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<h3 class="section" id="Quadratic-Programming-1"><span>25.2 Quadratic Programming<a class="copiable-link" href="#Quadratic-Programming-1"> ¶</a></span></h3>
<p>Octave can also solve Quadratic Programming problems, this is
</p>
<div class="example">
<pre class="example-preformatted">min 0.5 x'*H*x + x'*q
</pre></div>
<p>subject to
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> A*x = b
lb <= x <= ub
A_lb <= A_in*x <= A_ub
</pre></div></div>
<a class="anchor" id="XREFqp"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-qp"><span><code class="def-type">[<var class="var">x</var>, <var class="var">obj</var>, <var class="var">info</var>, <var class="var">lambda</var>] =</code> <strong class="def-name">qp</strong> <code class="def-code-arguments">(<var class="var">x0</var>, <var class="var">H</var>)</code><a class="copiable-link" href="#index-qp"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-qp-1"><span><code class="def-type">[<var class="var">x</var>, <var class="var">obj</var>, <var class="var">info</var>, <var class="var">lambda</var>] =</code> <strong class="def-name">qp</strong> <code class="def-code-arguments">(<var class="var">x0</var>, <var class="var">H</var>, <var class="var">q</var>)</code><a class="copiable-link" href="#index-qp-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-qp-2"><span><code class="def-type">[<var class="var">x</var>, <var class="var">obj</var>, <var class="var">info</var>, <var class="var">lambda</var>] =</code> <strong class="def-name">qp</strong> <code class="def-code-arguments">(<var class="var">x0</var>, <var class="var">H</var>, <var class="var">q</var>, <var class="var">A</var>, <var class="var">b</var>)</code><a class="copiable-link" href="#index-qp-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-qp-3"><span><code class="def-type">[<var class="var">x</var>, <var class="var">obj</var>, <var class="var">info</var>, <var class="var">lambda</var>] =</code> <strong class="def-name">qp</strong> <code class="def-code-arguments">(<var class="var">x0</var>, <var class="var">H</var>, <var class="var">q</var>, <var class="var">A</var>, <var class="var">b</var>, <var class="var">lb</var>, <var class="var">ub</var>)</code><a class="copiable-link" href="#index-qp-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-qp-4"><span><code class="def-type">[<var class="var">x</var>, <var class="var">obj</var>, <var class="var">info</var>, <var class="var">lambda</var>] =</code> <strong class="def-name">qp</strong> <code class="def-code-arguments">(<var class="var">x0</var>, <var class="var">H</var>, <var class="var">q</var>, <var class="var">A</var>, <var class="var">b</var>, <var class="var">lb</var>, <var class="var">ub</var>, <var class="var">A_lb</var>, <var class="var">A_in</var>, <var class="var">A_ub</var>)</code><a class="copiable-link" href="#index-qp-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-qp-5"><span><code class="def-type">[<var class="var">x</var>, <var class="var">obj</var>, <var class="var">info</var>, <var class="var">lambda</var>] =</code> <strong class="def-name">qp</strong> <code class="def-code-arguments">(…, <var class="var">options</var>)</code><a class="copiable-link" href="#index-qp-5"> ¶</a></span></dt>
<dd><p>Solve a quadratic program (QP).
</p>
<p>Solve the quadratic program defined by
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">min 0.5 x'*H*x + x'*q
x
</pre></div></div>
<p>subject to
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">A*x = b
lb <= x <= ub
A_lb <= A_in*x <= A_ub
</pre></div></div>
<p>using a null-space active-set method.
</p>
<p>Any bound (<var class="var">A</var>, <var class="var">b</var>, <var class="var">lb</var>, <var class="var">ub</var>, <var class="var">A_in</var>, <var class="var">A_lb</var>,
<var class="var">A_ub</var>) may be set to the empty matrix (<code class="code">[]</code>) if not present. The
constraints <var class="var">A</var> and <var class="var">A_in</var> are matrices with each row representing
a single constraint. The other bounds are scalars or vectors depending on
the number of constraints. The algorithm is faster if the initial guess is
feasible.
</p>
<p><var class="var">options</var> is a structure specifying additional parameters which
control the algorithm. Currently, <code class="code">qp</code> recognizes these options:
<code class="code">"MaxIter"</code>, <code class="code">"TolX"</code>.
</p>
<p><code class="code">"MaxIter"</code> proscribes the maximum number of algorithm iterations
before optimization is halted. The default value is 200.
The value must be a positive integer.
</p>
<p><code class="code">"TolX"</code> specifies the termination tolerance for the unknown variables
<var class="var">x</var>. The default is <code class="code">sqrt (eps)</code> or approximately 1e-8.
</p>
<p>On return, <var class="var">x</var> is the location of the minimum and <var class="var">fval</var> contains
the value of the objective function at <var class="var">x</var>.
</p>
<dl class="table">
<dt><var class="var">info</var></dt>
<dd><p>Structure containing run-time information about the algorithm. The
following fields are defined:
</p>
<dl class="table">
<dt><code class="code">solveiter</code></dt>
<dd><p>The number of iterations required to find the solution.
</p>
</dd>
<dt><code class="code">info</code></dt>
<dd><p>An integer indicating the status of the solution.
</p>
<dl class="table">
<dt>0</dt>
<dd><p>The problem is feasible and convex. Global solution found.
</p>
</dd>
<dt>1</dt>
<dd><p>The problem is not convex. Local solution found.
</p>
</dd>
<dt>2</dt>
<dd><p>The problem is not convex and unbounded.
</p>
</dd>
<dt>3</dt>
<dd><p>Maximum number of iterations reached.
</p>
</dd>
<dt>6</dt>
<dd><p>The problem is infeasible.
</p></dd>
</dl>
</dd>
</dl>
</dd>
</dl>
<p><strong class="strong">See also:</strong> <a class="ref" href="Nonlinear-Programming.html#XREFsqp">sqp</a>.
</p></dd></dl>
<a class="anchor" id="XREFpqpnonneg"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-pqpnonneg"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">pqpnonneg</strong> <code class="def-code-arguments">(<var class="var">c</var>, <var class="var">d</var>)</code><a class="copiable-link" href="#index-pqpnonneg"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-pqpnonneg-1"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">pqpnonneg</strong> <code class="def-code-arguments">(<var class="var">c</var>, <var class="var">d</var>, <var class="var">x0</var>)</code><a class="copiable-link" href="#index-pqpnonneg-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-pqpnonneg-2"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">pqpnonneg</strong> <code class="def-code-arguments">(<var class="var">c</var>, <var class="var">d</var>, <var class="var">x0</var>, <var class="var">options</var>)</code><a class="copiable-link" href="#index-pqpnonneg-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-pqpnonneg-3"><span><code class="def-type">[<var class="var">x</var>, <var class="var">minval</var>] =</code> <strong class="def-name">pqpnonneg</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-pqpnonneg-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-pqpnonneg-4"><span><code class="def-type">[<var class="var">x</var>, <var class="var">minval</var>, <var class="var">exitflag</var>] =</code> <strong class="def-name">pqpnonneg</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-pqpnonneg-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-pqpnonneg-5"><span><code class="def-type">[<var class="var">x</var>, <var class="var">minval</var>, <var class="var">exitflag</var>, <var class="var">output</var>] =</code> <strong class="def-name">pqpnonneg</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-pqpnonneg-5"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-pqpnonneg-6"><span><code class="def-type">[<var class="var">x</var>, <var class="var">minval</var>, <var class="var">exitflag</var>, <var class="var">output</var>, <var class="var">lambda</var>] =</code> <strong class="def-name">pqpnonneg</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-pqpnonneg-6"> ¶</a></span></dt>
<dd>
<p>Minimize <code class="code"> (1/2 * <var class="var">x</var>' * <var class="var">c</var> * <var class="var">x</var> + <var class="var">d</var>' * <var class="var">x</var>) </code>
subject to <code class="code"><var class="var">x</var> >= 0</code>.
</p>
<p><var class="var">c</var> and <var class="var">d</var> must be real matrices, and <var class="var">c</var> must be symmetric and
positive definite.
</p>
<p><var class="var">x0</var> is an optional initial guess for the solution <var class="var">x</var>.
</p>
<p><var class="var">options</var> is an options structure to change the behavior of the
algorithm (see <a class="pxref" href="Linear-Least-Squares.html#XREFoptimset"><code class="code">optimset</code></a>). <code class="code">pqpnonneg</code>
recognizes one option: <code class="code">"MaxIter"</code>.
</p>
<p>Outputs:
</p>
<dl class="table">
<dt><var class="var">x</var></dt>
<dd><p>The solution matrix
</p>
</dd>
<dt><var class="var">minval</var></dt>
<dd><p>The minimum attained model value,
<code class="code">1/2*<var class="var">xmin</var>'*<var class="var">c</var>*<var class="var">xmin</var> + <var class="var">d</var>'*<var class="var">xmin</var></code>
</p>
</dd>
<dt><var class="var">exitflag</var></dt>
<dd><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.)
</p>
</dd>
<dt><var class="var">output</var></dt>
<dd><p>A structure with two fields:
</p>
<ul class="itemize mark-bullet">
<li><code class="code">"algorithm"</code>: The algorithm used (<code class="code">"nnls"</code>)
</li><li><code class="code">"iterations"</code>: The number of iterations taken.
</li></ul>
</dd>
<dt><var class="var">lambda</var></dt>
<dd><p>Lagrange multipliers. If these are nonzero, the corresponding <var class="var">x</var>
values should be zero, indicating the solution is pressed up against a
coordinate plane. The magnitude indicates how much the residual would
improve if the <code class="code"><var class="var">x</var> >= 0</code> constraints were relaxed in that
direction.
</p>
</dd>
</dl>
<p><strong class="strong">See also:</strong> <a class="ref" href="Linear-Least-Squares.html#XREFlsqnonneg">lsqnonneg</a>, <a class="ref" href="#XREFqp">qp</a>, <a class="ref" href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
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