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<span class="path"><a href="index.html">Symphony Toolbox</a> >> <a href="section_19f4f1e5726c01d683e8b82be0a7e910.html">Symphony Toolbox</a> > qpipoptmat</span>
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<div class="refnamediv"><h1 class="refname">qpipoptmat</h1>
<p class="refpurpose">Solves a linear quadratic problem.</p></div>
<div class="refsynopsisdiv"><h3 class="title">Calling Sequence</h3>
<div class="synopsis"><pre><span class="default">xopt</span><span class="default"> = </span><span class="functionid">qpipoptmat</span><span class="default">(</span><span class="default">H</span><span class="default">,</span><span class="default">f</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">qpipoptmat</span><span class="default">(</span><span class="default">H</span><span class="default">,</span><span class="default">f</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">qpipoptmat</span><span class="default">(</span><span class="default">H</span><span class="default">,</span><span class="default">f</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">qpipoptmat</span><span class="default">(</span><span class="default">H</span><span class="default">,</span><span class="default">f</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">qpipoptmat</span><span class="default">(</span><span class="default">H</span><span class="default">,</span><span class="default">f</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">,</span><span class="default">x0</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">qpipoptmat</span><span class="default">(</span><span class="default">H</span><span class="default">,</span><span class="default">f</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">,</span><span class="default">x0</span><span class="default">,</span><span class="default">param</span><span class="default">)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">,</span><span class="default">exitflag</span><span class="default">,</span><span class="default">output</span><span class="default">,</span><span class="default">lamda</span><span class="default">] = </span><span class="functionid">qpipoptmat</span><span class="default">( ... )</span></pre></div></div>
<div class="refsection"><h3 class="title">Parameters</h3>
<dl><dt><span class="term">H :</span>
<dd><p class="para">a symmetric matrix of double, represents coefficients of quadratic in the quadratic problem.</p></dd></dt>
<dt><span class="term">f :</span>
<dd><p class="para">a vector of double, represents coefficients of linear in the quadratic problem</p></dd></dt>
<dt><span class="term">A :</span>
<dd><p class="para">a matrix of double, represents the linear coefficients in the inequality constraints A⋅x ≤ b.</p></dd></dt>
<dt><span class="term">b :</span>
<dd><p class="para">a vector of double, represents the linear coefficients in the inequality constraints A⋅x ≤ b.</p></dd></dt>
<dt><span class="term">Aeq :</span>
<dd><p class="para">a matrix of double, represents the linear coefficients in the equality constraints Aeq⋅x = beq.</p></dd></dt>
<dt><span class="term">beq :</span>
<dd><p class="para">a vector of double, represents the linear coefficients in the equality constraints Aeq⋅x = beq.</p></dd></dt>
<dt><span class="term">lb :</span>
<dd><p class="para">a vector of double, contains lower bounds of the variables.</p></dd></dt>
<dt><span class="term">ub :</span>
<dd><p class="para">a vector of double, contains upper bounds of the variables.</p></dd></dt>
<dt><span class="term">x0 :</span>
<dd><p class="para">a vector of double, contains initial guess of variables.</p></dd></dt>
<dt><span class="term">param :</span>
<dd><p class="para">a list containing the parameters to be set.</p></dd></dt>
<dt><span class="term">xopt :</span>
<dd><p class="para">a vector of double, the computed solution of the optimization problem.</p></dd></dt>
<dt><span class="term">fopt :</span>
<dd><p class="para">a double, the value of the function at x.</p></dd></dt>
<dt><span class="term">residual :</span>
<dd><p class="para">a vector of double, solution residuals returned as the vector d-C*x.</p></dd></dt>
<dt><span class="term">exitflag :</span>
<dd><p class="para">The exit status. See below for details.</p></dd></dt>
<dt><span class="term">output :</span>
<dd><p class="para">The structure consist of statistics about the optimization. See below for details.</p></dd></dt>
<dt><span class="term">lambda :</span>
<dd><p class="para">The structure consist of the Lagrange multipliers at the solution of problem. See below for details.</p></dd></dt></dl></div>
<div class="refsection"><h3 class="title">Description</h3>
<p class="para">Search the minimum of a constrained linear quadratic optimization problem specified by :</p>
<p class="para"><span><img src='./_LaTeX_qpipoptmat.xml_1.png' style='position:relative;top:41px;width:277px;height:90px'/></span></p>
<p class="para">The routine calls Ipopt for solving the quadratic problem, Ipopt is a library written in C++.</p>
<p class="para">The exitflag allows to know the status of the optimization which is given back by Ipopt.
<ul class="itemizedlist"><li>exitflag=0 : Optimal Solution Found</li>
<li>exitflag=1 : Maximum Number of Iterations Exceeded. Output may not be optimal.</li>
<li>exitflag=2 : Maximum CPU Time exceeded. Output may not be optimal.</li>
<li>exitflag=3 : Stop at Tiny Step.</li>
<li>exitflag=4 : Solved To Acceptable Level.</li>
<li>exitflag=5 : Converged to a point of local infeasibility.</li></ul></p>
<p class="para">For more details on exitflag see the ipopt documentation, go to http://www.coin-or.org/Ipopt/documentation/</p>
<p class="para">The output data structure contains detailed informations about the optimization process.
It has type "struct" and contains the following fields.
<ul class="itemizedlist"><li>output.iterations: The number of iterations performed during the search</li>
<li>output.constrviolation: The max-norm of the constraint violation.</li></ul></p>
<p class="para">The lambda data structure contains the Lagrange multipliers at the end
of optimization. In the current version the values are returned only when the the solution is optimal.
It has type "struct" and contains the following fields.
<ul class="itemizedlist"><li>lambda.lower: The Lagrange multipliers for the lower bound constraints.</li>
<li>lambda.upper: The Lagrange multipliers for the upper bound constraints.</li>
<li>lambda.eqlin: The Lagrange multipliers for the linear equality constraints.</li>
<li>lambda.ineqlin: The Lagrange multipliers for the linear inequality constraints.</li></ul></p>
<p class="para"></p></div>
<div class="refsection"><h3 class="title">Examples</h3>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Ref : example 14 :</span>
<span class="scilabcomment">//https://www.me.utexas.edu/~jensen/ORMM/supplements/methods/nlpmethod/S2_quadratic.pdf</span>
<span class="scilabcomment">// min. -8*x1*x1 -16*x2*x2 + x1 + 4*x2</span>
<span class="scilabcomment">// such that</span>
<span class="scilabcomment">// x1 + x2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 5,</span>
<span class="scilabcomment">// x1 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 3,</span>
<span class="scilabcomment">// x1 </span><span class="scilabcomment">></span><span class="scilabcomment">= 0,</span>
<span class="scilabcomment">// x2 </span><span class="scilabcomment">></span><span class="scilabcomment">= 0</span>
<span class="scilabid">H</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">2</span> <span class="scilabnumber">0</span>
<span class="scilabnumber">0</span> <span class="scilabnumber">8</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">f</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilaboperator">-</span><span class="scilabnumber">8</span><span class="scilabdefault">;</span> <span class="scilaboperator">-</span><span class="scilabnumber">16</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">A</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1</span> <span class="scilabnumber">1</span><span class="scilabdefault">;</span><span class="scilabnumber">1</span> <span class="scilabnumber">0</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">b</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">5</span><span class="scilabdefault">;</span><span class="scilabnumber">3</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">lb</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span> <span class="scilabnumber">0</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">ub</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabconstants">%inf</span><span class="scilabdefault">;</span> <span class="scilabconstants">%inf</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabopenclose">[</span><span class="scilabid">xopt</span><span class="scilabdefault">,</span><span class="scilabid">fopt</span><span class="scilabdefault">,</span><span class="scilabid">exitflag</span><span class="scilabdefault">,</span><span class="scilabid">output</span><span class="scilabdefault">,</span><span class="scilabid">lambda</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span> <span class="scilabid">qpipoptmat</span><span class="scilabopenclose">(</span><span class="scilabid">H</span><span class="scilabdefault">,</span><span class="scilabid">f</span><span class="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabid">lb</span><span class="scilabdefault">,</span><span class="scilabid">ub</span><span class="scilabopenclose">)</span>
<span class="scilabcomment">// Press ENTER to continue</span></pre></td><td valign="top"><a href="scilab://scilab.execexample/"><img src="ScilabExecute.png" border="0"/></a></td><td valign="top"><a href="scilab://scilab.editexample/"><img src="ScilabEdit.png" border="0"/></a></td><td></td></tr></table></div></div>
<div class="refsection"><h3 class="title">Examples</h3>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Find x in R^6 such that:</span>
<span class="scilabid">Aeq</span><span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">3</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span>
<span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">3</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">4</span><span class="scilabdefault">,</span><span class="scilabnumber">5</span><span class="scilabdefault">,</span><span class="scilabnumber">6</span><span class="scilabdefault">;</span>
<span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">5</span><span class="scilabdefault">,</span><span class="scilabnumber">3</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">beq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span> <span class="scilabnumber">2</span><span class="scilabdefault">;</span> <span class="scilabnumber">3</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">A</span><span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span>
<span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">b</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span> <span class="scilabnumber">2.5</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">lb</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilaboperator">-</span><span class="scilabnumber">1000</span><span class="scilabdefault">;</span> <span class="scilaboperator">-</span><span class="scilabnumber">10000</span><span class="scilabdefault">;</span> <span class="scilabnumber">0</span><span class="scilabdefault">;</span> <span class="scilaboperator">-</span><span class="scilabnumber">1000</span><span class="scilabdefault">;</span> <span class="scilaboperator">-</span><span class="scilabnumber">1000</span><span class="scilabdefault">;</span> <span class="scilaboperator">-</span><span class="scilabnumber">1000</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">ub</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">10000</span><span class="scilabdefault">;</span> <span class="scilabnumber">100</span><span class="scilabdefault">;</span> <span class="scilabnumber">1.5</span><span class="scilabdefault">;</span> <span class="scilabnumber">100</span><span class="scilabdefault">;</span> <span class="scilabnumber">100</span><span class="scilabdefault">;</span> <span class="scilabnumber">1000</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">x0</span> <span class="scilaboperator">=</span> <a class="scilabmacro" href="scilab://repmat">repmat</a><span class="scilabopenclose">(</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">6</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabid">param</span> <span class="scilaboperator">=</span> <a class="scilabcommand" href="scilab://list">list</a><span class="scilabopenclose">(</span><span class="scilabstring">"</span><span class="scilabstring">MaxIter</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabnumber">300</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">CpuTime</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabnumber">100</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//and minimize 0.5*x</span><span class="scilabcomment">'</span><span class="scilabcomment">*H*x + f</span><span class="scilabcomment">'</span><span class="scilabcomment">*x with</span>
<span class="scilabid">f</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span> <span class="scilabnumber">2</span><span class="scilabdefault">;</span> <span class="scilabnumber">3</span><span class="scilabdefault">;</span> <span class="scilabnumber">4</span><span class="scilabdefault">;</span> <span class="scilabnumber">5</span><span class="scilabdefault">;</span> <span class="scilabnumber">6</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> <span class="scilabid">H</span><span class="scilaboperator">=</span><a class="scilabcommand" href="scilab://eye">eye</a><span class="scilabopenclose">(</span><span class="scilabnumber">6</span><span class="scilabdefault">,</span><span class="scilabnumber">6</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabopenclose">[</span><span class="scilabid">xopt</span><span class="scilabdefault">,</span><span class="scilabid">fopt</span><span class="scilabdefault">,</span><span class="scilabid">exitflag</span><span class="scilabdefault">,</span><span class="scilabid">output</span><span class="scilabdefault">,</span><span class="scilabid">lambda</span><span class="scilabopenclose">]</span><span class="scilaboperator">=</span><span class="scilabid">qpipoptmat</span><span class="scilabopenclose">(</span><span class="scilabid">H</span><span class="scilabdefault">,</span><span class="scilabid">f</span><span class="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</span><span class="scilabdefault">,</span><span class="scilabid">Aeq</span><span class="scilabdefault">,</span><span class="scilabid">beq</span><span class="scilabdefault">,</span><span class="scilabid">lb</span><span class="scilabdefault">,</span><span class="scilabid">ub</span><span class="scilabdefault">,</span><span class="scilabid">x0</span><span class="scilabdefault">,</span><span class="scilabid">param</span><span class="scilabopenclose">)</span></pre></td><td valign="top"><a href="scilab://scilab.execexample/"><img src="ScilabExecute.png" border="0"/></a></td><td valign="top"><a href="scilab://scilab.editexample/"><img src="ScilabEdit.png" border="0"/></a></td><td></td></tr></table></div></div>
<div class="refsection"><h3 class="title">Authors</h3>
<ul class="itemizedlist"><li class="member">Keyur Joshi, Saikiran, Iswarya, Harpreet Singh</li></ul></div>
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