From 2b2df13940011c0a22910e8e539acdcca41be795 Mon Sep 17 00:00:00 2001 From: Christos Choutouridis Date: Wed, 27 Nov 2024 16:09:37 +0200 Subject: [PATCH] version 2 --- Work2/report/Work2_report.pdf | Bin 1106536 -> 1828959 bytes Work2/report/Work2_report.tex | 182 +++++++++++++----- Work2/scripts/GivenEnv.m | 4 + Work2/scripts/Script_1_Plots.m | 4 +- Work2/scripts/Script_2_Steepest_descent.m | 4 +- Work2/scripts/Script_3_Newton.m | 1 - Work2/scripts/Script_4_LevMar.m | 15 +- Work2/scripts/figures/FunctionContour.png | Bin 74551 -> 0 bytes Work2/scripts/figures/FunctionPlot.png | Bin 261511 -> 0 bytes .../scripts/figures/LevMar_Iter_o_gamma_2.png | Bin 0 -> 58660 bytes Work2/scripts/figures/LevMar_armijo_2.png | Bin 0 -> 100347 bytes Work2/scripts/figures/LevMar_armijo_3.png | Bin 0 -> 88404 bytes Work2/scripts/figures/LevMar_compare_2.png | Bin 0 -> 73995 bytes Work2/scripts/figures/LevMar_fixed_2.png | Bin 109065 -> 102342 bytes Work2/scripts/figures/LevMar_fixed_3.png | Bin 93598 -> 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zS-1>*AJ^r0i=Au$&wuS>c*~0D>fME^->m;3y0{jfNy&KrpVj2%ZszLl?P_I>&dtrm Q$Is7?PERkZDu@2R0D`mU)c^nh diff --git a/Work2/report/Work2_report.tex b/Work2/report/Work2_report.tex index 24d2f6c..5cb3830 100644 --- a/Work2/report/Work2_report.tex +++ b/Work2/report/Work2_report.tex @@ -36,9 +36,17 @@ \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} +\usepackage{commath} \usepackage{float} +\hypersetup{ + colorlinks=true, + linkcolor=red, % Internal links, those generated by cross-referenced elements + filecolor=blue, % Links to local files + urlcolor=blue % Links to web sites +} + \begin{document} \setlist[itemize]{topsep=0pt, partopsep=0pt, itemsep=3pt, parsep=3pt} @@ -74,7 +82,7 @@ \subsection{Κλήση μεθόδων επιλογής βήματος $\gamma_k$} \label{subsec:polymorphic-calls} Δεδομένου ότι οι μέθοδοι θα πρέπει να καλεστούν και εκτελεστούν με παραπάνω από μία τεχνική επιλογής βήματος $\gamma_k$, δημιουργήσαμε εσωτερικά της κάθε μεθόδου ένα κοινό interface για τις μεθόδους επιλογής βήματος. -Αυτό έχει τη μορφή: \textit{\textbf{gamma\_(f, dk, xk)}}, όπου το \textbf{f} είναι η αντικειμενική συνάρτηση, \textbf{dk} η τιμή της συνάρτησης κλίσης στο xk και \textbf{xk} το σημείο ενδιαφέροντος. +Αυτό έχει τη μορφή: \textit{\textbf{gamma\_(f, grad\_f, dk, xk)}}, όπου το \textbf{f} είναι η αντικειμενική συνάρτηση, \textbf{grad\_f} η συνάρτηση κλίσης της, \textbf{dk} η τιμή της συνάρτησης κλίσης στο xk και \textbf{xk} το σημείο ενδιαφέροντος. Για την κάθε μία από αυτές δημιουργήσαμε ξεχωριστή συνάρτηση που υλοποιεί το παραπάνω interface. Μία για σταθερό βήμα, μία για επιλογή βήματος που ελαχιστοποιεί την $f(x_k + \gamma_k d_k)$ και μία με τη μέθοδο Armijo. Για την επιλογή και κλήση των μεθόδων επιλογής βήματος εισαγάγαμε μία νέα παράμετρο string που χρησιμοποιείται ως enumerator και με βάση αυτή γίνεται η τελική επιλογή. @@ -93,7 +101,7 @@ 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} +\section{Απεικόνιση της συνάρτησης - Θέμα 1} Η συνάρτηση με την οποία ασχολούμαστε στην παρούσα εργασία είναι η: \boldmath \begin{equation} @@ -102,11 +110,11 @@ \label{eq:ObjectiveFunction} \unboldmath Στο παρακάτω σχήμα \ref{fig:plot3dFunction} φαίνεται η τρισδιάστατη απεικόνιση της συνάρτησης. -\InsertFigure{!h}{0.8}{fig:plot3dFunction}{../scripts/figures/FunctionPlot.png}{Γραφική παράσταση της f} +\InsertFigure{!h}{0.8}{fig:plot3dFunction}{../scripts/figures/Plot_Function.png}{Γραφική παράσταση της f} Από το σχήμα μπορούμε πολύ εύκολα να διακρίνουμε ότι η συνάρτηση έχει ένα ευκρινές μέγιστο και ένα ελάχιστο στο διάστημα $x,y \in [-3, 3]$. Για να πάρουμε μια καλύτερη αίσθηση για το που βρίσκονται αυτά τα τοπικά ακρότατα, παρακάτω παραθέτουμε ένα γράφημα με τις ισοβαρείς καμπύλες της $f$. -\InsertFigure{H}{0.8}{fig:plotContour}{../scripts/figures/FunctionContour.png}{Ισοβαρείς της f} +\InsertFigure{H}{0.8}{fig:plotContour}{../scripts/figures/Plot_Contour.png}{Ισοβαρείς της f} Από το παραπάνω σχήμα \ref{fig:plotContour} φαίνεται ότι το ελάχιστο της f βρίσκεται στο αρνητικό ημιεπίπεδο των χ, κοντά στο $y = 0$ \par @@ -116,10 +124,10 @@ Πριν προχωρήσουμε στα επόμενα θέματα της εργασίας και στην ανάλυση των μεθόδων υπολογισμού του ελάχιστου, θέλουμε να αναφερθούμε στις διαφορετικές τεχνικές επιλογής βήματος $\gamma_k$ και ειδικότερα για αυτή της ελαχιστοποίησης της $f(x_k + \gamma_k d_k)$ και την Armijo. \subsection{Ελαχιστοποίηση της $f(x_k + \gamma_k d_k)$} -Η μέθοδος αυτή αναζητά την τιμή $\gamma_k$ που ελαχιστοποιεί την τιμή της συνάρτησης κατά μήκος της κατεύθυνσης $d_k$. +Η μέθοδος αυτή αναζητά την τιμή $\gamma_k$ που ελαχιστοποιεί την τιμή της συνάρτησης κατά μήκος της κατεύθυνσης $d_k$ (αρχείο: \textbf{gamma\_minimized.m}). Δηλαδή, λύνουμε το μονοδιάστατο πρόβλημα: \boldmath -\[\displaystyle \min{\gamma_k} f(x_k + \gamma_k d_k) \] +\[\displaystyle \min{\gamma_k} \{f(x_k + \gamma_k d_k)\} \] Η κατεύθυνση $d_k$ μπορεί να είναι: \begin{itemize} \item Η αρνητική κλίση $-\nabla f(x_k)$ (Steepest Descent). @@ -130,127 +138,195 @@ Στα μειονεκτήματα μπορούμε να αναφέρουμε το \textbf{υπολογιστικό κόστος}, καθώς ο υπολογισμός του $\gamma_k$ απαιτεί πολλαπλές αξιολογήσεις της $f(x)$. \subsection{Armijo rule} -Η Armijo rule είναι μια προσαρμοστική τεχνική που επιλέγει το $\gamma_k$ για να εξασφαλίσει επαρκή μείωση της συνάρτησης. +Η Armijo rule είναι μια προσαρμοστική τεχνική που επιλέγει το $\gamma_k$ για να εξασφαλίσει επαρκή μείωση της συνάρτησης (αρχείο: \textbf{gamma\_armijo.m}). Η βασική ιδέα είναι ότι η συνάρτηση πρέπει να μειώνεται "αρκετά" σε κάθε βήμα, χωρίς να χρειάζεται να υπολογίζεται το ακριβές ελάχιστο. Η συνθήκη του Armijo είναι: \boldmath -\[ f(x_k + \gamma_k d_k) \leq f(x_k) + \sigma\gamma_k {d_k}^T*\nabla f(x_k) \] +\[ f(x_k + \gamma_k d_k) \leq f(x_k) + \sigma\gamma_k {d_k}^T\nabla f(x_k) \] Όπου $\sigma \in (0, 0.1)$ είναι μια σταθερά (τυπικά $\sigma = 0.1$) και $\gamma_k$ αρχικά να ορίζεται ως 1 και να μειώνεται προοδευτικά (π.χ., $\gamma_k = \beta \cdot \gamma_k$) έως ότου ικανοποιηθεί η συνθήκη. \unboldmath \par Πλεονεκτήματα της μεθόδου είναι η \textbf{σταθερότητα}, καθώς αποτρέπει πολύ μεγάλα βήματα που μπορεί να αυξήσουν την τιμή της $f(x)$, αλλά και η \textbf{ανθεκτικότητα}, καθώς λειτουργεί καλά ακόμα και όταν η $f(x)$ δεν συμπεριφέρεται πολύ καλά. Στα μειονεκτήματα μπορούμε να αναφέρουμε την \textbf{εξάρτησή της από τις παραμέτρους} \boldmath $\sigma$ και $\beta$\unboldmath, μια κακή επιλογή των οποίων μπορεί να οδηγήσει σε αργή σύγκλιση. -\section{Μέθοδος Μέγιστης Καθόδου - Steepest Descent} -Η πρώτη μέθοδος που χρησιμοποιούμε στην εργασία (Θέμα 2), είναι η μέθοδος μέγιστης καθόδου. +\section{Μέθοδος Μέγιστης Καθόδου - Θέμα 2} +Η πρώτη μέθοδος που χρησιμοποιούμε στην εργασία, είναι η μέθοδος μέγιστης καθόδου (Steepest descent) (αρχείο: \textbf{method\_steepest\_descent.m}). Είναι μια μέθοδος πρώτης τάξης που χρησιμοποιεί την κατεύθυνση της αρνητικής κλίσης $\nabla f(x,y)$ της $f$ ως κατεύθυνση καθόδου. Η μέθοδος θεωρείται βασική και συχνά χρησιμοποιείται ως εισαγωγή στις μεθόδους βελτιστοποίησης. \par Η μέθοδος επιλέγει την κατεύθυνση $d_k = -\nabla f(x_k)$, η οποία είναι η κατεύθυνση της μέγιστης τοπικής μείωσης της συνάρτησης. Στη συνέχεια, υπολογίζεται το βήμα $\gamma_k$​ για να βρεθεί το επόμενο σημείο $x_{k+1} = x_k + \gamma_k d_k$. Για να χρησιμοποιήσουμε τη μέθοδο, η συνάρτηση $f$ \textbf{πρέπει να είναι συνεχής και διαφορίσιμη} και η κλίση $\nabla f$ να είναι υπολογίσιμη. -Επίσης για την εφαρμογή της μεθόδου το αρχικό σημείο θα πρέπει να \textbf{μην είναι ακρότατο} της $f$, δηλαδή \boldmath$\nabla f(x_0) \neq 0$\unboldmath. -\par +Επίσης για την εφαρμογή της μεθόδου, για το αρχικό σημείο θα πρέπει \boldmath$\nabla f(x_0) \neq 0$\unboldmath. Όλοι οι υπολογισμοί και τα διαγράμματα για τη μέθοδο βρίσκονται στο αρχείο \textbf{Script\_2\_Steepest\_descent.m} \subsection{Σημείο εκκίνησης (0,0)} Για το σημείο (0, 0) η κλίση της $f$ είναι: $\nabla f(0,0) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$, με αποτέλεσμα η μέθοδος να μην μπορεί να εφαρμοστεί για κανένα τρόπο υπολογισμού βήματος. \subsection{Σημείο εκκίνησης (-1,1)} -Για το σημείο (-1, 1) η τιμή της $f$ είναι: $f(-1, 1) = -0.135335$ και το διάνυσμα της κλίσης: $\nabla f(0,0) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$, επομένως μπορούμε να εφαρμόσουμε τη μέθοδο. +Για το σημείο (-1, 1) η τιμή της $f$ είναι: $f(-1, 1) = -0.1353$ και το διάνυσμα της κλίσης: $\nabla f(0,0) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$, επομένως μπορούμε να εφαρμόσουμε τη μέθοδο. + \par \underline{Σταθερό βήμα} \\ Επιλέγοντας ακρίβεια $\epsilon = 0.0001$, εκτελούμε την μέθοδο \textit{method\_steepest\_descent()} και υπολογίζουμε τον αριθμό επαναλήψεων για διαφορετικές τιμές $\gamma_k$. -\InsertFigure{H}{0.8}{fig:point2ItersOverGamma}{../scripts/figures/StDes_Iter_o_gamma_2.png}{Αριθμός επαναλήψεων για διαφορετικές τιμές $\gamma_k$ [Μέγιστη Κάθοδος].} +\InsertFigure{H}{0.8}{fig:StDes_Iter_o_gamma_2}{../scripts/figures/StDes_Iter_o_gamma_2.png}{Αριθμός επαναλήψεων για διαφορετικές τιμές $\gamma_k$ [Μέγιστη Κάθοδος].} -Στο παραπάνω σχήμα \ref{fig:point2ItersOverGamma} παρατηρούμε ότι για τιμές του $\gamma_k > 0.61$ η μέθοδος αποκλίνει. +Στο παραπάνω σχήμα \ref{fig:StDes_Iter_o_gamma_2} παρατηρούμε ότι για τιμές του $\gamma_k > 0.61$ η μέθοδος αποκλίνει. Από την παραπάνω διαδικασία επίσης υπολογίζουμε το $\gamma_k = 0,46768$ για το οποίο η μέθοδος συγκλίνει με τα λιγότερα βήματα. Στο παρακάτω σχήμα \ref{fig:StDes_fixed_2} αναπαριστούμε την πορεία των σημείων καθώς συγκλίνουν στο ελάχιστο. \InsertFigure{H}{0.8}{fig:StDes_fixed_2}{../scripts/figures/StDes_fixed_2.png}{Σύγκλιση της μεθόδου Steepest descent [fixed $\gamma_k$].} + \par \underline{Ελαχιστοποίηση της $f(x_k + \gamma_k d_k$)} \\ Για την ελαχιστοποίηση της $f$, χρησιμοποιήθηκε η bisection από την προηγούμενη εργασία, η οποία τροποποιήθηκε ώστε δέχεται functions και όχι symbolic expressions. \InsertFigure{H}{0.8}{fig:StDes_minimized_2}{../scripts/figures/StDes_minimized_2.png}{Σύγκλιση της μεθόδου Steepest descent [minimized f].} Από το γράφημα φαίνεται τόσο ότι η μέθοδος συγκλίνει κοντά στο ελάχιστο γρηγορότερα, όσο και ότι πραγματοποιεί “διορθώσεις πορείας”. + \par \underline{Armijo rule} \\ Για τη μέθοδο η βασική ιδέα είναι να ξεκινήσει ο αλγόριθμος από ένα μεγάλο $\gamma_k = 1$ και συνεχώς να μειώνεται με βάση τον κανόνα Armijo. Μετά από ένα tuning των παραμέτρων της μεθόδου καταλήξαμε στα $\beta=0.4, \sigma=0.1$ \InsertFigure{H}{0.8}{fig:StDes_armijo_2}{../scripts/figures/StDes_armijo_2.png}{Σύγκλιση της μεθόδου Steepest descent [armijo rule].} +\par +\underline{Σύγκριση} \\ +Στο παρακάτω διάγραμμα φαίνονται τα βήματα αλλά ο ρυθμός σύγκλισης της κάθε μεθόδου. +\InsertFigure{H}{0.8}{fig:StDes_compare_2}{../scripts/figures/StDes_compare_2.png}{Ευκλείδεια απόσταση από το ελάχιστο για κάθε μέθοδο υπολογισμού $\gamma_k$ [Steepest descent].} + \subsection{Σημείο εκκίνησης (1,-1)} -Για το σημείο (1, -1) η τιμή της $f$ είναι: $f(1, -1) = -0.135335$ και το διάνυσμα της κλίσης: $\nabla f(0,0) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$, επομένως μπορούμε να εφαρμόσουμε τη μέθοδο. +Για το σημείο (1, -1) η τιμή της $f$ είναι: $f(1, -1) = -0.1353$ και το διάνυσμα της κλίσης: $\nabla f(0,0) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$, επομένως μπορούμε να εφαρμόσουμε τη μέθοδο. + \par \underline{Σταθερό βήμα} \\ -Για σταθερό βήμα εκτελέσαμε διαδοχικά τη μέθοδο \textit{method\_steepest\_descent()} για να υπολογίσουμε τον αριθμό επαναλήψεων για διαφορετικές τιμές $\gamma_k$, όμως σε καμία τιμή ο αλγόριθμος δεν συγκλίνει. -Ακόμα και για μεγάλες τιμές του $\gamma_k$, ο αλγόριθμος εγκλωβίζεται στο δεξιό ημιεπίπεδο. +Για σταθερό βήμα εκτελέσαμε διαδοχικά τη μέθοδο \textit{method\_steepest\_descent()} για να υπολογίσουμε τον αριθμό επαναλήψεων για διαφορετικές τιμές $\gamma_k$, όμως σε καμία τιμή ο αλγόριθμος δεν καταφέρνει να συγκλίνει. +Ακόμα και για μεγάλες τιμές βήματος, όπως για παράδειγμα $\gamma_k = 1$, ο αλγόριθμος εγκλωβίζεται στο δεξιό ημιεπίπεδο, όπως φαίνεται και στο διάγραμμα \ref{fig:StDes_fixed_3} παρακάτω. \InsertFigure{H}{0.8}{fig:StDes_fixed_3}{../scripts/figures/StDes_fixed_3.png}{Μη σύγκλιση της μεθόδου Steepest descent [Fixed step].} + \par \underline{Ελαχιστοποίηση της $f(x_k + \gamma_k d_k$)} \\ \InsertFigure{H}{0.8}{fig:StDes_minimized_3}{../scripts/figures/StDes_minimized_3.png}{Σύγκλιση της μεθόδου Steepest descent [minimized f].} -Από το γράφημα φαίνεται ότι η μέθοδος συγκλίνει, καταφέρνοντας να περάσει την περιοχή με μηδενικές κλίσεις κοντά στον άξονα των $\psi$. +Για υπολογισμό βήματος ελαχιστοποιώντας τη $f(x_k + \gamma_k d_k)$, όπως φαίνεται και από το γράφημα \ref{fig:StDes_minimized_3}, η μέθοδος συγκλίνει, καταφέρνοντας να περάσει την περιοχή με μηδενικές κλίσεις κοντά στον άξονα των y. + \par \underline{Armijo rule} \\ \InsertFigure{H}{0.8}{fig:StDes_armijo_3}{../scripts/figures/StDes_armijo_3.png}{Μη σύγκλιση της μεθόδου Steepest descent [armijo rule].} -Αντίθετα η μέθοδος armijo δεν συγκλίνει, καθώς εγκλωβίζεται στο δεξιό ημιεπίπεδο. +Αντίθετα η μέθοδος armijo δεν συγκλίνει, καθώς και αυτή εγκλωβίζεται στο δεξιό ημιεπίπεδο. -\section{Μέθοδος Newton} -Η δεύτερη μέθοδος που χρησιμοποιούμε στην εργασία (Θέμα 3), είναι η μέθοδος Newton. -Η μέθοδος χρησιμοποιεί πληροφορίες δεύτερης τάξης (Hessian) για τη βελτίωση της κατεύθυνσης καθόδου. -Αν η συνάρτηση είναι τετραγωνική, η μέθοδος συγκλίνει σε ένα βήμα. +\section{Μέθοδος Newton - Θέμα 3} +Η δεύτερη μέθοδος που χρησιμοποιούμε στην εργασία, είναι η μέθοδος Newton (αρχείο: \textbf{method\_newton.m}). +Η μέθοδος χρησιμοποιεί πληροφορίες δεύτερης τάξης (Εσσιανό - Hessian) για τη βελτίωση της κατεύθυνσης καθόδου. Η μέθοδος ορίζει την κατεύθυνση \boldmath\[d_k = -{H_k}^{-1}\nabla f(x_k)\]\unboldmath Όπου $H_k$ είναι ο Εσσιανός πίνακας της $f$ στο $x_k$. Το επόμενο σημείο υπολογίζεται ως \boldmath\[x_{k+1} = x_k + \gamma_k d_k\]\unboldmath -Με κατάλληλο υπολογισμό του βήματος. -Για να λειτουργήσει η μέθοδος η $f$ πρέπει να είναι \textbf{δύο φορές διαφορίσιμη} και ο Εσσιανός \boldmath$H_k$\unboldmath να είναι \textbf{θετικά ορισμένος και αντιστρέψιμος}. +Ο υπολογισμός βήματος γίνεται παλι είτε με σταθερό βήμα, είτε με ελαχιστοποίηση της $f(x_k + \gamma_k d_k)$, είτε με τον κανόνα armijo. +Για να λειτουργήσει η μέθοδος, η $f$ πρέπει να είναι \textbf{δύο φορές διαφορίσιμη} και ο Εσσιανός \boldmath$H_k$\unboldmath να είναι \textbf{θετικά ορισμένος και αντιστρέψιμος}. \par Στα πλεονεκτήματα της μεθόδου είναι η \textbf{ταχύτερη σύγκλιση} από την Steepest Descent για κυρτές συναρτήσεις και το γεγονός ότι εκμεταλλεύεται την \textbf{πληροφορία καμπυλότητας} της συνάρτησης. Όμως είναι υπολογιστικά δαπανηρή και δεν είναι ανθεκτική σε μη κυρτές συναρτήσεις ή σε περιπτώσεις όπου ο Εσσιανός είναι κακώς ορισμένος. - Όλοι οι υπολογισμοί για τη μέθοδο βρίσκονται στο αρχείο \textbf{Script\_3\_Newton.m} \subsection{Σημείο εκκίνησης (0,0)} Για το σημείο $x_k = (0, 0)$ η κλίση της $f$ είναι: $\nabla f(x_k) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ και ο εσσιανός $H(x_k) = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ με αποτέλεσμα η μέθοδος και εδώ να μην μπορεί να εφαρμοστεί για κανένα τρόπο υπολογισμού βήματος. \subsection{Σημείο εκκίνησης (-1,1)} -Για το σημείο $x_k = (-1, 1)$ η κλίση της $f$ είναι: $\nabla f(x_k) = \begin{bmatrix} 0.406 \\ 0.270 \end{bmatrix}$ και ο εσσιανός $H(x_k) = \begin{bmatrix} -0.270 & -0.812 \\ -0.812 & -0.270 \end{bmatrix}$ με ιδιοτιμές $\lambda = \begin{bmatrix} -1.082 \\ 0.541 \end{bmatrix}$. +Για το σημείο $x_k = (-1, 1)$ η κλίση της $f$ είναι: $\nabla f(x_k) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$ και ο εσσιανός $H(x_k) = \begin{bmatrix} -0.2707 & -0.8120 \\ -0.8120 & -0.2707 \end{bmatrix}$ με ιδιοτιμές $\lambda = \begin{bmatrix} -1.0827 \\ 0.5413 \end{bmatrix}$. Από τα παραπάνω προκύπτει πως ο Εσσιανός είναι αόριστος και άρα δεν μπορεί να εφαρμοστεί η μέθοδος, για κανένα τρόπο υπολογισμού βήματος. \subsection{Σημείο εκκίνησης (1,-1)} -Για το σημείο $x_k = (1, -1)$ η κλίση της $f$ είναι: $\nabla f(x_k) = \begin{bmatrix} 0.406 \\ 0.270 \end{bmatrix}$ και ο εσσιανός $H(x_k) = \begin{bmatrix} 0.270 & 0.812 \\ 0.812 & 0.270 \end{bmatrix}$ με ιδιοτιμές $\lambda = \begin{bmatrix} -0.541 \\ 1.082 \end{bmatrix}$. +Για το σημείο $x_k = (1, -1)$ η κλίση της $f$ είναι: $\nabla f(x_k) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$ και ο εσσιανός $H(x_k) = \begin{bmatrix} 0.2707 & 0.8120 \\ 0.8120 & 0.2707 \end{bmatrix}$ με ιδιοτιμές $\lambda = \begin{bmatrix} -0.5413 \\ 1.0827 \end{bmatrix}$. Και εδώ, από τα παραπάνω προκύπτει πως ο Εσσιανός είναι αόριστος και άρα δεν μπορεί να εφαρμοστεί η μέθοδος, για κανένα τρόπο υπολογισμού βήματος. -\section{Μέθοδος Levenberg-Marquardt} -Η τελευταία μέθοδος που χρησιμοποιούμε στην εργασία (Θέμα 4), είναι η μέθοδος Levenberg-Marquardt. +\section{Μέθοδος Levenberg-Marquardt - Θέμα 4} +Η τελευταία μέθοδος που χρησιμοποιούμε στην εργασία, είναι η μέθοδος Levenberg-Marquardt (αρχείο: \textbf{method\_lev\_mar.m}). Πρόκειται για μια τροποποιημένη έκδοση της μεθόδου Newton, η οποία εισάγει έναν παράγοντα απόσβεσης για τη σταθεροποίηση όταν ο εσσιανός δεν είναι θετικά ορισμένος. -Για το λόγο αυτό χρησιμοποιεί ένας προσαρμοσμένος εσσιανός $H_k' = H_k + \mu_k I$, όπου $\mu_k > 0$ ένας παράγοντας απόσβεσης. -Για μεγάλες τιμές του $\mu_k$ η μέθοδος συμπεριφέρεται σαν Gradient Descent. - -%Απαιτήσεις: -%Η f(x)f(x) πρέπει να είναι δύο φορές διαφορίσιμη. -%Υπολογισμός του λλ απαιτεί προσεκτική επιλογή παραμέτρων. -%Περιορισμοί: -%Η απόδοση εξαρτάται από την επιλογή του αρχικού λλ. -%Μπορεί να παρουσιάσει αργή σύγκλιση σε προβλήματα χωρίς κυρτότητα. -%Πλεονεκτήματα: -%Σταθερή ακόμη και για κακώς ορισμένα Hessians. -%Λειτουργεί καλά σε προβλήματα που συνδυάζουν γραμμικές και μη γραμμικές εξαρτήσεις. -%Μειονεκτήματα: -%Υψηλότερο υπολογιστικό κόστος σε σχέση με το Steepest Descent. - +\boldmath +Για το λόγο αυτό χρησιμοποιείται ένας προσαρμοσμένος εσσιανός $H_k' = H_k + \mu_k I$, όπου $\mu_k > 0$ ένας παράγοντας, τέτοιος ώστε ο $H_k'$ να είναι θετικά ορισμένος. +Έτσι σε κάθε βήμα, γίνεται έλεγχος των ιδιοτιμών του εσσιανού. +Αν υπάρχει ιδιοτιμή μικρότερη ή ίση με το 0, τότε υπολογίζεται ο παράγοντας \[\mu_k = \abs{\min\{\lambda(H_k)\}} + \epsilon \] +Όπου $\epsilon > 0$ ένας παράγοντας προσαύξησης ώστε να βεβαιωθούμε πως ο παραγόμενος προσαρμοσμένος Εσσιανός $H_k' = H_k + \mu_k I$ είναι θετικά ορισμένος και δεν έχει κάποια μηδενική ιδιοτιμή. +\unboldmath +Να σημειώσουμε εδώ πως για μεγάλες τιμές του $\mu_k$ η μέθοδος συμπεριφέρεται σαν Gradient Descent, ενώ για μικρές προσεγγίζει τη μέθοδο Newton. +\par +Για να λειτουργήσει η μέθοδος, η $f$ πρέπει να είναι \textbf{δύο φορές διαφορίσιμη}. +Επίσης σημαντικό ρόλο στην ευστάθεια της μεθόδου παίζει η επιλογή το $\epsilon$ και άρα κατ' επέκταση και του $\mu_k$. +Στα πλεονεκτήματα της μεθόδου έχουμε την σταθερότητα ακόμα και για κακώς ορισμένους Εσσιανούς πίνακες, η οποία όμως έρχεται με υπολογιστικό κόστος. Όλοι οι υπολογισμοί για τη μέθοδο βρίσκονται στο αρχείο \textbf{Script\_4\_LevMar.m} -\section{Σύγκριση των μεθόδων} -Εκτελώντας όλους του αλγόριθμους για τα ίδια δεδομένα, \textbf{για τον αριθμό επαναλήψεων} έχουμε: \\ +\subsection{Σημείο εκκίνησης (0,0)} +Για το σημείο $x_k = (0, 0)$ η κλίση της $f$ είναι: $\nabla f(x_k) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ και ο εσσιανός $H(x_k) = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ με αποτέλεσμα η μέθοδος και εδώ να μην μπορεί να εφαρμοστεί για κανένα τρόπο υπολογισμού βήματος. -\begin{itemize} - \item ... -\end{itemize} +\subsection{Σημείο εκκίνησης (-1,1)} +Για το σημείο $x_k = (0, 0)$ η κλίση της $f$ είναι: $\nabla f(x_k) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$ και η μέθοδος μπορεί να εφαρμοστεί. +\par +\underline{Σταθερό βήμα} \\ +Επιλέγοντας ακρίβεια $\epsilon = 0.0001$, προσπαθούμε να βρούμε τον συντελεστή $\epsilon$ για τον οποίο η μέθοδος είναι σταθερή. +Κατόπιν εκτελούμε την μέθοδο \textit{method\_lev\_mar()} και υπολογίζουμε τον αριθμό επαναλήψεων για διαφορετικές τιμές $\gamma_k$. +Μετά από tuning, καταλήξαμε στην τιμή $\epsilon = 0.3$ για την οποία αναζητήσαμε και υπολογίσαμε το βήμα $\gamma_k = 1.4152$ με τη γρηγορότερη σύγκλιση. +\InsertFigure{H}{0.8}{fig:LevMar_Iter_o_gamma_2}{../scripts/figures/LevMar_Iter_o_gamma_2.png}{Αριθμός επαναλήψεων για διαφορετικές τιμές $\gamma_k$ [Μέθοδος L-M].} + +Στο παρακάτω σχήμα \ref{fig:LevMar_fixed_2} αναπαριστούμε την πορεία των σημείων καθώς συγκλίνουν στο ελάχιστο. +Παρατηρούμε ωστόσο ότι η μέθοδος κάνει αρκετή "διόρθωση πορείας" μέχρι τη σύγκλιση. +\InsertFigure{H}{0.8}{fig:LevMar_fixed_2}{../scripts/figures/LevMar_fixed_2.png}{Σύγκλιση της μεθόδου Levenberg-Marquardt [fixed $\gamma_k$].} + +\par +\underline{Ελαχιστοποίηση της $f(x_k + \gamma_k d_k$)} \\ +Για την ελαχιστοποίηση της $f$, χρησιμοποιήθηκε και εδώ η bisection από την προηγούμενη εργασία. +\InsertFigure{H}{0.8}{fig:LevMar_minimized_2}{../scripts/figures/LevMar_minimized_2.png}{Σύγκλιση της μεθόδου Levenberg-Marquardt [minimized f].} +Από το γράφημα φαίνεται τόσο ότι η μέθοδος συγκλίνει κοντά στο ελάχιστο, όσο και ότι πραγματοποιεί “διορθώσεις πορείας”, μικρότερες όμως από αυτές του σταθερού βήματος. + +\par +\underline{Armijo rule} \\ +Για τη μέθοδο η βασική ιδέα και εδώ είναι να ξεκινήσει ο αλγόριθμος από ένα μεγάλο $\gamma_k = 1$ και συνεχώς να μειώνεται με βάση τον κανόνα Armijo. +Μετά από ένα tuning των παραμέτρων της μεθόδου καταλήξαμε στα $\beta=0.4, \sigma=0.1$ +\InsertFigure{H}{0.8}{fig:LevMar_armijo_2}{../scripts/figures/LevMar_armijo_2.png}{Σύγκλιση της μεθόδου Levenberg-Marquardt [armijo rule].} + +\par +\underline{Σύγκριση} \\ +Στο παρακάτω διάγραμμα φαίνονται τα βήματα αλλά ο ρυθμός σύγκλισης της κάθε μεθόδου. +\InsertFigure{H}{0.8}{fig:LevMar_compare_2}{../scripts/figures/LevMar_compare_2.png}{Ευκλείδεια απόσταση από το ελάχιστο για κάθε μέθοδο υπολογισμού $\gamma_k$ [Levenberg-Marquardt].} + +\subsection{Σημείο εκκίνησης (1,-1)} +Για το σημείο (1, -1) η τιμή της $f$ είναι: $f(1, -1) = -0.1353$ και το διάνυσμα της κλίσης: $\nabla f(0,0) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$, επομένως μπορούμε να εφαρμόσουμε τη μέθοδο. + +\par +\underline{Σταθερό βήμα} \\ +Για σταθερό βήμα εκτελέσαμε διαδοχικά τη μέθοδο \textit{method\_lev\_mar()} για να υπολογίσουμε τον αριθμό επαναλήψεων για διαφορετικές τιμές $\gamma_k$, όμως σε καμία τιμή ο αλγόριθμος δεν καταφέρνει να συγκλίνει. +Παρακάτω (σχήμα \ref{fig:LevMar_fixed_3}), παραθέτουμε ένα παράδειγμα $\gamma_k = 0.1$, όπου ο αλγόριθμος εγκλωβίζεται στο δεξιό ημιεπίπεδο. +\InsertFigure{H}{0.8}{fig:LevMar_fixed_3}{../scripts/figures/LevMar_fixed_3.png}{Μη σύγκλιση της μεθόδου Levenberg-Marquardt [Fixed step].} +\par +\underline{Ελαχιστοποίηση της $f(x_k + \gamma_k d_k$)} \\ +\InsertFigure{H}{0.8}{fig:LevMar_minimized_3}{../scripts/figures/LevMar_minimized_3.png}{Σύγκλιση της μεθόδου Levenberg-Marquardt [minimized f].} +Για υπολογισμό βήματος ελαχιστοποιώντας τη $f(x_k + \gamma_k d_k)$, όπως φαίνεται και από το γράφημα \ref{fig:LevMar_minimized_3}, η μέθοδος συγκλίνει, καταφέρνοντας να περάσει την περιοχή με μηδενικές κλίσεις κοντά στον άξονα των $\psi$. -\section{Συμπεράσματα} -Οι μέθοδοι της παρούσας εργασίας αποτελούν βασικές τεχνικές για την εύρεση του τοπικού ελαχίστου ... +\par +\underline{Armijo rule} \\ +\InsertFigure{H}{0.8}{fig:LevMar_armijo_3}{../scripts/figures/LevMar_armijo_3.png}{Μη σύγκλιση της μεθόδου Levenberg-Marquardt [armijo rule].} +Αντίθετα η μέθοδος armijo δεν συγκλίνει, καθώς και αυτή εγκλωβίζεται στο δεξιό ημιεπίπεδο. + +\section{Σύγκριση των μεθόδων - Συμπεράσματα} +Εκτελώντας όλους του αλγόριθμους σε συνδυασμό με όλες τις μεθόδους, για το σημείο (-1, 1), \textbf{για τον αριθμό επαναλήψεων} έχουμε: \\ + +\noindent +\renewcommand{\arraystretch}{1.2} +\begin{tabularx}{\textwidth}{% + >{\raggedleft\arraybackslash}m{0.21\textwidth} | + >{\centering\arraybackslash}m{0.21\textwidth} | + >{\centering\arraybackslash}m{0.21\textwidth} | + >{\centering\arraybackslash}m{0.21\textwidth} + } + Μέθοδος & Σταθερό βήμα & Ελαχιστοποίηση της f & Armijo \\ + \hline + St. Descent & 11 & 10 & 11 \\ + Newton & - & - & - \\ + Lev-Mar & 7 & 7 & 9 +\end{tabularx} \\ [2ex] +Από τον παραπάνω πίνακα, αλλά και από τη πορεία σύγκλισης συμπεραίνουμε πως ο αλγόριθμος Lev-Mar είναι ο πιο αποδοτικός. +Επίσης σε συνδυασμό με τον τρόπο επιλογής βήματος το οποίο ελαχιστοποιεί τη $f(x_k + \gamma_kd_k)$ ο αλγόριθμος εκτός από γρηγορότερος, καταφέρνει να απεγκλωβίζεται από περιοχές με πολύ μικρή κλίση. +Αντίθετα οι μέθοδοι με σταθερό βήμα και armijo, φαίνεται ότι παρουσιάζουν τοπικό χαρακτήρα, καθώς εγκλωβίζονται εύκολα, με αποτέλεσμα να μην υπάρχει εγγύηση ότι το σημείο που βρέθηκε είναι ολικό ελάχιστο. +Ακόμα η μέθοδος Newton δεν φάνηκε να μπορεί να χρησιμοποιηθεί στην παρούσα εργασία, καθώς η επιλογή των σημείων ήταν τέτοια που είτε η κλίση ήταν 0, είτε ο Εσσιανός κακώς ορισμένος. +Αυτό φυσικά αναδεικνύει απλώς την μικρότερη εφαρμοσιμότητα του αλγορίθμου. \end{document} diff --git a/Work2/scripts/GivenEnv.m b/Work2/scripts/GivenEnv.m index ec517e4..39a7099 100644 --- a/Work2/scripts/GivenEnv.m +++ b/Work2/scripts/GivenEnv.m @@ -15,6 +15,10 @@ fun = matlabFunction(fexpr, 'Vars', [x, y]); % Function grad_fun = matlabFunction(grad_fexpr, 'Vars', [x, y]); % Gradient hessian_fun = matlabFunction(hessian_fexpr, 'Vars', [x, y]); % Hessian +% Minimum reference +Freference = @(x) x(1).^5 .* exp(-x(1).^2 - x(2).^2); +[Xmin, Fmin] = fminsearch(Freference, [-1, -1]); + % Amijo globals global amijo_beta; % Step reduction factor in [0.1, 0.5] (typical range: [0.1, 0.8]) global amijo_sigma; % Sufficient decrease constant in [1e-5, 0.1] (typical range: [0.01, 0.3]) diff --git a/Work2/scripts/Script_1_Plots.m b/Work2/scripts/Script_1_Plots.m index 7ab9e5a..523e6d5 100644 --- a/Work2/scripts/Script_1_Plots.m +++ b/Work2/scripts/Script_1_Plots.m @@ -8,7 +8,7 @@ GivenEnv % % 3d plot the function -plot3dFun(fun, [-3, 3], [-3, 3], 100, title_fun, "figures/FunctionPlot.png"); +plot3dFun(fun, [-3, 3], [-3, 3], 100, title_fun, "figures/Plot_Function.png"); % Plot isobaric lines -plotContour(fun, [-3, 3], [-3, 3], 100, title_fun, "figures/FunctionContour.png"); \ No newline at end of file +plotContour(fun, [-3, 3], [-3, 3], 100, title_fun, "figures/Plot_Contour.png"); \ No newline at end of file diff --git a/Work2/scripts/Script_2_Steepest_descent.m b/Work2/scripts/Script_2_Steepest_descent.m index 71aefbe..0b5408f 100644 --- a/Work2/scripts/Script_2_Steepest_descent.m +++ b/Work2/scripts/Script_2_Steepest_descent.m @@ -63,6 +63,9 @@ fprintf('Armijo step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), plotPointsOverContour(x_armijo, fun, [-2, 0], [-2, 2], 100, point_str + ": Steepest descent Armijo method", "figures/StDes_armijo_" + point + ".png"); disp(' '); +% Compare methods +plotConvCompare(x_fixed, "Fixed", x_minimized, "Minimized", x_armijo, "Armijo", Xmin, "Convergence compare", "figures/StDes_compare_" + point + ".png"); + % Point x0 = (1, -1) % ========================================================================= point = 3; @@ -111,4 +114,3 @@ fprintf('Armijo step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), plotPointsOverContour(x_armijo, fun, [-1, 2], [-2, 2], 100, point_str + ": Steepest descent Armijo method", "figures/StDes_armijo_" + point + ".png"); - diff --git a/Work2/scripts/Script_3_Newton.m b/Work2/scripts/Script_3_Newton.m index dba9111..946d768 100644 --- a/Work2/scripts/Script_3_Newton.m +++ b/Work2/scripts/Script_3_Newton.m @@ -40,4 +40,3 @@ ev = eig(hf); fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Eigenvalues= [%f, %f], Can NOT use method\n', x0, f, gf, hf, ev); - diff --git a/Work2/scripts/Script_4_LevMar.m b/Work2/scripts/Script_4_LevMar.m index 57e5b82..a80326a 100644 --- a/Work2/scripts/Script_4_LevMar.m +++ b/Work2/scripts/Script_4_LevMar.m @@ -43,6 +43,7 @@ for g = n end j = j + 1; end +plotItersOverGamma(n, k, "Iteration for different $\gamma$ values", "figures/LevMar_Iter_o_gamma_" + point + ".png"); [~, j] = min(k); gamma_fixed_step = n(j); @@ -51,9 +52,9 @@ gamma_fixed_step = n(j); fprintf('Fixed step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_fixed(end, :), f_fixed(end)); plotPointsOverContour(x_fixed, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt $\gamma$ = " + gamma_fixed_step, "figures/LevMar_fixed_" + point + ".png"); -[x_fixed, f_fixed, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'minimized'); -fprintf('Minimized f(g): Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_fixed(end, :), f_fixed(end)); -plotPointsOverContour(x_fixed, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt minimized $f(x_k + \gamma_kd_k)$", "figures/LevMar_minimized_" + point + ".png"); +[x_minimized, f_minimized, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'minimized'); +fprintf('Minimized f(g): Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_minimized(end, :), f_minimized(end)); +plotPointsOverContour(x_minimized, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt minimized $f(x_k + \gamma_kd_k)$", "figures/LevMar_minimized_" + point + ".png"); % Armijo Rule @@ -63,9 +64,11 @@ amijo_sigma = 0.1; % typical range: [0.01, 0.3] [x_armijo, f_armijo, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'armijo'); fprintf('Armijo step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_armijo(end, :), f_armijo(end)); -plotPointsOverContour(x_armijo, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/StDes_armijo_" + point + ".png"); +plotPointsOverContour(x_armijo, fun, [-3, 0], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/LevMar_armijo_" + point + ".png"); disp(' '); +% Compare methods +plotConvCompare(x_fixed, "Fixed", x_minimized, "Minimized", x_armijo, "Armijo", Xmin, "Convergence compare", "figures/LevMar_compare_" + point + ".png"); % Point x0 = (1, -1) % ========================================================================= @@ -112,5 +115,5 @@ amijo_sigma = 0.1; % typical range: [0.01, 0.3] [x_armijo, f_armijo, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'armijo'); fprintf('Armijo step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), f(x,y)=%f\n', x0, kk, x_armijo(end, :), f_armijo(end)); -plotPointsOverContour(x_armijo, fun, [-3, 2], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/StDes_armijo_" + point + ".png"); -disp(' '); +plotPointsOverContour(x_armijo, fun, [-3, 2], [-2, 2], 100, point_str + ": Levenberg-Marquardt Armijo method", "figures/LevMar_armijo_" + point + ".png"); +disp(' '); \ No newline at end of file diff --git a/Work2/scripts/figures/FunctionContour.png b/Work2/scripts/figures/FunctionContour.png deleted file mode 100644 index 530ae8284a0cf343c4aa2b5bd76937d6a30bffde..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 74551 zcmeFac{rBs+BSY`QfW|W5TY_B$t<%d9NM=$R5E)8l$t+1o<|aZ!$~={s z%pxSd{W3kR_gm|?-fz9nw*9_8zI%JtdT`(Ob)Ca8?8koW=k?rCQjlJ|di!b;iL~~p z%wZK0i9&)zB6nW71b?Z}ll+N)t-368(tD>x`4~}I2!@ehcYfA4hZO@PWM0x(p zrn?*2m6fR;;~(c7=Lj<8y;NjmY$J{}Whu=zBsQM6MrWt12kc)~2c}hL3iR@H7B-|j zI2fl!tjOZ)p4YX8Rg1sjhc4mOf`9(5dxVVqkKdh3Sik=9E9oBfa@s$C=YH(G^pD>q z^r_`t}=5MlF9b+;db4`vNHUwXk^EcU5)8^X`f$*(jC%Woe4*O{s~=D(zZ5ZMJf5+GmPCejNMpBUs$Y(QAjO+OMm9 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