From 3b40c20ec04c472ee6767a9b5bbe75653a472358 Mon Sep 17 00:00:00 2001 From: Christos Choutouridis Date: Wed, 27 Nov 2024 02:48:17 +0200 Subject: [PATCH] WIP --- Work2/report/Work2_report.pdf | Bin 577450 -> 1106536 bytes Work2/report/Work2_report.tex | 89 ++++++++++-- Work2/scripts/GivenEnv.m | 3 +- Work2/scripts/Script_2_Steepest_descent.m | 82 +++++++++-- Work2/scripts/Script_3_Newton.m | 80 +++++------ Work2/scripts/Script_4_LevMar.m | 131 +++++++++++++----- Work2/scripts/figures/LevMar_fixed_2.png | Bin 0 -> 109065 bytes Work2/scripts/figures/LevMar_fixed_3.png | Bin 0 -> 93598 bytes Work2/scripts/figures/LevMar_minimized_2.png | Bin 0 -> 111419 bytes Work2/scripts/figures/LevMar_minimized_3.png | Bin 0 -> 100661 bytes .../scripts/figures/StDes_Iter_o_gamma_2.png | Bin 55792 -> 59427 bytes Work2/scripts/figures/StDes_armijo_2.png | Bin 0 -> 107053 bytes Work2/scripts/figures/StDes_armijo_3.png | Bin 0 -> 94226 bytes Work2/scripts/figures/StDes_fixed_2.png | Bin 97037 -> 103360 bytes Work2/scripts/figures/StDes_fixed_3.png | Bin 0 -> 91330 bytes Work2/scripts/figures/StDes_minimized_2.png | Bin 0 -> 106330 bytes Work2/scripts/figures/StDes_minimized_3.png | Bin 0 -> 99584 bytes Work2/scripts/fmin_bisection.m | 49 +++++++ Work2/scripts/gamma_armijo.m | 26 ++-- Work2/scripts/gamma_fixed.m | 2 +- Work2/scripts/gamma_minimized.m | 32 +++-- Work2/scripts/method_lev_mar.m | 37 +++-- Work2/scripts/method_newton.m | 22 +-- Work2/scripts/method_steepest_descent.m | 22 +-- Work2/scripts/plotPointsOverContour.m | 2 +- 25 files changed, 414 insertions(+), 163 deletions(-) create mode 100644 Work2/scripts/figures/LevMar_fixed_2.png create mode 100644 Work2/scripts/figures/LevMar_fixed_3.png create mode 100644 Work2/scripts/figures/LevMar_minimized_2.png create mode 100644 Work2/scripts/figures/LevMar_minimized_3.png create mode 100644 Work2/scripts/figures/StDes_armijo_2.png create mode 100644 Work2/scripts/figures/StDes_armijo_3.png create mode 100644 Work2/scripts/figures/StDes_fixed_3.png create mode 100644 Work2/scripts/figures/StDes_minimized_2.png create mode 100644 Work2/scripts/figures/StDes_minimized_3.png create mode 100644 Work2/scripts/fmin_bisection.m diff --git a/Work2/report/Work2_report.pdf b/Work2/report/Work2_report.pdf index b7720d3bbd8a8500c622691548d5cade3d62e737..8aa5e0f65a16b9da37384f00442b76bcc2826d8f 100644 GIT binary patch delta 754438 zcmc$FV{~MHyJb2_M;+U?ZFIVWj%{~r8>c(AZKq>f9ox2TJDK{w_nnzLUuHhtxwTfU zQ&s1O2Yc`5Id!T^vd9j5zgIbH18iJ^f^d#b4#o!7aBj;HDnmA_%<%0Gbhi+yBB8V3 zkb>yezwa@fY($X=`FJ>}J&rGu^2t}&gn#`c{*llK-tD=;EnkY1Zs}7cu;+6t2V|7o%dABY7PBDH(#wEUBnl{E1(p@e$ zCc8al!T8<%X`l2(vn$v47nPIA!Jcf?1mRA-RNpy*$>{MBuN=*x!5J1@OI^kdyvKmW z^YR1K-c29%tvYFIC*PEMEx_d9=B(^Hksev9cmC4F%L^|^VPvdBQA#at#B^2H6i>cM z{1~<^Za*d!mS}X~oXINCkj-5u-v3T5thla{5MIBQx@!^00rweP6Hgk>s02Ke0bcKh zcu`xaHeH#>RH|CIs%M@_a-{ZL*+f}W4v`Q4#lyy(qe|+FuY+ga0TAOl$J}b-*uYPV ztdp2ox3XKv75zu2#pTFu)s+DuV-g}Jjr>)S_W(2VNtfOibI$oeu{Eh6<`>!ip0s48 zD?<)@oxuS2Pe}n|vg7`VV3*5pBU2I$JCPq5;NhYk5SQA&c8&|d@*r?fN6XJj;iC>I zrxhdo!sl5X%Lw=%5CIa41d|~PV;udHJ4n-3;ll{tP_fdEI?l$#`F_8b{#u^^k`k_! zjNW+}`1Qy{{B=oN;&>WA3V&?$tg_Ow`pPs$LO_mtohGkQ40n#gzd~V-~+?u`WfXE-uGQ2EUk4Cf4tikxOZodGhuWmDmA_ zN5Ul@MWW`h?-uCqLdJGl!}KZ_h4k`bB#SVUq5SZaUl%qach{-)%t593xKw;%tH_VH>?w;E(i=vQfrY?Eh|q`k^?hZl5m1J% zXEc{NzC5$%KCWAIlQ)4aa8b+}6%#uADvZ@qaw-d3^@Df=ngIrbC+EW74f7|Wi5Y=eYM8|XPjogb0d$QURAw$lN> zh!fQGqRtQZM>=>D!V(IeqR&WC%S@UBV(%rhPsxfuui5-wL*|RhAw{*WzfKURb4=i1 z+VX!f8S}h8_kZ=zpxm{z<7bKjn{tAuTRlJtKx}#zzNR6w&S;mEtx)UtCyoCCN8o0@ zm#Tu9)B;5VQ+D_0F4-NXPIBl0%&nP4(xL3hy9Sxw)MT@}wpfM)G!~V9k`c-4lDph9 zmq4LJ6Td3W)xg9D3`U~GMn`S;2oGe5PAE{ z=;09qIiAK3)Tkle2;^k`sLU|AHKa4;jYVMOj+%?V^{E`b)~0?QF%*vQf4oHG&yx_w z7jQK-Sa2RZ=hiEiEXdZ#FJa`SD;<$6Iucv8Zz~$4OvXSlP3I3%b|%Bmd>rCA3Ab*@ z7&-P+=3lq>Xb=q_TmY{?M{)oq`Y zk}>DZ1cz^IIdxPA`w2|jH^XI0I!bOlX#Jz1=^@ z3C(YFsC5{=tU!@NQ@|mf`sT-5E0@CVX1DDol(^Gbru{VWKOa&C>YHJl33`{5J#W{T z?SDcaEror-gNKaKw2j{|%h@I-qKM>Jb>=cCD2Quj(@Zvi`@nQp_7HEypH!~8wjC2I zIQ_;~E0S^h)l40d-R=A=Au=p9acp z7g2_CrZ4!Ao@%KLV3I@j+l$V#!J{SoIG)vU7tvGMhh^O;(U_}%<6OF^P1huq-NmM1 z0-?>Ey#*(@7?pm2gh4jw$k*9161FnXm!LW0k1BDFIZd4}nqT=~LOB7puNF(RQQC0z z3|p~QhRV<^?0k@s2#1@=3Dc1o|1nEhZY81oiPC^(1mVyS@cmK5coeogz2N|+pEJ97 zHf3m8MeHCi`{oMf;!%J{m7wg$(^THLI75|C3#YMTng5-9XXeBSywWT)5gxOm-Y2tmaltB=hQ(>8D=RY5KfHaNhoe%LVy+Htr>6BD_0^PK+M6kt zcl{q0Z(@(xK)%Nsf01t1c;HtnM5!4z$eO9$3EZx?K<(CKm5ExChddH-GXHA4MKI2D zRbG@M7s&k*?fbPDSq4pm!T!wBz~Cr>_Z*1M`{LUxS*tWyK29`$q|fSB3ko6)9vBRE zf#%{E#MKGDB2`-E9&kY`j^A2OEiZ`0+wYvKI>1H|fm|MOD%j$*Pp>>G7w{DJ6S^^I z9Z8e$9p0TnH9itz7nkB}2y8YIB@d#!k64N2?$Zq9lRkAW%sl}Y#Iy8(IZ7A0@h_3T zGLg)F*Hdh7@U~#&5zn$jJy? 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εσωτερικά της κάθε μεθόδου ένα κοινό interface για τις μεθόδους επιλογής βήματος. -Αυτό έχει τη μορφή: \textit{\textbf{gamma\_(f, grad\_f, x0)}}, όπου το \textbf{f} είναι η αντικειμενική συνάρτηση, \textbf{grad\_f} η συνάρτηση κλίσης και \textbf{x0} το σημείο ενδιαφέροντος. +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. 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. @@ -134,8 +134,8 @@ 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 -\[ f(x_k + \gamma_k d_k) \leq f(x_k) + \sigma\gamma_k\nabla f(x_k)^Td_k\] -Όπου $\sigma \in (0,1)$ 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. +\[ f(x_k + \gamma_k d_k) \leq f(x_k) + \sigma\gamma_k {d_k}^T*\nabla f(x_k) \] +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. @@ -157,25 +157,94 @@ 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 η τιμή της κλίσης: $\nabla f(0,0) = \begin{bmatrix} 0.4060 \\ 0.2707 \end{bmatrix}$. -Επιλέγοντας ακρίβεια $\epsilon = 0.0001$, εκτελούμε την μέθοδο 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 επαναήψεων για διαφορετικές τιμές $\gamma_k [Μέγιστη Κάθοδος]$}. +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. +\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].} Στο παραπάνω σχήμα \ref{fig:point2ItersOverGamma} 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{ref:StDes_fixed_2} +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].} -\InsertFigure{H}{0.8}{StDes_fixed_2}{../scripts/figures/StDes_fixed_2.png}{Σύγκλιση της μεθόδου}. +\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. +\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. +\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 $\psi$. +\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. \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. +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}. +\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 $\lambda = \begin{bmatrix} -1.082 \\ 0.541 \end{bmatrix}$. +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 $\lambda = \begin{bmatrix} -0.541 \\ 1.082 \end{bmatrix}$. +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. +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. + +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: \\ -\par -\underline{Παρατηρήσεις:} \begin{itemize} \item ... \end{itemize} diff --git a/Work2/scripts/GivenEnv.m b/Work2/scripts/GivenEnv.m index 046b4c2..ec517e4 100644 --- a/Work2/scripts/GivenEnv.m +++ b/Work2/scripts/GivenEnv.m @@ -16,7 +16,8 @@ grad_fun = matlabFunction(grad_fexpr, 'Vars', [x, y]); % Gradient hessian_fun = matlabFunction(hessian_fexpr, 'Vars', [x, y]); % Hessian % Amijo globals -global amijo_beta amijo_sigma +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 diff --git a/Work2/scripts/Script_2_Steepest_descent.m b/Work2/scripts/Script_2_Steepest_descent.m index 5afb532..71aefbe 100644 --- a/Work2/scripts/Script_2_Steepest_descent.m +++ b/Work2/scripts/Script_2_Steepest_descent.m @@ -5,22 +5,19 @@ GivenEnv max_iter = 300; % Maximum iterations tol = 1e-4; % Tolerance -% Methods tuning -amijo_beta = 0.5; % Armijo reduction factor -amijo_sigma = 0.1; % Armijo condition constant - % Point x0 = (0, 0) +% ========================================================================= point = 1; x0 = [0, 0]; f = fun(x0(1), x0(2)); gf = grad_fun(x0(1), x0(2)); hf = hessian_fun(x0(1), x0(2)); -fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Can not use method\n\n', x0, f, gf, hf); +fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Can NOT use method\n', x0, f, gf, hf); +disp(' '); -% Points x0 = (-1, 1), (1, -1) -%x0s = [-1, 1 ; 1, -1]; % Initial points % Point x0 = (-1, 1) +% ========================================================================= point = 2; x0 = [-1, 1]; point_str = "[" + x0(1) + ", " + x0(2) + "]"; @@ -49,14 +46,69 @@ fprintf('Fixed step: Initial point (%f, %f), steps:%d, Final (x,y)=(%f, %f), plotPointsOverContour(x_fixed, fun, [-2, 0], [-2, 2], 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, "figures/StDes_fixed_" + point + ".png"); - % Minimized f - [x_minimized, f_minimized, kk] = method_steepest_descent(fun, grad_fun, 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, [-2, 2], [-2, 2], 100, point_str + ": Steepest descent minimized $f(x_k + \gamma_kd_k)$", "StDes_minimized_" + i + ".png"); +% Minimized f +[x_minimized, f_minimized, kk] = method_steepest_descent(fun, grad_fun, 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, [-2, 0], [-2, 2], 100, point_str + ": Steepest descent minimized $f(x_k + \gamma_kd_k)$", "figures/StDes_minimized_" + point + ".png"); + + +% Armijo Rule + +% Methods tuning +amijo_beta = 0.4; % typical range: [0.1, 0.8] +amijo_sigma = 0.1; % typical range: [0.01, 0.3] + +[x_armijo, f_armijo, kk] = method_steepest_descent(fun, grad_fun, 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, [-2, 0], [-2, 2], 100, point_str + ": Steepest descent Armijo method", "figures/StDes_armijo_" + point + ".png"); +disp(' '); + +% Point x0 = (1, -1) +% ========================================================================= +point = 3; +x0 = [1, -1]; +point_str = "[" + x0(1) + ", " + x0(2) + "]"; + +f = fun(-1, 1); +gf = grad_fun(x0(1), x0(2)); +hf = hessian_fun(x0(1), x0(2)); +fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Can use method\n', x0, f, gf, hf); + +% Find the best fixed gamma +k = zeros(100, 1); +j = 1; +n = linspace(0.1, 1, 100); +for g = n + gamma_fixed_step = g; + [~, ~, k(j)] = method_steepest_descent(fun, grad_fun, x0, tol, max_iter, 'fixed'); + j = j + 1; +end +%if min(k) == max_iter +% fprintf('Fixed step: Initial point (%d, %d). Can NOT use method\n', x0); +%end +%plotItersOverGamma(n, k, "Iteration for different $\gamma$ values", "figures/StDes_Iter_o_gamma_" + point + ".png"); + +%gamma_fixed_step = 1; +[x_fixed, f_fixed, kk] = method_steepest_descent(fun, grad_fun, x0, tol, max_iter, 'fixed'); +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, [-1, 2], [-2, 2], 100, point_str + ": Steepest descent $\gamma$ = " + gamma_fixed_step, "figures/StDes_fixed_" + point + ".png"); + + +% Minimized f +[x_minimized, f_minimized, kk] = method_steepest_descent(fun, grad_fun, 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, [-2, 2], [-3, 2], 100, point_str + ": Steepest descent minimized $f(x_k + \gamma_kd_k)$", "figures/StDes_minimized_" + point + ".png"); + + +% Armijo Rule + +% Methods tuning +amijo_beta = 0.4; % typical range: [0.1, 0.8] +amijo_sigma = 0.1; % typical range: [0.01, 0.3] + +[x_armijo, f_armijo, kk] = method_steepest_descent(fun, grad_fun, 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, [-1, 2], [-2, 2], 100, point_str + ": Steepest descent Armijo method", "figures/StDes_armijo_" + point + ".png"); - % Armijo Rule - [x_armijo, f_armijo, kk] = method_steepest_descent(fun, grad_fun, 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, [-2, 2], [-2, 2], 100, point_str + ": Steepest descent Armijo method", "StDes_armijo_" + i + ".png"); diff --git a/Work2/scripts/Script_3_Newton.m b/Work2/scripts/Script_3_Newton.m index b776f98..dba9111 100644 --- a/Work2/scripts/Script_3_Newton.m +++ b/Work2/scripts/Script_3_Newton.m @@ -1,51 +1,43 @@ % Define environment (functions, gradients etc...) GivenEnv -% Define parameters -max_iter = 300; % Maximum iterations -tol = 1e-4; % Tolerance - -% Methods tuning -amijo_beta = 0.5; % Armijo reduction factor -amijo_sigma = 0.1; % Armijo condition constant % Point x0 = (0, 0) -% x0 = [0, 0]; +% ========================================================================= +point = 1; +x0 = [0, 0]; +f = fun(x0(1), x0(2)); +gf = grad_fun(x0(1), x0(2)); +hf = hessian_fun(x0(1), x0(2)); +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); +disp(' '); + + +% Point x0 = (-1, 1) +% ========================================================================= +point = 2; +x0 = [-1, 1]; +point_str = "[" + x0(1) + ", " + x0(2) + "]"; + +f = fun(-1, 1); +gf = grad_fun(x0(1), x0(2)); +hf = hessian_fun(x0(1), x0(2)); +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); +disp(' '); + +% Point x0 = (1, -1) +% ========================================================================= +point = 3; +x0 = [1, -1]; +point_str = "[" + x0(1) + ", " + x0(2) + "]"; + +f = fun(-1, 1); +gf = grad_fun(x0(1), x0(2)); +hf = hessian_fun(x0(1), x0(2)); +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); + -% Point x0 = (0, 0) -x0s = [0, 0; -1, 1 ; 1, -1]; % Initial points -for i = 1:size(x0s, 1) - x0 = x0s(i, :); - point_str = "[" + x0(1) + ", " + x0(2) + "]"; - - % Find the best fixed gamma - k = zeros(100, 1); - j = 1; - n = linspace(0.1, 1.5, 100); - for g = n - gamma_fixed_step = g; - [~, ~, k(j)] = newton(fun, grad_fun, hessian_fun, x0, tol, max_iter, 'fixed'); - j = j + 1; - end - plotIterationsOverGamma(n, k, "Iteration for different $\gamma$ values", "Newton_Iter_o_gamma_" + i + ".png"); - - [~, j] = min(k); - gamma_fixed_step = n(j); - - [x_fixed, f_fixed, kk] = newton(fun, grad_fun, hessian_fun, x0, tol, max_iter, 'fixed'); - 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, [-2, 2], [-3, 3], 100, point_str + ": Newton $\gamma$ = " + gamma_fixed_step, "Newton_fixed_" + i + ".png"); - - - % Minimized f - [x_minimized, f_minimized, kk] = newton(fun, grad_fun, hessian_fun, 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, [-2, 2], [-3, 3], 100, point_str + ": Newton minimized $f(x_k + \gamma_kd_k)$", "Newton_minimized_" + i + ".png"); - - - % Armijo Rule - [x_armijo, f_armijo, kk] = newton(fun, grad_fun, hessian_fun, 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, [-2, 2], [-3, 3], 100, point_str + ": Newton Armijo method", "Newton_armijo_" + i + ".png"); -end diff --git a/Work2/scripts/Script_4_LevMar.m b/Work2/scripts/Script_4_LevMar.m index 08d6cc0..57e5b82 100644 --- a/Work2/scripts/Script_4_LevMar.m +++ b/Work2/scripts/Script_4_LevMar.m @@ -5,49 +5,112 @@ GivenEnv max_iter = 300; % Maximum iterations tol = 1e-4; % Tolerance -% Methods tuning -amijo_beta = 0.5; % Armijo reduction factor -amijo_sigma = 0.1; % Armijo condition constant - -m = 0.01; % Point x0 = (0, 0) -% x0 = [0, 0]; +% ========================================================================= +point = 1; +x0 = [0, 0]; +f = fun(x0(1), x0(2)); +gf = grad_fun(x0(1), x0(2)); +hf = hessian_fun(x0(1), x0(2)); +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); +disp(' '); -% Point x0 = (0, 0) -x0s = [0, 0; -1, 1 ; 1, -1]; % Initial points -for i = 1:size(x0s, 1) - x0 = x0s(i, :); - point_str = "[" + x0(1) + ", " + x0(2) + "]"; - - % Find the best fixed gamma - k = zeros(100, 1); - j = 1; - n = linspace(0.1, 1.5, 100); - for g = n - gamma_fixed_step = g; - [~, ~, k(j)] = lev_mar(fun, grad_fun, hessian_fun, m, x0, tol, max_iter, 'fixed'); - j = j + 1; +% Point x0 = (-1, 1) +% ========================================================================= +point = 2; +x0 = [-1, 1]; +point_str = "[" + x0(1) + ", " + x0(2) + "]"; + +f = fun(-1, 1); +gf = grad_fun(x0(1), x0(2)); +hf = hessian_fun(x0(1), x0(2)); +ev = eig(hf); +fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Eigenvalues= [%f, %f], Can use method\n', x0, f, gf, hf, ev); + + +% Find the best fixed gamma +k = zeros(100, 1); +j = 1; +n = linspace(0.1, 1.5, 100); +for g = n + gamma_fixed_step = g; + [x, f, k(j)] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'fixed'); + if ~(x(end, 1) < -1.57 && x(end, 1) > -1.59 && x(end, 2) < 0.01 && x(end,2) > -0.01 && f(end) < -0.8 && f(end) > -0.82) + k(j) = 300; end - plotIterationsOverGamma(n, k, "Iteration for different $\gamma$ values", "LevMar_Iter_o_gamma_" + i + ".png"); + j = j + 1; +end + +[~, j] = min(k); +gamma_fixed_step = n(j); + +[x_fixed, f_fixed, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'fixed'); +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"); - [~, j] = min(k); - gamma_fixed_step = n(j); +[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_fixed, f_fixed, kk] = lev_mar(fun, grad_fun, hessian_fun, m, x0, tol, max_iter, 'fixed'); - 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, [-2, 2], [-3, 3], 100, point_str + ": LevMar $\gamma$ = " + gamma_fixed_step, "LevMar_fixed_" + i + ".png"); +% Armijo Rule + +% Methods tuning +amijo_beta = 0.4; % typical range: [0.1, 0.8] +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"); +disp(' '); - % Minimized f - [x_minimized, f_minimized, kk] = lev_mar(fun, grad_fun, hessian_fun, m, 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, [-2, 2], [-3, 3], 100, point_str + ": LevMar minimized $f(x_k + \gamma_kd_k)$", "LevMar_minimized_" + i + ".png"); +% Point x0 = (1, -1) +% ========================================================================= +point = 3; +x0 = [1, -1]; +point_str = "[" + x0(1) + ", " + x0(2) + "]"; - % Armijo Rule - [x_armijo, f_armijo, kk] = lev_mar(fun, grad_fun, hessian_fun, m, 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, [-2, 2], [-3, 3], 100, point_str + ": LevMar Armijo method", "LevMar_armijo_" + i + ".png"); +f = fun(-1, 1); +gf = grad_fun(x0(1), x0(2)); +hf = hessian_fun(x0(1), x0(2)); +ev = eig(hf); +fprintf('Initial point (%d, %d), f = %f, grad = [%f;%f], hessian = [%f %f ; %f %f]. Eigenvalues= [%f, %f], Can use method\n', x0, f, gf, hf, ev); + + +% Find the best fixed gamma +k = zeros(100, 1); +j = 1; +n = linspace(0.1, 1.5, 100); +for g = n + gamma_fixed_step = g; + [x, f, k(j)] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'fixed'); + if ~(x(end, 1) < -1.57 && x(end, 1) > -1.59 && x(end, 2) < 0.01 && x(end,2) > -0.01 && f(end) < -0.8 && f(end) > -0.82) + k(j) = 300; + end + j = j + 1; end + +[~, j] = min(k); +gamma_fixed_step = n(j); + +[x_fixed, f_fixed, kk] = method_lev_mar(fun, grad_fun, hessian_fun, 0.3, x0, tol, max_iter, 'fixed'); +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, 2], [-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, 2], [-2, 2], 100, point_str + ": Levenberg-Marquardt minimized $f(x_k + \gamma_kd_k)$", "figures/LevMar_minimized_" + point + ".png"); + +% Armijo Rule + +% Methods tuning +amijo_beta = 0.4; % typical range: [0.1, 0.8] +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(' '); diff --git a/Work2/scripts/figures/LevMar_fixed_2.png b/Work2/scripts/figures/LevMar_fixed_2.png new file mode 100644 index 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zmj;5yihE_P&A_wD!08a+aWDHjkN^L*y${%S0+!wvdt(%Vx)>t;AX$SEbn5DnPT|Gx zK*zNMC$D0U%>lNHfWt#QyMPf@4cxXUwbBlxqo@(GDCFk4EnB`U>egGeVueR(^_Lf* z%Lm`)egN(W{R&(Pyz~}uj1O3uWC81V>BU|kjhp5{V&y7ustedw2Cn>`08ahDeLZf# zd7tlQb-+Fu=z0s_Bu6l)p|TG+eGJU8y@A>w?KjUZ25s|bSOfHVzpLiVknPi^_~?jb zUtM(-G?%_&H*f`e{GUhSyb<6$ztb7wu{hwg+kfD-wc@3qy_9yfzutU~1@@ix0L!OE z?|=)P>;C_~Z|+eGvf+tHKWsA$g8~P27Kl-6Bo*X75EyLL)Y|R;mM}0dZ}fC=41u_i zfz(xR!##*bg9RJ{6f9>OZJL29KM(*OcmUgLM@dm|=s&B@HrrX>c^MyqO!aj2b6Mw< G&;$UDlQQN2 literal 0 HcmV?d00001 diff --git a/Work2/scripts/fmin_bisection.m b/Work2/scripts/fmin_bisection.m new file mode 100644 index 0000000..02d7fe8 --- /dev/null +++ b/Work2/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 diff --git a/Work2/scripts/gamma_armijo.m b/Work2/scripts/gamma_armijo.m index 9727cdc..8ee0b9e 100644 --- a/Work2/scripts/gamma_armijo.m +++ b/Work2/scripts/gamma_armijo.m @@ -1,25 +1,33 @@ -function [gamma] = gamma_armijo(f, grad_f, x0) +function [gamma] = gamma_armijo(f, grad_f, dk, xk) % Calculates the best step based on amijo method % - % f(xk​− γk*∇f(xk)) ≤ f(xk) − σ*γk*∥∇f(xk)∥^2 + % f(xk​+ γk*dk) ≤ f(xk) + σ * γk * dk^T * ∇f(xk) + % γk = β*γk_0 % % f: Objective function - % x0: Initial (x,y) point + % 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) - % signam: sigma factor in (0,1) + % 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(1), xk(2)); gamma = 1; % Start with a step size of 1 - grad = grad_f(x0(1), x0(2)); - % Perform Armijo line search - while f(x0(1) - gamma * grad(1), x0(2) - gamma * grad(2)) > ... - f(x0(1), x0(2)) - amijo_sigma * gamma * norm(grad)^2 + while f(xk(1) + gamma * dk(1), xk(2) + gamma * dk(2)) > ... + f(xk(1), xk(2)) + 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 diff --git a/Work2/scripts/gamma_fixed.m b/Work2/scripts/gamma_fixed.m index c9976bb..995e6f1 100644 --- a/Work2/scripts/gamma_fixed.m +++ b/Work2/scripts/gamma_fixed.m @@ -1,4 +1,4 @@ -function [gamma] = gamma_fixed(~, ~, ~) +function [gamma] = gamma_fixed(~, ~, ~, ~) % Return a fixed step % % This is for completion and code symmetry. diff --git a/Work2/scripts/gamma_minimized.m b/Work2/scripts/gamma_minimized.m index 48891d7..a3e3188 100644 --- a/Work2/scripts/gamma_minimized.m +++ b/Work2/scripts/gamma_minimized.m @@ -1,18 +1,24 @@ -function [gamma] = gamma_minimized(f, grad_f, x0) - % Calculates the step based on minimizing f(xk​− γ*∇f(xk)) +function [gamma] = gamma_minimized(f, ~, dk, xk) + % Calculates the step based on minimizing f(xk​− γk*dk) % % - % f: Objective function - % grad_f: Gradient of objective function - % x0: Initial (x,y) point + % 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 g(gamma) = f(x0 - gamma * grad) - grad = grad_f(x0(1), x0(2)); - g = @(gamma) f(x0(1) - gamma * grad(1), x0(2) - gamma * grad(2)); + % Define the line search function fmin(g) = f(xk - g * dk) + fmin = @(g) f(xk(1) + g*dk(1), xk(2) + g*dk(2)); - % Perform line search - gamma = fminbnd(g, 0, 1); - % ToDo: Check if we can use fmin_bisection_der - % from the previous assigment here! + % find g that minimizes fmin + e = 0.0001; + l = 0.001; + [a,b,k,~] = fmin_bisection(fmin, 0, 5, e, l); + 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 diff --git a/Work2/scripts/method_lev_mar.m b/Work2/scripts/method_lev_mar.m index c50b502..a22122a 100644 --- a/Work2/scripts/method_lev_mar.m +++ b/Work2/scripts/method_lev_mar.m @@ -1,8 +1,9 @@ -function [x_vals, f_vals, k] = method_lev_mar(f, grad_f, hessian_f, m, x0, tol, max_iter, mode) +function [x_vals, f_vals, k] = method_lev_mar(f, grad_f, hessian_f, e, xk, tol, max_iter, mode) % f: Objective function % grad_f: Gradient of the function % hessian_f: Hessian of the function - % x0: Initial point [x0, y0] + % e: mu offset for hessian damping H' = H_k + mI + % xk: Initial point [xk, yk] % tol: Tolerance for stopping criterion % max_iter: Maximum number of iterations @@ -12,36 +13,46 @@ function [x_vals, f_vals, k] = method_lev_mar(f, grad_f, hessian_f, m, x0, tol, if strcmp(mode, 'armijo') == 1 - gamma_f = @(f, grad_f, x0) gamma_armijo(f, grad_f, x0); + gamma_f = @(f, grad_f, dk, xk) gamma_armijo(f, grad_f, dk, xk); elseif strcmp(mode, 'minimized') == 1 - gamma_f = @(f, grad_f, x0) gamma_minimized(f, grad_f, x0); + gamma_f = @(f, grad_f, dk, xk) gamma_minimized(f, grad_f, dk, xk); else % mode == 'fixed' - gamma_f = @(f, grad_f, x0) gamma_fixed(f, grad_f, x0); + gamma_f = @(f, grad_f, dk, xk) gamma_fixed(f, grad_f, dk, xk); end - x_vals = x0; % Store iterations - f_vals = f(x0(1), x0(2)); + x_vals = xk; % Store iterations + f_vals = f(xk(1), xk(2)); for k = 1:max_iter - grad = grad_f(x0(1), x0(2)); + grad = grad_f(xk(1), xk(2)); % Check for convergence if norm(grad) < tol break; end - hess = hessian_f(x0(1), x0(2)); - mI = m * eye(size(hess)); + hess = hessian_f(xk(1), xk(2)); + + % Check if hessian is not positive defined + lmin = min(eig(hess)); + if lmin <= 0 + m = abs(lmin) + e; + mI = m * eye(size(hess)); + nev = eig(hess + mI); + if min(nev) <= 0 + warning('Can not normalize hessian matrix.'); + end + end % Solve for search direction using Newton's step dk = - inv(hess + mI) * grad; % Calculate gamma - gamma = gamma_f(f, grad_f, x0); + gk = gamma_f(f, grad_f, dk, xk); - x_next = x0 + gamma * dk'; % Update step + x_next = xk + gk * dk'; % Update step f_next = f(x_next(1), x_next(2)); - x0 = x_next; % Update point + xk = x_next; % Update point x_vals = [x_vals; x_next]; % Store values f_vals = [f_vals; f_next]; % Store function values end diff --git a/Work2/scripts/method_newton.m b/Work2/scripts/method_newton.m index 4554dad..de219db 100644 --- a/Work2/scripts/method_newton.m +++ b/Work2/scripts/method_newton.m @@ -1,4 +1,4 @@ -function [x_vals, f_vals, k] = method_newton(f, grad_f, hessian_f, x0, tol, max_iter, mode) +function [x_vals, f_vals, k] = method_newton(f, grad_f, hessian_f, xk, tol, max_iter, mode) % f: Objective function % grad_f: Gradient of the function % hessian_f: Hessian of the function @@ -12,35 +12,35 @@ function [x_vals, f_vals, k] = method_newton(f, grad_f, hessian_f, x0, tol, max_ if strcmp(mode, 'armijo') == 1 - gamma_f = @(f, grad_f, x0) gamma_armijo(f, grad_f, x0); + gamma_f = @(f, grad_f, dk, xk) gamma_armijo(f, grad_f, dk, xk); elseif strcmp(mode, 'minimized') == 1 - gamma_f = @(f, grad_f, x0) gamma_minimized(f, grad_f, x0); + gamma_f = @(f, grad_f, dk, xk) gamma_minimized(f, grad_f, dk, xk); else % mode == 'fixed' - gamma_f = @(f, grad_f, x0) gamma_fixed(f, grad_f, x0); + gamma_f = @(f, grad_f, dk, xk) gamma_fixed(f, grad_f, dk, xk); end - x_vals = x0; % Store iterations - f_vals = f(x0(1), x0(2)); + x_vals = xk; % Store iterations + f_vals = f(xk(1), xk(2)); for k = 1:max_iter - grad = grad_f(x0(1), x0(2)); + grad = grad_f(xk(1), xk(2)); % Check for convergence if norm(grad) < tol break; end - hess = hessian_f(x0(1), x0(2)); + hess = hessian_f(xk(1), xk(2)); % Solve for search direction using Newton's step dk = - inv(hess) * grad; % Calculate gamma - gamma = gamma_f(f, grad_f, x0); + gk = gamma_f(f, grad_f, dk, xk); - x_next = x0 + gamma * dk'; % Update step + x_next = xk + gk * dk'; % Update step f_next = f(x_next(1), x_next(2)); - x0 = x_next; % Update point + xk = x_next; % Update point x_vals = [x_vals; x_next]; % Store values f_vals = [f_vals; f_next]; % Store function values end diff --git a/Work2/scripts/method_steepest_descent.m b/Work2/scripts/method_steepest_descent.m index 91724e1..877f77d 100644 --- a/Work2/scripts/method_steepest_descent.m +++ b/Work2/scripts/method_steepest_descent.m @@ -1,7 +1,7 @@ -function [x_vals, f_vals, k] = method_steepest_descent(f, grad_f, x0, tol, max_iter, mode) +function [x_vals, f_vals, k] = method_steepest_descent(f, grad_f, xk, tol, max_iter, mode) % f: Objective function % grad_f: Gradient of the function - % x0: Initial point [x0, y0] + % xk: Initial point [x0, y0] % tol: Tolerance for stopping criterion % max_iter: Maximum number of iterations @@ -10,19 +10,19 @@ function [x_vals, f_vals, k] = method_steepest_descent(f, grad_f, x0, tol, max_i % k: Number of iterations if strcmp(mode, 'armijo') == 1 - gamma_f = @(f, grad_f, x0) gamma_armijo(f, grad_f, x0); + gamma_f = @(f, grad_f, dk, xk) gamma_armijo(f, grad_f, dk, xk); elseif strcmp(mode, 'minimized') == 1 - gamma_f = @(f, grad_f, x0) gamma_minimized(f, grad_f, x0); + gamma_f = @(f, grad_f, dk, xk) gamma_minimized(f, grad_f, dk, xk); else % mode == 'fixed' - gamma_f = @(f, grad_f, x0) gamma_fixed(f, grad_f, x0); + 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 = x0; - f_vals = f(x0(1), x0(2)); + x_vals = xk; + f_vals = f(xk(1), xk(2)); for k = 1:max_iter - grad = grad_f(x0(1), x0(2)); + grad = grad_f(xk(1), xk(2)); % Check for convergence if norm(grad) < tol @@ -31,12 +31,12 @@ function [x_vals, f_vals, k] = method_steepest_descent(f, grad_f, x0, tol, max_i dk = - grad; % Calculate gamma - gk = gamma_f(f, grad_f, x0); + gk = gamma_f(f, grad_f, dk, xk); - x_next = x0 + gk * dk'; % Update step + x_next = xk + gk * dk'; % Update step f_next = f(x_next(1), x_next(2)); - x0 = x_next; % Update point + xk = x_next; % Update point x_vals = [x_vals; x_next]; % Store values f_vals = [f_vals; f_next]; % Store function values end diff --git a/Work2/scripts/plotPointsOverContour.m b/Work2/scripts/plotPointsOverContour.m index 57dc863..fbd60de 100644 --- a/Work2/scripts/plotPointsOverContour.m +++ b/Work2/scripts/plotPointsOverContour.m @@ -21,7 +21,7 @@ function plotPointsOverContour(points, contour_fun, x_lim, y_lim, size, plot_tit Z = contour_fun(X, Y); % 2D plot - figure('Name', '(x,y)', 'NumberTitle', 'off'); + figure('Name', '(x,y) convergence', 'NumberTitle', 'off'); set(gcf, 'Position', [100, 100, image_width, image_height]); % Set the figure size plot(points(:, 1), points(:, 2), '-or'); hold on