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previews/PR2572/.documenter-siteinfo.json

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-{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2024-10-30T22:04:12","documenter_version":"1.7.0"}}
+{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2024-10-30T22:27:15","documenter_version":"1.7.0"}}

previews/PR2572/background/duality/index.html

@@ -77,4 +77,4 @@
        \max & \sum b_k y_k \\
 \text{s.t.} \;\; & C+C^\top - \sum (A_k+A_k^\top) y_k \in \mathcal{S}_+ \\
             & C-C^\top - \sum(A_k-A_k^\top) y_k = 0
-\end{align}\]</p><p>and we recover <span>$Z = X + X^\top$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../manual/modification/">« Problem modification</a><a class="docs-footer-nextpage" href="../infeasibility_certificates/">Infeasibility certificates »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Wednesday 30 October 2024 22:04">Wednesday 30 October 2024</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{align}\]</p><p>and we recover <span>$Z = X + X^\top$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../manual/modification/">« Problem modification</a><a class="docs-footer-nextpage" href="../infeasibility_certificates/">Infeasibility certificates »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Wednesday 30 October 2024 22:27">Wednesday 30 October 2024</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

previews/PR2572/background/infeasibility_certificates/index.html

@@ -25,4 +25,4 @@
 \end{align}\]</p><p>and:</p><p class="math-container">\[-\sum_{i=1}^m b_i^\top (y_i + \eta d_i) &gt; -\sum_{i=1}^m b_i^\top y_i,\]</p><p>for any feasible dual solution <span>$y$</span>. The latter simplifies to <span>$-\sum_{i=1}^m b_i^\top d_i &gt; 0$</span>. For a maximization problem, the inequality is <span>$\sum_{i=1}^m b_i^\top d_i &lt; 0$</span>. (Note that these are the same inequality, modulo a <code>-</code> sign.)</p><p>If the solver has found a certificate of primal infeasibility:</p><ul><li><a href="../../reference/models/#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a> must be <code>INFEASIBLE</code></li><li><a href="../../reference/models/#MathOptInterface.DualStatus"><code>DualStatus</code></a> must be <code>INFEASIBILITY_CERTIFICATE</code></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintDual"><code>ConstraintDual</code></a> must be the corresponding value of <span>$d$</span></li><li><a href="../../reference/models/#MathOptInterface.DualObjectiveValue"><code>DualObjectiveValue</code></a> must be the value <span>$-\sum_{i=1}^m b_i^\top d_i$</span> for minimization problems and <span>$\sum_{i=1}^m b_i^\top d_i$</span> for maximization problems.</li></ul><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The choice of whether to scale the ray <span>$d$</span> to have magnitude <code>1</code> is left to the solver.</p></div></div><h3 id="Infeasibility-certificates-of-variable-bounds"><a class="docs-heading-anchor" href="#Infeasibility-certificates-of-variable-bounds">Infeasibility certificates of variable bounds</a><a id="Infeasibility-certificates-of-variable-bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Infeasibility-certificates-of-variable-bounds" title="Permalink"></a></h3><p>Many linear solvers (for example, Gurobi) do not provide explicit access to the primal infeasibility certificate of a variable bound. However, given a set of linear constraints:</p><p class="math-container">\[\begin{align}
 l_A \le A x \le u_A \\
 l_x \le x \le u_x,
-\end{align}\]</p><p>the primal certificate of the variable bounds can be computed using the primal certificate associated with the affine constraints, <span>$d$</span>. (Note that <span>$d$</span> will have one element for each row of the <span>$A$</span> matrix, and that some or all of the elements in the vectors <span>$l_A$</span> and <span>$u_A$</span> may be <span>$\pm \infty$</span>. If both <span>$l_A$</span> and <span>$u_A$</span> are finite for some row, the corresponding element in `<code>d</code>  must be <code>0</code>.)</p><p>Given <span>$d$</span>, compute <span>$\bar{d} = d^\top A$</span>. If the bound is finite, a certificate for the lower variable bound of <span>$x_i$</span> is <span>$\max\{\bar{d}_i, 0\}$</span>, and a certificate for the upper variable bound is <span>$\min\{\bar{d}_i, 0\}$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../duality/">« Duality</a><a class="docs-footer-nextpage" href="../naming_conventions/">Naming conventions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Wednesday 30 October 2024 22:04">Wednesday 30 October 2024</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{align}\]</p><p>the primal certificate of the variable bounds can be computed using the primal certificate associated with the affine constraints, <span>$d$</span>. (Note that <span>$d$</span> will have one element for each row of the <span>$A$</span> matrix, and that some or all of the elements in the vectors <span>$l_A$</span> and <span>$u_A$</span> may be <span>$\pm \infty$</span>. If both <span>$l_A$</span> and <span>$u_A$</span> are finite for some row, the corresponding element in `<code>d</code>  must be <code>0</code>.)</p><p>Given <span>$d$</span>, compute <span>$\bar{d} = d^\top A$</span>. If the bound is finite, a certificate for the lower variable bound of <span>$x_i$</span> is <span>$\max\{\bar{d}_i, 0\}$</span>, and a certificate for the upper variable bound is <span>$\min\{\bar{d}_i, 0\}$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../duality/">« Duality</a><a class="docs-footer-nextpage" href="../naming_conventions/">Naming conventions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Wednesday 30 October 2024 22:27">Wednesday 30 October 2024</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>