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<html lang="en">
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<title>Quadratic Programming - GNU Octave</title>
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<a name="Quadratic-Programming"></a>
<p>
Next: <a rel="next" accesskey="n" href="Nonlinear-Programming.html#Nonlinear-Programming">Nonlinear Programming</a>,
Previous: <a rel="previous" accesskey="p" href="Linear-Programming.html#Linear-Programming">Linear Programming</a>,
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<h3 class="section">25.2 Quadratic Programming</h3>
<p>Octave can also solve Quadratic Programming problems, this is
<pre class="example"> min 0.5 x'*H*x + x'*q
</pre>
<p>subject to
<pre class="example"> A*x = b
lb <= x <= ub
A_lb <= A_in*x <= A_ub
</pre>
<!-- qp scripts/optimization/qp.m -->
<p><a name="doc_002dqp"></a>
<div class="defun">
— Function File: [<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] = <b>qp</b> (<var>x0, H</var>)<var><a name="index-qp-2374"></a></var><br>
— Function File: [<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] = <b>qp</b> (<var>x0, H, q</var>)<var><a name="index-qp-2375"></a></var><br>
— Function File: [<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] = <b>qp</b> (<var>x0, H, q, A, b</var>)<var><a name="index-qp-2376"></a></var><br>
— Function File: [<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] = <b>qp</b> (<var>x0, H, q, A, b, lb, ub</var>)<var><a name="index-qp-2377"></a></var><br>
— Function File: [<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] = <b>qp</b> (<var>x0, H, q, A, b, lb, ub, A_lb, A_in, A_ub</var>)<var><a name="index-qp-2378"></a></var><br>
— Function File: [<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] = <b>qp</b> (<var><small class="dots">...</small>, options</var>)<var><a name="index-qp-2379"></a></var><br>
<blockquote><p>Solve the quadratic program
<pre class="example"> min 0.5 x'*H*x + x'*q
x
</pre>
<p>subject to
<pre class="example"> A*x = b
lb <= x <= ub
A_lb <= A_in*x <= A_ub
</pre>
<p class="noindent">using a null-space active-set method.
<p>Any bound (<var>A</var>, <var>b</var>, <var>lb</var>, <var>ub</var>, <var>A_lb</var>,
<var>A_ub</var>) may be set to the empty matrix (<code>[]</code>) if not
present. If the initial guess is feasible the algorithm is faster.
<dl>
<dt><var>options</var><dd>An optional structure containing the following
parameter(s) used to define the behavior of the solver. Missing elements
in the structure take on default values, so you only need to set the
elements that you wish to change from the default.
<dl>
<dt><code>MaxIter (default: 200)</code><dd>Maximum number of iterations.
</dl>
</dl>
<dl>
<dt><var>info</var><dd>Structure containing run-time information about the algorithm. The
following fields are defined:
<dl>
<dt><code>solveiter</code><dd>The number of iterations required to find the solution.
<br><dt><code>info</code><dd>An integer indicating the status of the solution.
<dl>
<dt>0<dd>The problem is feasible and convex. Global solution found.
<br><dt>1<dd>The problem is not convex. Local solution found.
<br><dt>2<dd>The problem is not convex and unbounded.
<br><dt>3<dd>Maximum number of iterations reached.
<br><dt>6<dd>The problem is infeasible.
</dl>
</dl>
</dl>
</p></blockquote></div>
<!-- pqpnonneg scripts/optimization/pqpnonneg.m -->
<p><a name="doc_002dpqpnonneg"></a>
<div class="defun">
— Function File: <var>x</var> = <b>pqpnonneg</b> (<var>c, d</var>)<var><a name="index-pqpnonneg-2380"></a></var><br>
— Function File: <var>x</var> = <b>pqpnonneg</b> (<var>c, d, x0</var>)<var><a name="index-pqpnonneg-2381"></a></var><br>
— Function File: [<var>x</var>, <var>minval</var>] = <b>pqpnonneg</b> (<var><small class="dots">...</small></var>)<var><a name="index-pqpnonneg-2382"></a></var><br>
— Function File: [<var>x</var>, <var>minval</var>, <var>exitflag</var>] = <b>pqpnonneg</b> (<var><small class="dots">...</small></var>)<var><a name="index-pqpnonneg-2383"></a></var><br>
— Function File: [<var>x</var>, <var>minval</var>, <var>exitflag</var>, <var>output</var>] = <b>pqpnonneg</b> (<var><small class="dots">...</small></var>)<var><a name="index-pqpnonneg-2384"></a></var><br>
— Function File: [<var>x</var>, <var>minval</var>, <var>exitflag</var>, <var>output</var>, <var>lambda</var>] = <b>pqpnonneg</b> (<var><small class="dots">...</small></var>)<var><a name="index-pqpnonneg-2385"></a></var><br>
<blockquote><p>Minimize <code>1/2*x'*c*x + d'*x</code> subject to <var>x</var><code> >= 0</code>. <var>c</var>
and <var>d</var> must be real, and <var>c</var> must be symmetric and positive
definite. <var>x0</var> is an optional initial guess for <var>x</var>.
<p>Outputs:
<ul>
<li>minval
<p>The minimum attained model value, 1/2*xmin'*c*xmin + d'*xmin
<li>exitflag
<p>An indicator of convergence. 0 indicates that the iteration count
was exceeded, and therefore convergence was not reached; >0 indicates
that the algorithm converged. (The algorithm is stable and will
converge given enough iterations.)
<li>output
<p>A structure with two fields:
<ul>
<li>"algorithm": The algorithm used ("nnls")
<li>"iterations": The number of iterations taken.
</ul>
<li>lambda
<p>Not implemented.
</ul>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002doptimset.html#doc_002doptimset">optimset</a>, <a href="doc_002dlsqnonneg.html#doc_002dlsqnonneg">lsqnonneg</a>, <a href="doc_002dqp.html#doc_002dqp">qp</a>.
</p></blockquote></div>
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