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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. @@ -164,16 +164,15 @@ \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} + \begin{equation} 0 < \gamma_k < \frac{2}{L} \label{eq:gammaLimmit} \end{equation} 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} + L = \max_{x} \{\lambda_{max} (H(x))\} + \label{eq:Lipschitz} \end{equation} - \label{eq:Lipschitz} \end{enumerate} \par -Έτσι για τη δική μας περίπτωση έχουμε: -Η κλίση της $f(x)$ είναι: +Έτσι για τη δική μας περίπτωση η κλίση της $f(x)$ είναι: \[ \nabla f(x) = \begin{bmatrix} \frac{\partial f}{\partial x1} \\ \frac{\partial f}{\partial x2} \end{bmatrix} = \begin{bmatrix} \frac{2}{3}x_1 \\ 6x_2 \end{bmatrix} \] @@ -191,22 +190,61 @@ H(x) = \] Και εφόσον είναι διαγώνιος οι ιδιοτιμές του είναι οι τιμές της διαγωνίου. Δηλαδή: \[ -\det(H - \lambda I) = 0 \Leftrightarrow \\ -\det(\begin{bmatrix}\frac{2}{3} - \lambda & 0 \\ 0 & 6 - \lambda \end{bmatrix}) = 0 \Leftrightarrow \\ -\left(\frac{2}{3} - \lambda\right)(6 - \lambda) = 0 \Leftrightarrow \\ -\lambda_{\min} = \frac{2}{3}, \quad \lambda_{\max} = 6. + \det(H - \lambda I) = 0 \Leftrightarrow + \det(\begin{bmatrix}\frac{2}{3} - \lambda & 0 \\ 0 & 6 - \lambda \end{bmatrix}) = 0 \Leftrightarrow + \left(\frac{2}{3} - \lambda\right)(6 - \lambda) = 0 \Leftrightarrow + \lambda_{\min} = \frac{2}{3}, \quad \lambda_{\max} = 6. \] -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. +Έτσι από τις εξισώσεις (\ref{eq:gammaLimmit}) και (\ref{eq:Lipschitz}) προκύπτει τελικά ότι $\frac{2}{L} = \frac{2}{\lambda_{max}} = \frac{1}{3}$ και επομένως για σύγκλιση της μεθόδου πρέπει να ισχύει: +\boldmath +\begin{equation} + 0 < \gamma_k < \frac{1}{3} + \label{eq:gammaConvergence} +\end{equation} +\unboldmath +Βλέπουμε ότι από την ανάλυσή μας, η τιμή που βρήκαμε εμπειρικά για την απόκλιση από την εκτέλεση του αλγορίθμου για το $\gamma > 0.33$ επιβεβαιώνεται. +\par +Επίσης το γεγονός ότι ο Εσσιανός είναι διαγώνιος, μας δίνει επίσης πληροφορία ότι: +\boldmath +\begin{itemize} + \item Η ιδιοτιμή $\lambda_1 = \frac{2}{3}$ εφαρμοσμένη στη σχέση (\ref{eq:gammaLimmit}) μας δίνει: \begin{equation} 0<\gamma_k<3 \label{eq:gammaLimmit_x1} \end{equation} + που είναι η τιμή για την οποία η μέθοδος συγκλίνει στη διάσταση $x_1$ + \item Η ιδιοτιμή $\lambda_2 = 6$ εφαρμοσμένη στη σχέση (\ref{eq:gammaLimmit}) μας δίνει: + \begin{equation} 0<\gamma_k<\frac{1}{3}\label{eq:gammaLimmit_x2} \end{equation} + που είναι η τιμή για την οποία η μέθοδος συγκλίνει στη διάσταση $x_2$ +\end{itemize} +\unboldmath -\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: +\par +\underline{Εναλλακτικά} \\[0.5ex] +Αν θέλαμε να βρούμε το κριτήριο σύγκλισης για το βήμα $\gamma_k$ ξεχωριστά για την κάθε διάσταση θα μπορούσαμε να θεωρήσουμε ισοδύναμα με την παραπάνω ανάλυση, πως για να συγκλίνει η μέθοδος θα πρέπει να ισχύει: \[ - \norm{\frac{x_{k+1}}{x_k}} < 1 + \abs{\frac{x_{k+1}}{x_k}} < 1 \] -Από την παραπάνω εξίσωση προκείπτει: - +Από την παραπάνω εξίσωση προκύπτει πάλι ένα σύστημα εξισώσεων για την κάθε διάσταση: +\[ +\left\{ +\begin{aligned} + \abs{\dfrac{x_{1,k} - \frac{2}{3}\gamma_k x_{1,k}}{x_{1,k}}} < 1 \\ + \abs{\dfrac{x_{2,k} - 6\gamma_k x_{2,k}}{x_{2,k}}} < 1 +\end{aligned} +\right. +\Leftrightarrow +\left\{ +\begin{aligned} + \abs{1 - \frac{2}{3}\gamma_k} < 1 \\ + \abs{1 - 6\gamma_k} < 1 +\end{aligned} +\right. +\Leftrightarrow +\left\{ +\begin{aligned} + 0 < \gamma_k < 3 \\ + 0 < \gamma_k < \frac{1}{3} +\end{aligned} +\right. +\] +Που επιβεβαιώνει την προηγούμενη ανάλυσή μας για τη συνολική σύγκλιση, αλλά και για την κάθε διάσταση ξεχωριστά. \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}). @@ -215,32 +253,59 @@ H(x) = Αυτό σημαίνει ότι μπορεί να χρησιμοποιηθεί και για σημεία εκκίνησης εκτός του συνόλου των περιορισμών. Ο αλγόριθμος είναι παρόμοιος με αυτόν της προηγούμενης εργασίας με τη διαφορά ότι η διεύθυνση $d_k$ επιλέγεται από τη σχέση: \[ - d_k = Pr_X\{ x_k - s_k \nabla f(x_k)\} - x_k + d_k = Pr_X\{ x_k - s_k \nabla f(x_k)\} - x_k = \bar{x_k} - x_k +\] +και τελικά έχουμε: +\[ + x_{k+1} = x_k + \gamma_k d_k \] Δηλαδή εφαρμόζουμε πρώτα τη μέθοδο μέγιστης καθόδου με βήμα $s_k$ στην κατεύθυνση $-\nabla f$ και έπειτα προβάλουμε το σημείο στο σύνολο $X$ και χρησιμοποιούμε αυτό ως διεύθυνση με βήμα $\gamma_k$. +\par +Αξίζει να παρατηρήσουμε πως \textbf{αν το σημείο $x_k$ είναι εντός του συνόλου περιορισμών} τότε η προβολή του σημείου στο σύνολο, είναι το ίδιο το σημείο και έτσι: +\[ +\begin{aligned} + x_{k+1} & = x_k + \gamma_k \left( Pr_X\{ x_k - s_k \nabla f(x_k)\} - x_k \right) \\ + & = x_k + \gamma_k \left( ( x_k - s_k \nabla f(x_k) ) - x_k \right) \\ + & = x_k - \gamma_k s_k \nabla f(x_k) \\ + & = x_k - \gamma_k' \nabla f(x_k) +\end{aligned} +\] +Βλέπουμε λοιπόν ότι για σημεία εσωτερικά του Χ, έχουμε ουσιαστικά τη μέθοδο της μέγιστης καθόδου με βήμα $\gamma_k' = \gamma_k s_k$. +Εφόσον όμως για μη εφικτά σημεία, πριν εφαρμόσουμε τη μέθοδο, τα προβάλουμε στο σύνολο των περιορισμών Χ, βλέπουμε πως η \textbf{μέθοδος τελικά προσεγγίζει τη μέγιστη κάθοδο με βήμα $\gamma_k'$, που είναι και το βήμα για το οποίο ελέγχουμε αν ισχύουν τα κριτήρια σύγκλισης}. -\section{Μέθοδος Μέγιστης Καθόδου με προβολή $s_k = 5, \gamma_k = 0.5$ - Θέμα 2} -Εφαρμόζοντας τη μέθοδο για ακρίβεια $\epsilon = 0.01$, $s_k = 5$ και $\gamma_k = 0.5$ έχουμε: +\section{Θέμα 2 - Σημείο (5, -5), $s_k = 5, \gamma_k = 0.5$} +Εφαρμόζοντας τη μέθοδο για ακρίβεια $\epsilon = 0.01$, $s_k = 5$ και $\gamma_k = 0.5$ έχουμε $\gamma_k' = 2.5 > \frac{1}{3}$, άρα το κριτήριο σύγκλισης δεν πληρείται. \InsertFigure{H}{0.8}{fig:StDesProj_sk_5_gamma_0.5}{../scripts/figures/StDesProj_sk_5_gamma_0.5.png}{Μέθοδος μέγιστης καθόδου με προβολή για $s_k = 5, \gamma_k = 0.5$.} Παρατηρούμε πως ενώ η μέθοδος ταλαντώνει και δεν συγκλίνει στο ελάχιστο \textbf{όπως και η αντίστοιχη εκτέλεση της μέγιστης καθόδου χωρίς περιορισμούς με το ίδιο} \boldmath$\gamma_k$\unboldmath. Παρόλα αυτά όμως, η ταλάντωση λαμβάνει χώρα \textbf{εντός του συνόλου των περιορισμών} της εκφώνησης. -\section{Μέθοδος Μέγιστης Καθόδου με προβολή $s_k = 15, \gamma_k = 0.1$ - Θέμα 3} -Εφαρμόζοντας τη μέθοδο για ακρίβεια $\epsilon = 0.01$, $s_k = 15$ και $\gamma_k = 0.1$ έχουμε: +\section{Θέμα 3 - Σημείο (-5, 10), $s_k = 15, \gamma_k = 0.1$} +Εφαρμόζοντας τη μέθοδο για ακρίβεια $\epsilon = 0.01$, $s_k = 15$ και $\gamma_k = 0.1$ έχουμε $\gamma_k' = 1.5 > \frac{1}{3}$, άρα το κριτήριο σύγκλισης δεν πληρείται και πάλι. \InsertFigure{H}{0.8}{fig:StDesProj_sk_15_gamma_0.1}{../scripts/figures/StDesProj_sk_15_gamma_0.1.png}{Μέθοδος μέγιστης καθόδου με προβολή για $s_k = 15, \gamma_k = 0.1$.} -Εδώ παρατηρούμε πως ενώ το $\gamma_k$ έχει επιλεγεί στο εύρος που οδηγεί σε σύγκλιση, το αντίστοιχο βήμα $s_k$ είναι πολύ μεγάλο, με αποτέλεσμα η μέθοδος να ταλαντώνει και πάλι. +Και εδώ παρατηρούμε πως ενώ το $\gamma_k$ έχει επιλεγεί θεωρητικά στο εύρος που οδηγεί σε σύγκλιση, το αντίστοιχο βήμα $s_k$ είναι πολύ μεγάλο, με αποτέλεσμα το γινόμενό τους $\gamma_k' = 15 * 0.1 = 1.5$ συνολικά να μην πληροί το κριτήριο $\gamma_k' < \frac{1}{3}$ και η μέθοδος να ταλαντώνει και πάλι. Αυτή τη φορά μόνο στον άξονα $x_2$. +Αυτό βέβαια εξηγείται από την παραπάνω ανάλυσή καθώς βλέπουμε πως ενώ το γινόμενο 1.5 δεν πληροί τις προϋποθέσεις για σύγκλιση, πληροί όμως τις προϋποθέσεις της εξίσωσης (\ref{eq:gammaLimmit_x1}) με αποτέλεσμα να ταλαντώνει μόνο στον άξονα $x_2$ +Αυτό φυσικά είναι αληθές και για την προηγούμενη περίπτωση όπου το $\gamma_k' = 5 * 0.5 = 2.5$ και όπου πάλι η σύγκλιση ήταν μερική, μόνο για την διάσταση $x_1$. -\section{Μέθοδος Μέγιστης Καθόδου με προβολή $s_k = 0.1, \gamma_k = 0.2$ - Θέμα 4} -Αρχικά παρατηρούμε πως το σημείο δεν είναι εφικτό, καθώς είναι εκτός του συνόλου των περιορισμών της εκφώνησης. +\section{Θέμα 4 - Σημείο (8, -10), $s_k = 0.1, \gamma_k = 0.2$ - Θέμα 4} +Αρχικά παρατηρούμε πως το σημείο \textbf{δεν είναι εφικτό}, καθώς είναι εκτός του συνόλου των περιορισμών της εκφώνησης. Αυτό βέβαια δεν μας αποτρέπει από την εφαρμογή της μεθόδου, καθώς αρχικά μπορούμε να προβάλουμε το σημείο στο σύνολο και να εφαρμόσουμε τη μέθοδο έπειτα. -Ακόμα, αυτή τη φορά οι τιμές των βημάτων $s_k, \gamma_k$, έχουν επιλεγεί μέσα στο εύρος για το οποίο έχουμε σύγκλιση, επομένως αναμένουμε η μέθοδος να συγκλίνει στο ελάχιστο. +Ακόμα, αυτή τη φορά οι τιμές των βημάτων $s_k, \gamma_k$, έχουν επιλεγεί μέσα στο εύρος για το οποίο έχουμε σύγκλιση, καθώς: $\gamma_k' = 0.1 * 0.2 = 0.02 < \frac{1}{3}$, επομένως αναμένουμε η μέθοδος να συγκλίνει στο ελάχιστο. Εφαρμόζοντας τη μέθοδο για ακρίβεια $\epsilon = 0.01$ έχουμε: \InsertFigure{H}{0.8}{fig:StDesProj_sk_0.1_gamma_0.2}{../scripts/figures/StDesProj_sk_0.1_gamma_0.2.png}{Μέθοδος μέγιστης καθόδου με προβολή για $s_k = 0.1, \gamma_k = 0.2$.} +Όπου βλέπουμε πως η μέθοδος συγκλίνει επιβεβαιώνοντας την παραπάνω ανάλυση. \section{Συμπεράσματα} -... + +Η μέθοδος μέγιστης καθόδου, με και χωρίς προβολή, παρουσιάζει τις αναμενόμενες συμπεριφορές σύγκλισης ανάλογα με την επιλογή του βήματος $\gamma_k$. +Για την περίπτωση χωρίς προβολή, επιβεβαιώθηκε θεωρητικά και εμπειρικά ότι η μέθοδος συγκλίνει μόνο όταν το $\gamma_k$ βρίσκεται εντός του εύρους που ορίζεται από την ανάλυση της Lipschitz σταθεράς, δηλαδή $0 < \gamma_k < \frac{1}{3}$. +​Επιπλέον, παρατηρήθηκε πως για μεγαλύτερες τιμές$\gamma_k$ η μέθοδος αποκλίνει, όπως προβλέπεται από την μαθηματική ανάλυση. + +Αντίστοιχα, για τη μέθοδο με προβολή, αποδεικνύεται πως το κριτήριο σύγκλισης εξαρτάται από τον συνδυασμό του $s_k$ (βήμα αντίθετο στην κατεύθυνση της κλίσης) και του $\gamma_k$ (βήμα κατά την προβολή). +Η ανάλυση επιβεβαίωσε ότι η μέθοδος προσεγγίζει τη μέθοδο χωρίς προβολή για εφικτά σημεία, ενώ για μη εφικτά σημεία εξασφαλίζει ότι οι διαδοχικές προσεγγίσεις παραμένουν εντός των περιορισμών. + +Τέλος, μέσα από τα παραδείγματα, αναδείχθηκε η σημασία της σωστής επιλογής των παραμέτρων $\gamma_k$ και $s_k$ για τη συνολική σταθερότητα και σύγκλιση της μεθόδου, καθώς και ο τρόπος με τον οποίο η ανάλυση των ιδιοτιμών του Εσσιανού επιτρέπει την καλύτερη κατανόηση της συμπεριφοράς της μεθόδου. \end{document}