8928 lines
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8928 lines
739 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>- · Optinum.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="index.html"><img src="assets/logo.png" alt="Optinum.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">Optinum.jl</span></div><form class="docs-search" action="search.html"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="index.html">Accueil</a></li><li><a class="tocitem" href="Sujet.html">Sujet</a></li><li><span class="tocitem">Algorithmes</span><ul><li><a class="tocitem" href="Algorithme_de_newton.html">L'algorithme de Newton local</a></li><li><a class="tocitem" href="Regions_de_confiance.html">La méthode des régions de confiance</a></li><li><a class="tocitem" href="Lagrangien_augmente.html">La méthode du Lagrangien augmenté</a></li></ul></li><li><a class="tocitem" href="fct_index.html">Index des fonctions</a></li><li><a class="tocitem" href="Annexes.html">Annexes</a></li><li><a class="tocitem" href="mise_en_place.html">Installation de Julia et tests unitaires</a></li><li><a class="tocitem" href="FAQ.html">Foire aux Questions</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href="Exemples.html">-</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="Exemples.html">-</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com//blob/master/docs/src/Exemples.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h2 id="Exemples-d'appels-1"><a class="docs-heading-anchor" href="#Exemples-d'appels-1">Exemples d'appels</a><a class="docs-heading-anchor-permalink" href="#Exemples-d'appels-1" title="Permalink"></a></h2><p>Dans les exemples suivants appliqués sur les fonctions :</p><ol><li><a href="fct_index.html#Optinum.Algorithme_De_Newton-Tuple{Function,Function,Function,Any,Any}"><code>Algorithme_De_Newton</code></a></li><li><a href="fct_index.html#Optinum.Pas_De_Cauchy-Tuple{Any,Any,Any}"><code>Pas_De_Cauchy</code></a></li><li><a href="fct_index.html#Optinum.Gradient_Conjugue_Tronque-Tuple{Any,Any,Any}"><code>Gradient_Conjugue_Tronque</code></a></li><li><a href="fct_index.html#Optinum.Regions_De_Confiance-Tuple{Any,Function,Function,Function,Any,Any}"><code>Regions_De_Confiance</code></a></li><li><a href="fct_index.html#Optinum.Lagrangien_Augmente-Tuple{Any,Function,Function,Function,Function,Function,Function,Any,Any}"><code>Lagrangien_Augmente</code></a></li></ol><p>nous allons utiliser la fonction suivante : <span>$\\$</span> <span>$\begin{aligned}\hspace*{1.5cm} f: \mathbb{R}^{2} \quad &\rightarrow \mathbb{R} \\ \hspace*{1.5cm} \left(x_{1}, x_{2}\right) &\rightarrow 100\left(x_{2}-x_{1}^{2}\right)^{2}+\left(1-x_{1}\right)^{2} \end{aligned} \\$</span> dont le gradient est : <span>$\hspace*{0.5cm}$</span> <span>$\nabla f(x) = \left[\begin{array}{l} -400 x_{1}(x_{2}-x_{1}^{2})-2(1-x_{1}) & 200 (x_{2}-x_{1}^{2}) \end{array}\right]^{T} \\$</span> et la hessienne est : <span>$\hspace*{0.5cm}$</span> <span>$\nabla^2 f(x) = \left[ \begin{array}{cc} -400 (x_{2}-3 x_{1}^{2})+2 & -400 x_{1} \\ -400 x_{1} & 200 \end{array}\right]$</span></p><pre><code class="language-julia">using OptinumProf
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using LinearAlgebra
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using Plots</code></pre><p>Voici la fonction <span>$f$</span></p><pre><code class="language-julia">f(x)=100*(x[2]-x[1]^2)^2+(1-x[1])^2
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x, y = -1.5:0.1:1, -2:0.1:3.5
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z = Plots.Surface((x,y)->f([x,y]), x, y)
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Plots.surface(x,y,z,camera=(85,43))</code></pre><?xml version="1.0" encoding="utf-8"?>
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<p>Ici on trace la norme de <span>$\nabla f$</span></p><pre><code class="language-julia">gradf(x)=[-400*x[1]*(x[2]-x[1]^2)-2*(1-x[1]) ; 200*(x[2]-x[1]^2)]
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z = Plots.Surface((x,y)->norm(gradf([x,y])), x, y)
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"/>
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</svg>
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<p>Et <span>$\nabla^2 f$</span></p><pre><code class="language-julia">hessf(x)=[-400*(x[2]-3*x[1]^2)+2 -400*x[1];-400*x[1] 200]
|
||
options = []
|
||
nothing # masquer la sortie</code></pre><h3 id="L'Algorithme-de-Newton-1"><a class="docs-heading-anchor" href="#L'Algorithme-de-Newton-1">L'Algorithme de Newton</a><a class="docs-heading-anchor-permalink" href="#L'Algorithme-de-Newton-1" title="Permalink"></a></h3><pre><code class="language-julia">x0 = [1; 0]
|
||
output = Algorithme_De_Newton(f,gradf,hessf,x0,options)
|
||
println(output) # (xmin,f_min,flag,nb_iters)</code></pre><pre><code class="language-none">([1.0, 1.0], 0.0, 0, 1)</code></pre><h3 id="Le-pas-de-Cauchy-1"><a class="docs-heading-anchor" href="#Le-pas-de-Cauchy-1">Le pas de Cauchy</a><a class="docs-heading-anchor-permalink" href="#Le-pas-de-Cauchy-1" title="Permalink"></a></h3><pre><code class="language-julia">xk = [0; 0.5]
|
||
delta1 = 1
|
||
output = Pas_De_Cauchy(gradf(xk),hessf(xk),delta1)
|
||
println(output) # (sk, e)</code></pre><pre><code class="language-none">([0.010007963153408747, -0.5003981576704374], 1)</code></pre><h3 id="Algorithme-du-Gradient-Conjugué-Tronqué-1"><a class="docs-heading-anchor" href="#Algorithme-du-Gradient-Conjugué-Tronqué-1">Algorithme du Gradient Conjugué Tronqué</a><a class="docs-heading-anchor-permalink" href="#Algorithme-du-Gradient-Conjugué-Tronqué-1" title="Permalink"></a></h3><pre><code class="language-julia"># deltak = options[1]
|
||
# max_iter = options[2]
|
||
# tol = options[3]
|
||
options = [1,5,1e-3]
|
||
xk = [0; 0.5]
|
||
sk = Gradient_Conjugue_Tronque(gradf(xk),hessf(xk),options)
|
||
println(sk)</code></pre><pre><code class="language-none">[0.8558967421837433, -0.5171467554952414]</code></pre><h3 id="L'Algorithme-des-régions-de-confiance-1"><a class="docs-heading-anchor" href="#L'Algorithme-des-régions-de-confiance-1">L'Algorithme des régions de confiance</a><a class="docs-heading-anchor-permalink" href="#L'Algorithme-des-régions-de-confiance-1" title="Permalink"></a></h3><pre><code class="language-julia">algo="gct" # ou cauchy
|
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x0 = [1; 0]
|
||
options = []
|
||
output = Regions_De_Confiance(algo,f,gradf,hessf,x0,options)
|
||
println(output) # (xmin, fxmin, flag,nb_iters)</code></pre><pre><code class="language-none">([1.0, 1.0], 0.0, 0, 2)</code></pre><h3 id="Algorithme-du-Lagrangien-augmenté-pour-contraintes-d’égalité-1"><a class="docs-heading-anchor" href="#Algorithme-du-Lagrangien-augmenté-pour-contraintes-d’égalité-1">Algorithme du Lagrangien augmenté pour contraintes d’égalité</a><a class="docs-heading-anchor-permalink" href="#Algorithme-du-Lagrangien-augmenté-pour-contraintes-d’égalité-1" title="Permalink"></a></h3><p>Dans cet exemple nous allons prendre la contrainte suivante : <span>$\\$</span> <span>$c(x) = x_{1}^2 + x_{2}^2 -1.5 = 0 \\$</span> dont le gradient est : <span>$\hspace*{0.5cm}$</span> <span>$\left[\begin{array}{l} 2x_{1} & 2 x_{2} \end{array}\right]^{T} \\$</span> et la hessienne est :<span>$\hspace*{0.5cm}$</span> <span>$\left[ \begin{array}{cc} 2 & 0 \\ 0 & 2 \end{array}\right]$</span></p><pre><code class="language-julia">algo = "gct" # ou newton|gct
|
||
options = []
|
||
contrainte(x) = (x[1]^2) + (x[2]^2) -1.5
|
||
grad_contrainte(x) = [2*x[1] ;2*x[2]]
|
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hess_contrainte(x) = [2 0;0 2]
|
||
output = Lagrangien_Augmente(algo,f,contrainte,gradf,hessf,grad_contrainte,hess_contrainte,x0,options)
|
||
println(output) # (xmin1,fxmin1,flag,nbiters)</code></pre><pre><code class="language-none">([0.9072339558154577, 0.8227554477306978], 0.008615651535155151, 0, 2)</code></pre></article></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 8 November 2021 11:31">Monday 8 November 2021</span>. Using Julia version 1.4.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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