-<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Duality · SimplexTableaux</title><meta name="title" content="Duality · SimplexTableaux"/><meta property="og:title" content="Duality · SimplexTableaux"/><meta property="twitter:title" content="Duality · SimplexTableaux"/><meta name="description" content="Documentation for SimplexTableaux."/><meta property="og:description" content="Documentation for SimplexTableaux."/><meta property="twitter:description" content="Documentation for SimplexTableaux."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/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="../search_index.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/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-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"><div class="docs-package-name"><span class="docs-autofit"><a href="../">SimplexTableaux</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Overview</a></li><li><a class="tocitem" href="../tutorial/">Tutorial</a></li><li><a class="tocitem" href="../other/">Other Functions</a></li><li><a class="tocitem" href="../api/">API</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"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Duality</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Duality</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/scheinerman/SimplexTableaux.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/scheinerman/SimplexTableaux.jl/blob/main/docs/src/dual.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Duality"><a class="docs-heading-anchor" href="#Duality">Duality</a><a id="Duality-1"></a><a class="docs-heading-anchor-permalink" href="#Duality" title="Permalink"></a></h1><blockquote><p>The duality functionality is under construction. I don't know how you found your way here, but please understand that everything on this page is in pre-beta.</p></blockquote><h2 id="The-dual-function"><a class="docs-heading-anchor" href="#The-dual-function">The dual function</a><a id="The-dual-function-1"></a><a class="docs-heading-anchor-permalink" href="#The-dual-function" title="Permalink"></a></h2><p>If a tableau <code>T</code> is created from a canonical minimization problem, then <code>dual(T)</code> returns a new tableau for its dual linear program. However, <code>dual(T)</code> is also a minimization problem. The minimum value of <code>dual(T)</code> is negative that of <code>T</code>. </p><p>That is, if <code>T</code> represents the LP <span>$\min c^T x$</span> s.t. <span>$Ax \ge b$</span>, <span>$x\ge0$</span>, then <code>dual(T)</code> respresents the LP <span>$\min b^T y$</span> s.t. <span>$-A^T y \le -c$</span>, <span>$y\ge0$</span>. </p><h2 id="Example"><a class="docs-heading-anchor" href="#Example">Example</a><a id="Example-1"></a><a class="docs-heading-anchor-permalink" href="#Example" title="Permalink"></a></h2><pre><code class="nohighlight hljs">julia> A = [11 2 11; 8 6 9; 8 8 5; 6 5 8; 4 1 2; 2 -1 4];
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