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<h1 class="chapter"> 43. augmented_lagrangian </h1>

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<h2 class="section"> 43.1 Definitions for augmented_lagrangian </h2>

<dl>
<dt><u>Function:</u> <b>augmented_lagrangian_method</b><i> (<var>FOM</var>, <var>xx</var>, <var>C</var>, <var>yy</var>)</i>
<a name="IDX1383"></a>
</dt>
<dt><u>Function:</u> <b>augmented_lagrangian_method</b><i> (<var>FOM</var>, <var>xx</var>, <var>C</var>, <var>yy</var>, optional_args)</i>
<a name="IDX1384"></a>
</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><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(augmented_lagrangian)</code> loads this function.
</p>
<p>See also <code>lbfgs</code>.
</p>
<p>References:
</p>
<p>[1] http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html
</p>
<p>[2] http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (lbfgs);
(%o1) /home/robert/tmp/maxima-release-branch/maxima/share/lbfgs/\
lbfgs.mac
(%i2) load (augmented_lagrangian);
(%o2) /home/robert/tmp/maxima-release-branch/maxima/share/contri\
b/augmented_lagrangian.mac
(%i3) FOM: x^2 + 2*y^2;
                               2    2
(%o3)                       2 y  + x
(%i4) xx: [x, y];
(%o4)                        [x, y]
(%i5) C: [x + y - 1];
(%o5)                      [y + x - 1]
(%i6) yy: [1, 1];
(%o6)                        [1, 1]
(%i7) augmented_lagrangian_method (FOM, xx, C, yy, iprint = [-1, 0]);
(%o7) [[x = 0.6478349888525, y = 0.32391749442625], 
                                 %lambda = [- 1.267422460983745]]
</pre></td></tr></table>
</dd></dl>


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