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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Annexes · 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&#39;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 class="is-active"><a class="tocitem" href="Annexes.html">Annexes</a><ul class="internal"><li><a class="tocitem" href="#A.-Problèmes-sans-contraintes-1"><span>A. Problèmes sans contraintes</span></a></li><li><a class="tocitem" href="#B.-Cas-tests-pour-le-calcul-du-pas-de-Cauchy-1"><span>B. Cas tests pour le calcul du pas de Cauchy</span></a></li><li><a class="tocitem" href="#C.-Cas-tests-pour-la-résolution-du-sous-problème-par-l’algorithme-du-Gradient-Conjugué-Tronqué-1"><span>C. Cas tests pour la résolution du sous-problème par l’algorithme du Gradient Conjugué Tronqué</span></a></li><li><a class="tocitem" href="#D.-Problèmes-avec-contraintes-1"><span>D. Problèmes avec contraintes</span></a></li></ul></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="Annexes.html">Annexes</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="Annexes.html">Annexes</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com//blob/master/docs/src/Annexes.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="A.-Problèmes-sans-contraintes-1"><a class="docs-heading-anchor" href="#A.-Problèmes-sans-contraintes-1">A. Problèmes sans contraintes</a><a class="docs-heading-anchor-permalink" href="#A.-Problèmes-sans-contraintes-1" title="Permalink"></a></h2><p>Les problèmes de minimisation sans contraintes à résoudre sont les suivants :</p><h4 id="Problème-1-1"><a class="docs-heading-anchor" href="#Problème-1-1">Problème 1</a><a class="docs-heading-anchor-permalink" href="#Problème-1-1" title="Permalink"></a></h4><p><span>$\hspace*{1.5cm}$</span> <span>$\begin{aligned} f_{1}: \mathbb{R}^{3} &amp; \rightarrow \mathbb{R} \\ \left(x_{1}, x_{2}, x_{3}\right) &amp; \mapsto 2\left(x_{1}+x_{2}+x_{3}-3\right)^{2}+\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2} \end{aligned}$</span></p><p>On cherchera à minimiser <span>$f_{1}$</span> sur <span>$\mathbb{R}^{3}$</span> , en partant des points suivants : <span>$\\$</span></p><p><span>$\hspace*{2cm}$</span> <span>$x_{011}=\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right], \quad x_{012}=\left[\begin{array}{c} 10 \\ 3 \\ -2.2 \end{array}\right]$</span></p><h4 id="Problème-2-1"><a class="docs-heading-anchor" href="#Problème-2-1">Problème 2</a><a class="docs-heading-anchor-permalink" href="#Problème-2-1" title="Permalink"></a></h4><p><span>$\hspace*{1.5cm}$</span> <span>$\begin{aligned} f_{2}: \mathbb{R}^{2} &amp; \rightarrow \mathbb{R} \\ \left(x_{1}, x_{2}\right) &amp; \mapsto 100\left(x_{2}-x_{1}^{2}\right)^{2}+\left(1-x_{1}\right)^{2} \end{aligned}$</span></p><p>On cherchera à minimiser <span>$f_{2}$</span> sur <span>$\mathbb{R}^{2}$</span> , en partant des points suivants :<span>$\\$</span> </p><p><span>$x_{0 2 1}=\left[\begin{array}{c} -1.2 \\ 1 \end{array}\right]\\$</span></p><p><span>$x_{0 2 2}=\left[\begin{array}{c} 10 \\ 0 \end{array}\right]\\$</span></p><p><span>$x_{023}=\left[\begin{array}{c} 0 \\ \frac{1}{200}+\frac{1}{10^{12}} \end{array}\right]\\$</span></p><h2 id="B.-Cas-tests-pour-le-calcul-du-pas-de-Cauchy-1"><a class="docs-heading-anchor" href="#B.-Cas-tests-pour-le-calcul-du-pas-de-Cauchy-1">B. Cas tests pour le calcul du pas de Cauchy</a><a class="docs-heading-anchor-permalink" href="#B.-Cas-tests-pour-le-calcul-du-pas-de-Cauchy-1" title="Permalink"></a></h2><p>On considère des fonctions quadratiques de la forme <span>$q(s)=s^{\top} g+\frac{1}{2} s^{\top} H s$</span></p><h4 id="Quadratique-1-1"><a class="docs-heading-anchor" href="#Quadratique-1-1">Quadratique 1</a><a class="docs-heading-anchor-permalink" href="#Quadratique-1-1" title="Permalink"></a></h4><p><span>$g=\left[\begin{array}{l} 0 \\ 0 \end{array}\right]$</span>,<span>$H=\left[\begin{array}{ll} 7 &amp; 0 \\ 0 &amp; 2 \end{array}\right]\\$</span></p><h4 id="Quadratique-2-1"><a class="docs-heading-anchor" href="#Quadratique-2-1">Quadratique 2</a><a class="docs-heading-anchor-permalink" href="#Quadratique-2-1" title="Permalink"></a></h4><p><span>$g=\left[\begin{array}{l} 6 \\ 2 \end{array}\right]$</span>,<span>$H=\left[\begin{array}{ll} 7 &amp; 0 \\ 0 &amp; 2 \end{array}\right]\\$</span></p><h4 id="Quadratique-3-1"><a class="docs-heading-anchor" href="#Quadratique-3-1">Quadratique 3</a><a class="docs-heading-anchor-permalink" href="#Quadratique-3-1" title="Permalink"></a></h4><p><span>$g=\left[\begin{array}{l} -2 \\ 1 \end{array}\right]$</span>,<span>$H=\left[\begin{array}{ll} -2 &amp; 0 \\ 0 &amp; 10 \end{array}\right]\\$</span></p><h2 id="C.-Cas-tests-pour-la-résolution-du-sous-problème-par-l’algorithme-du-Gradient-Conjugué-Tronqué-1"><a class="docs-heading-anchor" href="#C.-Cas-tests-pour-la-résolution-du-sous-problème-par-l’algorithme-du-Gradient-Conjugué-Tronqué-1">C. Cas tests pour la résolution du sous-problème par l’algorithme du Gradient Conjugué Tronqué</a><a class="docs-heading-anchor-permalink" href="#C.-Cas-tests-pour-la-résolution-du-sous-problème-par-l’algorithme-du-Gradient-Conjugué-Tronqué-1" title="Permalink"></a></h2><p>On reprendra les 3 quadratiques testées avec le pas de Cauchy, auxquelles on ajoutera :</p><h4 id="Quadratique-4-1"><a class="docs-heading-anchor" href="#Quadratique-4-1">Quadratique 4</a><a class="docs-heading-anchor-permalink" href="#Quadratique-4-1" title="Permalink"></a></h4><p><span>$g=\left[\begin{array}{l} 0 \\ 0 \end{array}\right]$</span>, <span>$H=\left[\begin{array}{ll} -2 &amp; 0 \\ 0 &amp; 10 \end{array}\right]\\$</span></p><h4 id="Quadratique-5-1"><a class="docs-heading-anchor" href="#Quadratique-5-1">Quadratique 5</a><a class="docs-heading-anchor-permalink" href="#Quadratique-5-1" title="Permalink"></a></h4><p><span>$g=\left[\begin{array}{l} 2 \\ 3 \end{array}\right]$</span>,<span>$H=\left[\begin{array}{ll} 4 &amp; 6 \\ 6 &amp; 5 \end{array}\right]\\$</span></p><h4 id="Quadratique-6-1"><a class="docs-heading-anchor" href="#Quadratique-6-1">Quadratique 6</a><a class="docs-heading-anchor-permalink" href="#Quadratique-6-1" title="Permalink"></a></h4><p><span>$g=\left[\begin{array}{l} 2 \\ 0 \end{array}\right]$</span>, <span>$H=\left[\begin{array}{ll} 4 &amp; 0 \\ 0 &amp; -15 \end{array}\right]\\$</span></p><h2 id="D.-Problèmes-avec-contraintes-1"><a class="docs-heading-anchor" href="#D.-Problèmes-avec-contraintes-1">D. Problèmes avec contraintes</a><a class="docs-heading-anchor-permalink" href="#D.-Problèmes-avec-contraintes-1" title="Permalink"></a></h2><h3 id="Retour-sur-f_1-1"><a class="docs-heading-anchor" href="#Retour-sur-f_1-1">Retour sur <span>$f_1$</span></a><a class="docs-heading-anchor-permalink" href="#Retour-sur-f_1-1" title="Permalink"></a></h3><p>On s&#39;intéresse à la valeur minimale de <span>$f_1$</span> sur un ensemble défini par une contrainte linéaire. La formulation du problème sera alors </p><p><span>$\min _{x_{1}+x_{3}=1; x \in \mathbb{R}^{3}} f_{1}(x)$</span></p><p>On choisira comme point initial</p><p><span>$x_{c 11}=\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right]$</span> (réalisable) </p><p>ou </p><p><span>$x_{c 12}=\left[\begin{array}{l} 0.5 \\ 1.25 \\ 1 \end{array}\right]$</span> (non réalisable) .</p><h4 id="Retour-sur-f_2-1"><a class="docs-heading-anchor" href="#Retour-sur-f_2-1">Retour sur <span>$f_2$</span></a><a class="docs-heading-anchor-permalink" href="#Retour-sur-f_2-1" title="Permalink"></a></h4><p>On cherche à minimiser la fonction <span>$f_2$</span> décrite dans la partie précédente, en se restreignant maintenant à une sphère. Le problème s&#39;écrit :</p><p><span>$\min _{x_{1}^{2}+x_{2}^{2}=1.5; x \in \mathbb{R}^{2}} f_{2}(x)$</span></p><p>On choisira comme point initial</p><p><span>$x_{c 21}=\left[\begin{array}{c} 1 \\ 0 \end{array}\right]$</span> (non réalisable) </p><p>ou </p><p><span>$x_{c 22}=\left[\begin{array}{c} \sqrt{3} / 2 \\ \sqrt{3} / 2 \end{array}\right]$</span> (réalisable).</p><h3 id="Un-problème-avec-contraintes-d&#39;inégalité-(supplément)-1"><a class="docs-heading-anchor" href="#Un-problème-avec-contraintes-d&#39;inégalité-(supplément)-1">Un problème avec contraintes d&#39;inégalité (supplément)</a><a class="docs-heading-anchor-permalink" href="#Un-problème-avec-contraintes-d&#39;inégalité-(supplément)-1" title="Permalink"></a></h3><p><span>$\left\{\begin{array}{lll} \min _{(x, y) \in \mathbb{R}^{2}} f_{3}(x, y) &amp; = &amp; (x-1)^{2}+(y-2.5)^{2} \\ x-2 y+2 &amp; \geq &amp; 0 \\ -x-2 y+6 &amp; \geq &amp; 0 \\ -x+2 y+2 &amp; \geq &amp; 0 \\ x &amp; \geq &amp; 0 \\ y &amp; \geq &amp; 0 \end{array}\right.$</span></p><p>L&#39;origine pourra être prise comme point initial.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="fct_index.html">« Index des fonctions</a><a class="docs-footer-nextpage" href="mise_en_place.html">Installation de Julia et tests unitaires »</a></nav></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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