογίσουμε την τιμή μιας συνάρτησης σε κάποιο σημείο (subs(expr, number)). -Αυτό είναι κάτι που χρειάζεται εκτενώς στον κώδικά μας. -Για το λόγο αυτό, ενώ η συνάρτηση δίνεται ως symbolic expression, μέσω αυτής υπολογίζονται αυτόματα η κλίση, ο Εσσιανός αλλά και οι “κανονικές” συναρτήσεις MATLAB που τις υλοποιούν. -Έτσι έχουμε την ακριβή αναπαράσταση της κλίσης και του Εσσιανού ως συναρτήσεις χωρίς να πληρώνουμε το κόστος της subs(). +%\subsection{Symbolic expression functions} +%Μία ακόμη προγραμματιστική τεχνική που ακολουθήθηκε είναι η χρήση \textbf{symbolic expression} για την αναπαράσταση της αντικειμενικής συνάρτησης. +%Η εξήγηση υπάρχει και στις προηγούμενες εργασίες αλλά την παραθέτουμε εδώ για ολότητα. +%Ο λόγος που επιλέχθηκε είναι η \textbf{δυνατότητα εξαγωγής ενός symbolic expression που αναπαριστά την κλίση $\nabla f$ και τον Εσσιανό $\nabla^2f$ μιας συνάρτησης} από την MATLAB, κάνοντας χρήση των εντολών \textit{gradient()} και \textit{hessian()}. +%Αν αντίθετα χρησιμοποιούσαμε απλές συναρτήσεις, πολυώνυμα ή lambdas για την αναπαράσταση των αντικειμενικών συναρτήσεων, τότε για τον υπολογισμό της κλίσης και του Εσσιανού θα έπρεπε: +%\begin{itemize} +% \item Είτε να υπολογίζαμε αριθμητικά τις παραγώγους gradient και hessian μέσα στις μεθόδους, κάτι που θα εισήγαγε \textit{\textbf{αχρείαστο αριθμητικό σφάλμα}}. +% \item Είτε να κάναμε χρήση δύο επιπλέων συναρτήσεων (ή πολυωνύμων) για την αναπαράσταση τους, κάτι που ουσιαστικά θα δημιουργούσε \textit{\textbf{πλεονασμό πληροφορίας εισόδου}} και άρα μεγαλύτερη πιθανότητα να κάνουμε λάθος. +%\end{itemize} +%Η αναπαράσταση όμως με χρήση symbolic expression είναι πιο “βαριά” όταν χρειάζεται να υπολογίσουμε την τιμή μιας συνάρτησης σε κάποιο σημείο (subs(expr, number)). +%Αυτό είναι κάτι που χρειάζεται εκτενώς στον κώδικά μας. +%Για το λόγο αυτό, ενώ η συνάρτηση δίνεται ως symbolic expression, μέσω αυτής υπολογίζονται αυτόματα η κλίση, ο Εσσιανός αλλά και οι “κανονικές” συναρτήσεις MATLAB που τις υλοποιούν. +%Έτσι έχουμε την ακριβή αναπαράσταση της κλίσης και του Εσσιανού ως συναρτήσεις χωρίς να πληρώνουμε το κόστος της subs(). \section{Απεικόνιση της συνάρτησης} Η συνάρτηση με την οποία ασχολούμαστε στην παρούσα εργασία είναι η: @@ -110,15 +110,15 @@ Για να πάρουμε μια καλύτερη αίσθηση για το που βρίσκεται το τοπικό ελάχιστο της $f$, παρακάτω παραθέτουμε ένα γράφημα με τις ισοβαρείς καμπύλες της $f$. \InsertFigure{H}{0.8}{fig:plotContour}{../scripts/figures/Plot_Contour.png}{Ισοβαρείς της f} -Από το παραπάνω σχήμα \ref{fig:plotContour} επιβεβαιώνεται και γραφικά το ελάχιστο στο σημείο $(0,0)$. +Από το παραπάνω σχήμα \ref{fig:plotContour} φαίνονται και γραφικά οι μικρές κλίσης που παρουσιάζει η συνάρτηση κοντά στο ελάχιστο σημείο $(0,0)$. Τα διαγράμματα για τη μέθοδο δημιουργούνται εκτελώντας το αρχείο \textbf{Script\_0\_Plots.m} \section{Μέθοδος Μέγιστης Καθόδου χωρίς περιορισμούς - Θέμα 1} -Εφαρμόζοντας την μέθοδο μέγιστης καθόδου από την προηγούμενη εργασία, με ακρίβεια $\epsilon = 0.001$, για τα βήματα $\gamma_k$ της εκφώνησης, παρατηρούμε ότι η μέθοδος συγκλίνει στο ελάχιστο για μικρά $\gamma_k$ ενώ αποκλίνει για μεγάλα \boldmath$\gamma_k > 0.34$\unboldmath. +Εφαρμόζοντας την μέθοδο μέγιστης καθόδου από την προηγούμενη εργασία, με ακρίβεια $\epsilon = 0.001$, για τα βήματα $\gamma_k$ της εκφώνησης, παρατηρούμε ότι η μέθοδος συγκλίνει στο ελάχιστο για μικρά $\gamma_k$ ενώ αποκλίνει για μεγάλα \boldmath$\gamma_k \geq 0.34$\unboldmath. Από τις δοκιμές φαίνεται ότι το σημείο εκκίνησης δεν παίζει ρόλο και για αυτό επιλέξαμε να παραθέσουμε τα ευρήματά μας από το σημείο $(5,-5)$, για αντιπαραβολή με το επόμενο βήμα της εκφώνησης. \InsertFigure{H}{0.6}{fig:StDes_Iter_o_gamma_2}{../scripts/figures/StDes_Iter_o_gamma_1.png}{Αριθμός επαναλήψεων για διαφορετικές τιμές $\gamma_k$ [Μέγιστη Κάθοδος].} Επίσης παρατηρούμε ότι για μικρό \boldmath$\gamma_k = 0.1$ η σύγκλιση είναι ομαλή, ενώ για μεγάλο $\gamma_k = 0.3$ \unboldmath παρουσιάζει ταλάντωση κατά την σύγκλιση. -Παρακάτω παραθέτουμε την πορεία σύγκλισης και απόκλισης για τις διαφορετικές τιμές του $\gamma_k$. +Παρακάτω στο σχήμα \ref{fig:StDes_gamma} παραθέτουμε την πορεία σύγκλισης και απόκλισης για τις διαφορετικές τιμές του $\gamma_k$. \begin{figure}[ht] \centering @@ -157,7 +157,56 @@ \label{fig:StDes_gamma} \end{figure} -Απόδειξη... +\subsection{Μαθηματική ανάλυση} +Τα παραπάνω αποτελέσματα επιβεβαιώνονται και θεωρητικά. +Πιο συγκεκριμένα για τη σύγκλιση της μεθόδου μέγιστης καθόδου όπου το κάθε σημείο υπολογίζεται από την σχέση \boldmath$x_{k+1} = x_k - \gamma_k \nabla f(x_k)$, \unboldmath πρέπει να ισχύουν: +\begin{enumerate} + \item H $f$ να είναι κυρτή. + \item Η $f$ να είναι συνεχής και διαφορίσιμη και η κλίση της υπολογίσιμη. + \item Για το βήμα υπολογισμού να ισχύει η σχέση: + \begin{equation} 0 < \gamma_k < \frac{2}{L} \end{equation} \label{eq:gammaLimmit} + Όπου $L$ το άνω φράγμα της Lipschitz για την κλίση $\nabla f(x)$ (αν είναι γνωστή), η οποία είναι η μέγιστη ιδιοτιμή του Εσσιανού και δίνεται από τη σχέση: + \begin{equation} + L = \max_{x} \{\lambda_{max} (H(x))\}, \quad H(x) = \begin{bmatrix} \dfrac{\theta^2 f(x)}{\theta x_i \theta x_j} \end{bmatrix} + \end{equation} + \label{eq:Lipschitz} +\end{enumerate} +\par +Έτσι για τη δική μας περίπτωση έχουμε: +Η κλίση της $f(x)$ είναι: +\[ +\nabla f(x) = \begin{bmatrix} \frac{\partial f}{\partial x1} \\ \frac{\partial f}{\partial x2} \end{bmatrix} = \begin{bmatrix} \frac{2}{3}x_1 \\ 6x_2 \end{bmatrix} +\] +Ο Εσσιανός πίνακας $H(x)$ της $f(x)$ είναι: +\[ +H(x) = +\begin{bmatrix} + \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} \\ + \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} +\end{bmatrix} += +\begin{bmatrix} + \frac{2}{3} & 0 \\ 0 & 6 +\end{bmatrix} +\] +Και εφόσον είναι διαγώνιος οι ιδιοτιμές του είναι οι τιμές της διαγωνίου. Δηλαδή: +\[ +\det(H - \lambda I) = 0 \Leftrightarrow \\ +\det(\begin{bmatrix}\frac{2}{3} - \lambda & 0 \\ 0 & 6 - \lambda \end{bmatrix}) = 0 \Leftrightarrow \\ +\left(\frac{2}{3} - \lambda\right)(6 - \lambda) = 0 \Leftrightarrow \\ +\lambda_{\min} = \frac{2}{3}, \quad \lambda_{\max} = 6. +\] +Έτσι από τις εξισώσεις \ref{eq:gammaLimmit} και \ref{eq:Lipschitz} προκύπτει τελικά: +\boldmath\[ 0 < \gamma_k < \frac{1}{3} \]\unboldmath +Βλέπουμε ότι από την ανάλυσή μας επιβεβαιώνουμε τις τιμές που βρήκαμε εμπειρικά από την εκτέλεση του αλγορίθμου. + +\subsubsection{Εναλλακτικά} +Αν θέλαμε να βρούμε το κριτήριο σύγκλισης για το βήμα $\gamma_k$ ξεχωριστά για την κάθε διάσταση θα μπορούσαμε να θεωρήσουμε για 3ο κριτήριο πως για να συγκλίνει η μέθοδος θα πρέπει να ισχύει: +\[ + \norm{\frac{x_{k+1}}{x_k}} < 1 +\] +Από την παραπάνω εξίσωση προκείπτει: + \section{Μέθοδος Μέγιστης Καθόδου με προβολή} Πριν περάσουμε στις υπόλοιπες απαιτήσεις της εργασίας θα θέλαμε να παραθέσουμε κάποιες πληροφορίες για την υλοποίηση της μεθόδου μέγιστης καθόδου με προβολή (αρχείο: \textbf{method\_SteepDesc\_Proj.m}).