From d5c8ed5e68b59a65cb13aa231083709635abe0a0 Mon Sep 17 00:00:00 2001 From: Christos Choutouridis Date: Sun, 1 Dec 2024 22:08:21 +0200 Subject: [PATCH] First work3 code version and report initialization --- Work 3/report/Work3_report.pdf | Bin 0 -> 82332 bytes Work 3/report/Work3_report.tex | 93 +++++++++++++++++++++++ Work 3/scripts/BelongsTo.m | 19 +++++ Work 3/scripts/GivenEnv.asv | 33 ++++++++ Work 3/scripts/GivenEnv.m | 36 +++++++++ Work 3/scripts/ProjectionPoint.m | 18 +++++ Work 3/scripts/Script_0_Plots.m | 14 ++++ Work 3/scripts/Script_1_SteepDesc.m | 29 +++++++ Work 3/scripts/Script_2_SteepDesc_Proj.m | 32 ++++++++ Work 3/scripts/Script_3_SteepDesc_Proj.m | 27 +++++++ Work 3/scripts/Script_4_SteepDesc_Proj.m | 27 +++++++ Work 3/scripts/figures/Plot_Contour.png | Bin 0 -> 92696 bytes Work 3/scripts/figures/Plot_Function.png | Bin 0 -> 1166206 bytes Work 3/scripts/figures/StDes_fixed_1.png | Bin 0 -> 114536 bytes Work 3/scripts/figures/StDes_fixed_2.png | Bin 0 -> 112927 bytes Work 3/scripts/fmin_bisection.m | 49 ++++++++++++ Work 3/scripts/gamma_armijo.m | 32 ++++++++ Work 3/scripts/gamma_fixed.m | 12 +++ Work 3/scripts/gamma_minimized.m | 24 ++++++ Work 3/scripts/method_SteepDesc.m | 39 ++++++++++ Work 3/scripts/method_SteepDesc_Proj.m | 50 ++++++++++++ Work 3/scripts/plot3dFun.m | 42 ++++++++++ Work 3/scripts/plotContour.m | 41 ++++++++++ Work 3/scripts/plotConvCompare.m | 54 +++++++++++++ Work 3/scripts/plotItersOverGamma.m | 28 +++++++ Work 3/scripts/plotPointsOverContour.m | 48 ++++++++++++ Work 3/Εργασία3.pdf | Bin 0 -> 628026 bytes 27 files changed, 747 insertions(+) create mode 100644 Work 3/report/Work3_report.pdf create mode 100644 Work 3/report/Work3_report.tex create mode 100644 Work 3/scripts/BelongsTo.m create mode 100644 Work 3/scripts/GivenEnv.asv create mode 100644 Work 3/scripts/GivenEnv.m create mode 100644 Work 3/scripts/ProjectionPoint.m create mode 100644 Work 3/scripts/Script_0_Plots.m create mode 100644 Work 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0000000..5f2562d --- /dev/null +++ b/Work 3/report/Work3_report.tex @@ -0,0 +1,93 @@ +% +% Optimization Techniques Work 3 report +% +% authors: +% Χρήστος Χουτουρίδης ΑΕΜ 8997 +% cchoutou@ece.auth.gr + + +\documentclass[a4paper, 11pt]{AUTHReport} + +% Document configuration +\AuthorName{Χρήστος Χουτουρίδης} +\AuthorAEM{8997} +\AuthorMail{cchoutou@ece.auth.gr} + +%\CoAuthorName{CoAuthor Name} +%\CoAuthorAEM{AEM} +%\CoAuthorMail{CoAuthor Mail} + +% \WorkGroup{Ομάδα Χ} + +\DocTitle{3η Εργαστηριακή Άσκηση} +\DocSubTitle{Μέθοδος Μέγιστης Καθόδου με Προβολή} + +\Department{Τμήμα ΗΜΜΥ. Τομέας Ηλεκτρονικής} +\ClassName{Τεχνικές Βελτιστοποίησης} + +\InstructorName{Γ. Ροβιθάκης} +\InstructorMail{rovithak@auth.gr} + +\CoInstructorName{Θ. Αφορόζη} +\CoInstructorMail{taforozi@ece.auth.gr} + +\CurrentDate{\today} + + +\usepackage{enumitem} +\usepackage{tabularx} +\usepackage{array} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{commath} + +\usepackage{float} + + +\begin{document} + +\InsertTitle + +%\tableofcontents + +\sloppy + +\section{Εισαγωγή} +Η παρούσα εργασία αφορά το πρόβλημα της ελαχιστοποίησης μιας δοσμένης συνάρτησης πολλών μεταβλητών $f: \mathbb{R}^n \rightarrow \mathbb{R}$ χωρίς περιορισμούς. +Για το σκοπό αυτό κάνουμε χρήση τριών μεθόδων. +Της μεθόδου μέγιστης καθόδου (Steepest Descent), της μεθόδου Newton, και της Levenberg-Marquardt. +Ακόμα για κάθε μία από αυτές θα υλοποιήσουμε τρεις διαφορετικές τεχνικές υπολογισμού βήματος. + +\section{Παραδοτέα} +Τα παραδοτέα της εργασίας αποτελούνται από: +\begin{itemize} + \item Την παρούσα αναφορά. + \item Τον κατάλογο \textbf{scripts/}, που περιέχει τον κώδικα της MATLAB. + \item Το \href{https://git.hoo2.net/hoo2/OptimizationTechniques/src/branch/master/Work%203}{σύνδεσμο} με το αποθετήριο που περιέχει όλο το project με τον κώδικα της MATLAB, της αναφοράς και τα παραδοτέα. +\end{itemize} + +\section{Προγραμματιστική προσέγγιση} +Για τον προγραμματισμό και εκτέλεση των μεθόδων της παρούσας εργασίας έγινε χρήση της MATLAB. +Στον κατάλογο \textbf{scripts}, περιέχονται όλες οι μέθοδοι και οι τεχνικές υπολογισμού βημάτων με τη μορφή συναρτήσεων καθώς και scripts που τις καλούν. +Για κάθε μία μέθοδο (ένα θέμα της εργασίας), υπάρχει το αντίστοιχο script που περιέχει τους υπολογισμούς, τις κλήσεις των μεθόδων και τη δημιουργία των διαγραμμάτων. +Για το πρώτο θέμα το αρχείο Script\_1\_Plots.m για το δεύτερο το Script\_2\_Steepest\_descent.m και ούτω καθεξής. +Στην παρούσα εργασία η υλοποίηση του κώδικα ακολουθεί την τεχνική της προηγούμενης εργασίας και “ομαδοποιεί” αρκετές λειτουργίες. +Πιο συγκεκριμένα. + +\subsection{Symbolic expression functions} +Μία ακόμη προγραμματιστική τεχνική που ακολουθήθηκε είναι η χρήση \textbf{symbolic expression} για την αναπαράσταση των διαφορετικών αντικειμενικών συναρτήσεων. +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(). + + + +\end{document} diff --git a/Work 3/scripts/BelongsTo.m b/Work 3/scripts/BelongsTo.m new file mode 100644 index 0000000..57e0587 --- /dev/null +++ b/Work 3/scripts/BelongsTo.m @@ -0,0 +1,19 @@ +function [flag] = BelongsTo(x, SetLimmits) +%Checks if the x vector belongs to Set +% +% x: A vector to project +% SetLimmits: The set to project. Each line/dimension off the set has to contain the limits +% of the set to that particular dimension +% flag: True if belongs +% + flag = true; % Have faith + + for i = 1:size(SetLimmits, 2) + if x(i) < SetLimmits(i,1) + flag = false; + elseif x(i) > SetLimmits(i,2) + flag = false; + end + end +end + diff --git a/Work 3/scripts/GivenEnv.asv b/Work 3/scripts/GivenEnv.asv new file mode 100644 index 0000000..2979d11 --- /dev/null +++ b/Work 3/scripts/GivenEnv.asv @@ -0,0 +1,33 @@ +% Given environment + +clear; +% Setup the function under test +syms x [2 1] real; +fexpr = (1/3)*x(1)^2 +3*x(2)^2; +title_fun = "$f(x) = frac{1}{3}{x_1}^2 + 3{x_2}^2$"; + +% Calculate the gradient and Hessian +grad_fexpr = gradient(fexpr, x); % Gradient of f +hessian_fexpr = hessian(fexpr, x); % Hessian of f + +% Convert symbolic expressions to MATLAB functions +fun = matlabFunction(fexpr, 'Vars', {x}); % Function +grad_fun = matlabFunction(grad_fexpr, 'Vars', {x}); % Gradient +hessian_fun = matlabFunction(hessian_fexpr, 'Vars', {x}); % 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]) + +%fixed step size globals +global gamma_fixed_step + +global image_width, +global image_height; + +image_width = 960; +image_height = 640; \ No newline at end of file diff --git a/Work 3/scripts/GivenEnv.m b/Work 3/scripts/GivenEnv.m new file mode 100644 index 0000000..cf4fd2f --- /dev/null +++ b/Work 3/scripts/GivenEnv.m @@ -0,0 +1,36 @@ +% Given environment + +clear; +% Setup the function under test +syms x [2 1] real; +fexpr = (1/3)*x(1)^2 +3*x(2)^2; +title_fun = "$f(x) = \frac{1}{3}{x_1}^2 + 3{x_2}^2$"; + +XSetLimmits = [-10, 5 ; -8, 12]; + +% Calculate the gradient and Hessian +grad_fexpr = gradient(fexpr, x); % Gradient of f +hessian_fexpr = hessian(fexpr, x); % Hessian of f + +% Convert symbolic expressions to MATLAB functions +fun = matlabFunction(fexpr, 'Vars', {x}); % Function +grad_fun = matlabFunction(grad_fexpr, 'Vars', {x}); % Gradient +hessian_fun = matlabFunction(hessian_fexpr, 'Vars', {x}); % Hessian + +% Minimum reference +[Xmin, Fmin] = fminsearch(fun, [-1, -1]'); +Xmin = round(Xmin, 3); +Fmin = round(Fmin, 3); + +% 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]) + +%fixed step size globals +global gamma_fixed_step + +global image_width, +global image_height; + +image_width = 960; +image_height = 640; \ No newline at end of file diff --git a/Work 3/scripts/ProjectionPoint.m b/Work 3/scripts/ProjectionPoint.m new file mode 100644 index 0000000..8862865 --- /dev/null +++ b/Work 3/scripts/ProjectionPoint.m @@ -0,0 +1,18 @@ +function [x_p] = ProjectionPoint(x, SetLimmits) +%Project the x vector to Set, returns a point of Set close(st) to x +% +% x: A vector to project +% SetLimmits: The set to project. Each line/dimension off the set has to contain the limits +% of the set to that particular dimension +%x_p: The projected point +% + x_p = x; + for i = 1:size(SetLimmits, 2) + if x(i) < SetLimmits(i,1) + x_p(i) = SetLimmits(i,1); + elseif x(i) > SetLimmits(i,2) + x_p(i) = SetLimmits(i,2); + end + end +end + diff --git a/Work 3/scripts/Script_0_Plots.m b/Work 3/scripts/Script_0_Plots.m new file mode 100644 index 0000000..4c00a12 --- /dev/null +++ b/Work 3/scripts/Script_0_Plots.m @@ -0,0 +1,14 @@ +% Define environment (functions, gradients etc...) +GivenEnv + +% +% We plot the function in the domain of x,y in [-3, 3]. +% We also plot the contour in order to get a sense of the min and maximum +% points in the x-y plane +% + +% 3d plot the function +plot3dFun(fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, title_fun, "figures/Plot_Function.png"); + +% Plot isobaric lines +plotContour(fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, title_fun, "figures/Plot_Contour.png"); \ No newline at end of file diff --git a/Work 3/scripts/Script_1_SteepDesc.m b/Work 3/scripts/Script_1_SteepDesc.m new file mode 100644 index 0000000..6152f75 --- /dev/null +++ b/Work 3/scripts/Script_1_SteepDesc.m @@ -0,0 +1,29 @@ +% Define environment (functions, gradients etc...) +GivenEnv + +% Define parameters +max_iter = 1000; % Maximum iterations +tol = 1e-4; % Tolerance + +% Point x0 = (1, 1) +% ========================================================================= +point = 1; +x0 = [5, -5]'; +point_str = "[" + x0(1) + ", " + x0(2) + "]"; +f = fun(x0); +gf = grad_fun(x0); +hf = hessian_fun(x0); +fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]=> Method applicable\n', x0, f, gf, hf); + +for g=[0.1, 0.3, 0.5, 3, 5] + gamma_fixed_step = g; + + [x_fixed, f_fixed, kk] = method_SteepDesc(fun, grad_fun, x0, tol, max_iter, 'fixed'); + fprintf('Fixed step g=%f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', g, x0, kk, x_fixed(:, end), f_fixed(end)); + if g <= 1 + plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, ""); + else + plotPointsOverContour(x_fixed, fun, [-500, 500], [-500, 500], 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, ""); + end +end + diff --git a/Work 3/scripts/Script_2_SteepDesc_Proj.m b/Work 3/scripts/Script_2_SteepDesc_Proj.m new file mode 100644 index 0000000..3a9eabd --- /dev/null +++ b/Work 3/scripts/Script_2_SteepDesc_Proj.m @@ -0,0 +1,32 @@ +% Define environment (functions, gradients etc...) +GivenEnv + +% Settings +max_iter = 1000; % Maximum iterations + +% Define parameters +% ========================================================================= +x0 = [5, -5]'; +gamma = 0.5; +sk_step = 5; +tol = 0.01; % Tolerance + + + +point_str = "[" + x0(1) + ", " + x0(2) + "]"; +f = fun(x0); +gf = grad_fun(x0); +hf = hessian_fun(x0); +fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]=> Method applicable\n', x0, f, gf, hf); + + +gamma_fixed_step = gamma; +[x_fixed, f_fixed, kk] = method_SteepDesc_Proj(fun, grad_fun, x0, sk_step, XSetLimmits, tol, max_iter, 'fixed'); +fprintf('Fixed step g=%f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', gamma_fixed_step, x0, kk, x_fixed(:, end), f_fixed(end)); +plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, ""); + +%[x_minimized, f_minimized, kk] = method_SteepDesc_Proj(fun, grad_fun, x0, sk_step, XSetLimmits, tol, max_iter, 'minimized'); +%fprintf('Minimized f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', x0, kk, x_fixed(:, end), f_fixed(end)); +%plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent minimized", ""); + + diff --git a/Work 3/scripts/Script_3_SteepDesc_Proj.m b/Work 3/scripts/Script_3_SteepDesc_Proj.m new file mode 100644 index 0000000..144a1de --- /dev/null +++ b/Work 3/scripts/Script_3_SteepDesc_Proj.m @@ -0,0 +1,27 @@ +% Define environment (functions, gradients etc...) +GivenEnv + +% Settings +max_iter = 1000; % Maximum iterations + +% Define parameters +% ========================================================================= +x0 = [-5, 10]'; +gamma = 0.1; +sk_step = 55; +tol = 0.01; % Tolerance + + + +point_str = "[" + x0(1) + ", " + x0(2) + "]"; +f = fun(x0); +gf = grad_fun(x0); +hf = hessian_fun(x0); +fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]=> Method applicable\n', x0, f, gf, hf); + + +gamma_fixed_step = gamma; +[x_fixed, f_fixed, kk] = method_SteepDesc_Proj(fun, grad_fun, x0, sk_step, XSetLimmits, tol, max_iter, 'fixed'); +fprintf('Fixed step g=%f: Initial point (%f, %f), steps:%d, Final (x1,x2)=(%f, %f), f(x1,x2)=%f\n', gamma_fixed_step, x0, kk, x_fixed(:, end), f_fixed(end)); +plotPointsOverContour(x_fixed, fun, XSetLimmits(1, :), XSetLimmits(2, :), 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, ""); + diff --git a/Work 3/scripts/Script_4_SteepDesc_Proj.m b/Work 3/scripts/Script_4_SteepDesc_Proj.m new file mode 100644 index 0000000..760bb2e --- /dev/null +++ b/Work 3/scripts/Script_4_SteepDesc_Proj.m @@ -0,0 +1,27 @@ +% Define environment (functions, gradients etc...) +GivenEnv + +% Settings +max_iter = 1000; % Maximum iterations + +% Define parameters +% ========================================================================= +x0 = [8, -10]'; +gamma = 0.2; +sk_step = 0.1; +tol = 0.01; % Tolerance + + + +point_str = "[" + x0(1) + ", " + x0(2) + "]"; +f = fun(x0); +gf = grad_fun(x0); +hf = hessian_fun(x0); +fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]=> Method applicable\n', x0, f, gf, hf); + + +gamma_fixed_step = gamma; +[x_fixed, f_fixed, kk] = method_SteepDesc_Proj(fun, grad_fun, x0, sk_step, 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b/Work 3/scripts/fmin_bisection.m new file mode 100644 index 0000000..7f01dd5 --- /dev/null +++ b/Work 3/scripts/fmin_bisection.m @@ -0,0 +1,49 @@ +function [a, b, k, n] = fmin_bisection(fun, alpha, beta, epsilon, lambda) +% Bisection method for finding the local minimum of a function. +% +% fun: The objective function +% alpha: (number) The starting point of the interval in which we seek +% for minimum +% beta: (number) The ending point of the interval in which we seek +% for minimum +% epsilon: (number) The epsilon value (distance from midpoint) +% lambda: (number) The lambda value (accuracy) +% +% return: +% a: (vector) Starting points of the interval for each iteration +% b: (vector) Ending points of the interval for each iteration +% k: (number) The number of iterations +% n: (number) The calls of objective function fun_expr +% + + + +% Error checking +if alpha > beta || 2*epsilon >= lambda || lambda <= 0 + error ('Input criteria not met') +end + +% Init +a = alpha; +b = beta; +n = 0; + +k=1; +while b(k) - a(k) > lambda + % bisect [a,b] + mid = (a(k) + b(k)) / 2; + x_1 = mid - epsilon; + x_2 = mid + epsilon; + + % set new search interval + k = k + 1; + if fun(x_1) < fun(x_2) + a(k) = a(k-1); + b(k) = x_2; + else + a(k) = x_1; + b(k) = b(k-1); + end +end + +end \ No newline at end of file diff --git a/Work 3/scripts/gamma_armijo.m b/Work 3/scripts/gamma_armijo.m new file mode 100644 index 0000000..a46f38f --- /dev/null +++ b/Work 3/scripts/gamma_armijo.m @@ -0,0 +1,32 @@ +function [gamma] = gamma_armijo(f, grad_f, dk, xk) + % Calculates the best step based on amijo method + % + % f(xk​+ γk*dk) ≤ f(xk) + σ * γk * dk^T * ∇f(xk) + % γk = β*γk_0 + % + % f: Objective function + % grad_fun: Gradient function of f + % dk: Current value of selected direction -∇f or -inv{H}*∇f or -inv{H + lI}*∇f + % xk: Current point (x,y) + + + % beta: beta factor in [0.1, 0.5] + % signam: sigma factor in (0, 0.1] + global amijo_beta + global amijo_sigma + + gf = grad_f(xk); + gamma = 1; % Start with a step size of 1 + + % Perform Armijo line search + while f(xk + gamma * dk) > f(xk) + amijo_sigma * gamma * dk * gf + %while f(xk(1) + gamma * dk(1), xk(2) + gamma * dk(2)) > ... + % f(xk(1), xk(2)) + amijo_sigma * gamma * norm(dk)^2 + gamma = amijo_beta * gamma; % Reduce step size + if gamma < 1e-12 % Safeguard to prevent infinite reduction + warning('Armijo step size became too small.'); + break; + end + end + +end \ No newline at end of file diff --git a/Work 3/scripts/gamma_fixed.m b/Work 3/scripts/gamma_fixed.m new file mode 100644 index 0000000..995e6f1 --- /dev/null +++ b/Work 3/scripts/gamma_fixed.m @@ -0,0 +1,12 @@ +function [gamma] = gamma_fixed(~, ~, ~, ~) + % Return a fixed step + % + % This is for completion and code symmetry. + % + + global gamma_fixed_step + + % Perform line search + gamma = gamma_fixed_step; + +end \ No newline at end of file diff --git a/Work 3/scripts/gamma_minimized.m b/Work 3/scripts/gamma_minimized.m new file mode 100644 index 0000000..388dc48 --- /dev/null +++ b/Work 3/scripts/gamma_minimized.m @@ -0,0 +1,24 @@ +function [gamma] = gamma_minimized(f, ~, dk, xk) + % Calculates the step based on minimizing f(xk​− γk*dk) + % + % + % f: Objective function + % ~: Gradient function of f - Not used + % dk: Current value of selected direction -∇f or -inv{H}*∇f or -inv{H + lI}*∇f + % xk: Current point (x,y) + + % Define the line search function fmin(g) = f(xk - g * dk) + fmin = @(g) f(xk) + g * dk; + + % find g that minimizes fmin + e = 0.0001; + l = 0.001; + [a,b,k,~] = fmin_bisection(fmin, 0.0001, 1, e, l); % g in (0, 1] + gamma = 0.5*(a(k) + b(k)); + + % Define the line search function fmin(g) = f(xk - g * dk) + %fmin = @(g) f(xk(1) - gamma * dk(1), xk(2) - gamma * dk(2)); + + % find g that minimizes fmin + %gamma = fminbnd(g, 0, 1); +end \ No newline at end of file diff --git a/Work 3/scripts/method_SteepDesc.m b/Work 3/scripts/method_SteepDesc.m new file mode 100644 index 0000000..3148a9b --- /dev/null +++ b/Work 3/scripts/method_SteepDesc.m @@ -0,0 +1,39 @@ +function [x_vals, f_vals, k] = method_SteepDesc(f, grad_f, xk, tol, max_iter, mode) + % f: Objective function + % grad_f: Gradient of the function + % xk: Initial point [x0; y0] + % tol: Tolerance for stopping criterion + % max_iter: Maximum number of iterations + + % x_vals: Vector with the (x,y) values until minimum + % f_vals: Vector with f(x,y) values until minimum + % k: Number of iterations + + if strcmp(mode, 'armijo') == 1 + gamma_f = @(f, grad_f, dk, xk) gamma_armijo(f, grad_f, dk, xk); + elseif strcmp(mode, 'minimized') == 1 + gamma_f = @(f, grad_f, dk, xk) gamma_minimized(f, grad_f, dk, xk); + else % mode == 'fixed' + gamma_f = @(f, grad_f, dk, xk) gamma_fixed(f, grad_f, dk, xk); + end + + % Storage for iterations, begin with the first point + x_vals = xk; + f_vals = f(xk); + for k = 1:max_iter + % Check for convergence + if norm(grad_f(xk)) < tol + break; + end + dk = - grad_f(xk); % Steepset descent direction + + gk = gamma_f(f, grad_f, dk, xk); % Calculate gamma + + x_next = xk + gk * dk; % Update step + f_next = f(x_next); + + xk = x_next; % Update point + x_vals = [x_vals, x_next]; % Store values + f_vals = [f_vals, f_next]; % Store function values + end +end \ No newline at end of file diff --git a/Work 3/scripts/method_SteepDesc_Proj.m b/Work 3/scripts/method_SteepDesc_Proj.m new file mode 100644 index 0000000..1e11e8b --- /dev/null +++ b/Work 3/scripts/method_SteepDesc_Proj.m @@ -0,0 +1,50 @@ +function [x_vals, f_vals, k] = method_SteepDesc_Proj(f, grad_f, xk, sk, limmits, tol, max_iter, mode) + % f: Objective function + % grad_f: Gradient of the function + % xk: Initial point [x0; y0] + % sk: Step size (fixed positive scalar) + % limits: Bounds of the feasible set for each dimension + % tol: Tolerance for stopping criterion + % max_iter: Maximum number of iterations + + % x_vals: Vector with the (x,y) values until minimum + % f_vals: Vector with f(x,y) values until minimum + % k: Number of iterations + + if strcmp(mode, 'armijo') == 1 + gamma_f = @(f, grad_f, dk, xk) gamma_armijo(f, grad_f, dk, xk); + elseif strcmp(mode, 'minimized') == 1 + gamma_f = @(f, grad_f, dk, xk) gamma_minimized(f, grad_f, dk, xk); + else % mode == 'fixed' + gamma_f = @(f, grad_f, dk, xk) gamma_fixed(f, grad_f, dk, xk); + end + + % Project the first point if needed + xk = ProjectionPoint(xk, limmits); + + % Storage for iterations, begin with the first point + x_vals = xk; + f_vals = f(xk); + for k = 1:max_iter + % Check for convergence + if norm(grad_f(xk)) < tol + break; + end + dk = - grad_f(xk); % Steepset descent direction + + % First calculate xk-bar and project it if nessesary + xkbar = xk + sk * dk; + xkbar = ProjectionPoint(xkbar, limmits); + + dk = (xkbar - xk); % Steepest descent projection direction + + gk = gamma_f(f, grad_f, dk, xk); % Calculate gamma + + x_next = xk + gk * dk; % Update step + f_next = f(x_next); + + xk = x_next; % Update point + x_vals = [x_vals, x_next]; % Store values + f_vals = [f_vals, f_next]; % Store function values + end +end \ No newline at end of file diff --git a/Work 3/scripts/plot3dFun.m b/Work 3/scripts/plot3dFun.m new file mode 100644 index 0000000..79740f1 --- /dev/null +++ b/Work 3/scripts/plot3dFun.m @@ -0,0 +1,42 @@ +function plot3dFun(fun, x_lim, y_lim, size, plot_title, filename) + % 3D plots a function + % fun: The function to plot + % x_lim: The range for x axis. ex: [-2, 2] + % y_lim: The range for y axis. ex: [0, 2] + % size: The number of points for each axis + % plot_title: The latex title for the plot + % + + global image_width, + global image_height; + + % Generate a grid for x and y + x_space = linspace(x_lim(1), x_lim(2), size); + y_space = linspace(y_lim(1), y_lim(2), size); + [X, Y] = meshgrid(x_space, y_space); + + % Evaluate the function on the grid + for i = 1:size + for j = 1:size + % Pass each [x1; x2] as input to fun + Z(i, j) = fun([X(i, j); Y(i, j)]); + end + end + + % 3D plot + figure('Name', 'f(x,y)', 'NumberTitle', 'off'); + set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size + surf(X, Y, Z); + + % Customize the plot + xlabel('x1'); % Label for x-axis + ylabel('x2'); % Label for y-axis + zlabel('f(x, y)'); % Label for z-axis + title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot + colorbar; + + % save the figure + if strcmp(filename, '') == 0 + print(gcf, filename, '-dpng', '-r300'); + end +end \ No newline at end of file diff --git a/Work 3/scripts/plotContour.m b/Work 3/scripts/plotContour.m new file mode 100644 index 0000000..15fe90e --- /dev/null +++ b/Work 3/scripts/plotContour.m @@ -0,0 +1,41 @@ +function plotContour(fun, x_lim, y_lim, size, plot_title, filename) + % plot the contour of a function + % fun: The function to plot + % x_lim: The range for x axis. ex: [-2, 2] + % y_lim: The range for y axis. ex: [0, 2] + % size: The number of points for each axis + % plot_title: The latex title for the plot + % + + global image_width, +global image_height; + + % Generate a grid for x and y + x_space = linspace(x_lim(1), x_lim(2), size); + y_space = linspace(y_lim(1), y_lim(2), size); + [X, Y] = meshgrid(x_space, y_space); + + % Evaluate the function on the grid + for i = 1:size + for j = 1:size + % Pass each [x1; x2] as input to fun + Z(i, j) = fun([X(i, j); Y(i, j)]); + end + end + + % Contour + figure('Name', 'Contours of f(x1,x2)', 'NumberTitle', 'off'); + set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size + contour(X, Y, Z); + + % Customize the plot + xlabel('x1'); % Label for x-axis + ylabel('x2'); % Label for y-axis + title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot + colorbar; + + % save the figure + if strcmp(filename, '') == 0 + print(gcf, filename, '-dpng', '-r300'); + end +end \ No newline at end of file diff --git a/Work 3/scripts/plotConvCompare.m b/Work 3/scripts/plotConvCompare.m new file mode 100644 index 0000000..fff36f1 --- /dev/null +++ b/Work 3/scripts/plotConvCompare.m @@ -0,0 +1,54 @@ +function plotConvCompare(points_1, title_1, points_2, title_2, points_3, title_3, Min_point, plot_title, filename) + % 3D plots a function + % points: The points to plot + % contur_fun: The function for contour plot + % x_lim: The range for x axis. ex: [-2, 2] + % y_lim: The range for y axis. ex: [0, 2] + % size: The number of points for each axis + % plot_title: The latex title for the plot + % filename: The filename to save the plot (if exists) + % + + global image_width, + global image_height; + + distances_1 = sqrt((points_1(1, :) - Min_point(1)).^2 + (points_1(2, :) - Min_point(2)).^2); + distances_2 = sqrt((points_2(1, :) - Min_point(1)).^2 + (points_2(2, :) - Min_point(2)).^2); + distances_3 = sqrt((points_3(1, :) - Min_point(1)).^2 + (points_3(2, :) - Min_point(2)).^2); + + + % 2D plot + figure('Name', 'Convergence compare', 'NumberTitle', 'off'); + set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size + title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot + + % One + subplot(3, 1, 1); + plot(distances_1, '-o'); + % Customize the plot + ylabel(title_1, 'Interpreter', 'none'); + xlabel('Step'); + grid on + + % One + subplot(3, 1, 2); + plot(distances_2, '-o'); + % Customize the plot + ylabel(title_2, 'Interpreter', 'none'); + xlabel('Step'); + grid on + + % One + subplot(3, 1, 3); + plot(distances_3, '-o'); + % Customize the plot + ylabel(title_3, 'Interpreter', 'none'); + xlabel('Step'); + grid on + + + % save the figure + if strcmp(filename, '') == 0 + print(gcf, filename, '-dpng', '-r300'); + end +end \ No newline at end of file diff --git a/Work 3/scripts/plotItersOverGamma.m b/Work 3/scripts/plotItersOverGamma.m new file mode 100644 index 0000000..45ac1e8 --- /dev/null +++ b/Work 3/scripts/plotItersOverGamma.m @@ -0,0 +1,28 @@ +function plotItersOverGamma(gamma, iterations, plot_title, filename) + % 3D plots a function + % fun: The points to plot + % contur_fun: The function for contour plot + % x_lim: The range for x axis. ex: [-2, 2] + % y_lim: The range for y axis. ex: [0, 2] + % size: The number of points for each axis + % plot_title: The latex title for the plot + % filename: The filename to save the plot (if exists) + % + + global image_width, + global image_height; + + figure('Name', 'Iterations_over_gamma', 'NumberTitle', 'off'); + set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size + plot(gamma, iterations, '*r', 'LineWidth', 2); + + % Customize the plot + title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot + xlabel('\gamma') ; + ylabel('Iterations'); + + % save the figure + if strcmp(filename, '') == 0 + print(gcf, filename, '-dpng', '-r300'); + end +end \ No newline at end of file diff --git a/Work 3/scripts/plotPointsOverContour.m b/Work 3/scripts/plotPointsOverContour.m new file mode 100644 index 0000000..a81082c --- /dev/null +++ b/Work 3/scripts/plotPointsOverContour.m @@ -0,0 +1,48 @@ +function plotPointsOverContour(points, contour_fun, x_lim, y_lim, size, plot_title, filename) + % 3D plots a function + % points: The points to plot + % contur_fun: The function for contour plot + % x_lim: The range for x axis. ex: [-2, 2] + % y_lim: The range for y axis. ex: [0, 2] + % size: The number of points for each axis + % plot_title: The latex title for the plot + % filename: The filename to save the plot (if exists) + % + + global image_width, + global image_height; + + % Generate a grid for x and y + x_space = linspace(x_lim(1), x_lim(2), size); + y_space = linspace(y_lim(1), y_lim(2), size); + [X, Y] = meshgrid(x_space, y_space); + + % Evaluate the function on the grid + for i = 1:size + for j = 1:size + % Pass each [x1; x2] as input to fun + Z(i, j) = contour_fun([X(i, j); Y(i, j)]); + end + end + + % 2D plot + figure('Name', '(x1,x2) convergence', 'NumberTitle', 'off'); + set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size + plot(points(1, :), points(2, :), '-or'); + hold on + contour(X, Y, Z); + % Customize the plot + xlim(x_lim); + ylim(y_lim); + xlabel('x1'); % Label for x-axis + ylabel('x2'); % Label for y-axis + grid on + + title(plot_title, 'Interpreter', 'latex', 'FontSize', 16); % Title of the plot + colorbar; + + % save the figure + if strcmp(filename, '') == 0 + print(gcf, filename, '-dpng', '-r300'); + end +end \ No newline at end of file diff --git a/Work 3/Εργασία3.pdf b/Work 3/Εργασία3.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d96d178b9a7e689f4a0914d3e73e09eafaea6ae7 GIT binary patch literal 628026 zcmcG!1#Bfvvn6=V_BAszUNbW@Gqc;w%*@Qp%+!W9wwalsZMUJ#jBSSf-u&}Fz1`J} zW~3SEs1)a9WL121p}gX71o-?dA;vnmd^Nhr;pi z5s`nW`G3mX|6S(i2Qo=o+qs#$kT6NwnYfvYo0~eCnFE>R%^fV=tVmdyxp`RF{!Lj( zIJmhvS=oV1>fTP~Bur{%T0oyo(+ow~&ZEWi~ zH1#|Wj=rX|*<`d^pL=)SdcU6DSSPOE3$oeRERV7;vt4IHQ2hFKF><8 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