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<a name="Functions-and-Variables-for-augmented_005flagrangian"></a>
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Previous: <a href="maxima_186.html#augmented_005flagrangian_002dpkg" accesskey="p" rel="previous">augmented_lagrangian-pkg</a>, Up: <a href="maxima_186.html#augmented_005flagrangian_002dpkg" accesskey="u" rel="up">augmented_lagrangian-pkg</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Functions-and-Variables-for-augmented_005flagrangian-1"></a>
<h3 class="section">41.1 Functions and Variables for augmented_lagrangian</h3>

<a name="augmented_005flagrangian_005fmethod"></a><a name="Item_003a-augmented_005flagrangian_002fdeffn_002faugmented_005flagrangian_005fmethod"></a><dl>
<dt><a name="index-augmented_005flagrangian_005fmethod"></a>Function: <strong>augmented_lagrangian_method</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>augmented_lagrangian_method</tt> (<var>FOM</var>, <var>xx</var>, <var>C</var>, <var>yy</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>augmented_lagrangian_method</tt> (<var>FOM</var>, <var>xx</var>, <var>C</var>, <var>yy</var>, optional_args) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>augmented_lagrangian_method</tt> ([<var>FOM</var>, <var>grad</var>], <var>xx</var>, <var>C</var>, <var>yy</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>augmented_lagrangian_method</tt> ([<var>FOM</var>, <var>grad</var>], <var>xx</var>, <var>C</var>, <var>yy</var>, optional_args)</em></dt>
<dd>
<p>Returns an approximate minimum of the expression <var>FOM</var>
with respect to the variables <var>xx</var>,
holding the constraints <var>C</var> equal to zero.
<var>yy</var> is a list of initial guesses for <var>xx</var>.
The method employed is the augmented Lagrangian method (see Refs [1] and [2]).
</p>
<p><var>grad</var>, if present, is the gradient of <var>FOM</var> with respect to <var>xx</var>,
represented as a list of expressions,
one for each variable in <var>xx</var>.
If not present, the gradient is constructed automatically.
</p>
<p><var>FOM</var> and each element of <var>grad</var>, if present,
must be ordinary expressions, not names of functions or lambda expressions.
</p>
<p><code>optional_args</code> represents additional arguments,
specified as <code><var>symbol</var> = <var>value</var></code>.
The optional arguments recognized are:
</p>
<dl compact="compact">
<dt><code>niter</code></dt>
<dd><p>Number of iterations of the augmented Lagrangian algorithm
</p></dd>
<dt><code>lbfgs_tolerance</code></dt>
<dd><p>Tolerance supplied to LBFGS
</p></dd>
<dt><code>iprint</code></dt>
<dd><p>IPRINT parameter (a list of two integers which controls verbosity) supplied to LBFGS
</p></dd>
<dt><code>%lambda</code></dt>
<dd><p>Initial value of <code>%lambda</code> to be used for calculating the augmented Lagrangian
</p></dd>
</dl>

<p>This implementation minimizes the augmented Lagrangian by
applying the limited-memory BFGS (LBFGS) algorithm,
which is a quasi-Newton algorithm.
</p>
<p><code>load(&quot;augmented_lagrangian&quot;)</code> loads this function.
</p>
<p>See also <a href="maxima_302.html#lbfgs_002dpkg">lbfgs-pkg</a>
</p>
<p>References:
</p>
<p>[1] <a href="http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html">http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html</a>
</p>
<p>[2] <a href="http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf">http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf</a>
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) load (&quot;lbfgs&quot;);
(%o1)  /home/gunter/src/maxima-code/share/lbfgs/lbfgs.mac
</pre><pre class="example">(%i2) load (&quot;augmented_lagrangian&quot;);
(%o2) /home/gunter/src/maxima-code/share/contrib/augmented_lagra\
ngian.mac
</pre><pre class="example">(%i3) FOM: x^2 + 2*y^2;
                               2    2
(%o3)                       2 y  + x
</pre><pre class="example">(%i4) xx: [x, y];
(%o4)                        [x, y]
</pre><pre class="example">(%i5) C: [x + y - 1];
(%o5)                      [y + x - 1]
</pre><pre class="example">(%i6) yy: [1, 1];
(%o6)                        [1, 1]
</pre><pre class="example">(%i7) augmented_lagrangian_method(FOM, xx, C, yy, iprint=[-1,0]);
(%o7) [[x = 0.666659841080023, y = 0.333340272455448], 
                                 %lambda = [- 1.333337940892518]]
</pre></div>

<p>Same example as before, but this time the gradient is supplied as an argument.
</p>
<div class="example">
<pre class="example">(%i1) load (&quot;lbfgs&quot;)$
(%i2) load (&quot;augmented_lagrangian&quot;)$
</pre><pre class="example">(%i3) FOM: x^2 + 2*y^2;
                               2    2
(%o3)                       2 y  + x
</pre><pre class="example">(%i4) xx: [x, y];
(%o4)                        [x, y]
</pre><pre class="example">(%i5) grad : [2*x, 4*y];
(%o5)                      [2 x, 4 y]
</pre><pre class="example">(%i6) C: [x + y - 1];
(%o6)                      [y + x - 1]
</pre><pre class="example">(%i7) yy: [1, 1];
(%o7)                        [1, 1]
</pre><pre class="example">(%i8) augmented_lagrangian_method ([FOM, grad], xx, C, yy,
                             iprint = [-1, 0]);
(%o8) [[x = 0.6666598410800247, y = 0.3333402724554464], 
                                 %lambda = [- 1.333337940892525]]
</pre></div>

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