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<h1 class="chapter"> 42. augmented_lagrangian </h1>
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<tr><td align="left" valign="top"><a href="#SEC188">42.1 Functions and Variables for augmented_lagrangian</a></td><td> </td><td align="left" valign="top">
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<h2 class="section"> 42.1 Functions and Variables for augmented_lagrangian </h2>
<p><a name="Item_003a-augmented_005flagrangian_005fmethod"></a>
</p><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="IDX1501"></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="IDX1502"></a>
</dt>
<dt><u>Function:</u> <b>augmented_lagrangian_method</b><i> ([<var>FOM</var>, <var>grad</var>], <var>xx</var>, <var>C</var>, <var>yy</var>)</i>
<a name="IDX1503"></a>
</dt>
<dt><u>Function:</u> <b>augmented_lagrangian_method</b><i> ([<var>FOM</var>, <var>grad</var>], <var>xx</var>, <var>C</var>, <var>yy</var>, optional_args)</i>
<a name="IDX1504"></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><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(augmented_lagrangian)</code> loads this function.
</p>
<p>See also <code>lbfgs</code>.
</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>
<pre class="example">(%i1) load (lbfgs);
(%o1) /maxima/share/lbfgs/lbfgs.mac
(%i2) load (augmented_lagrangian);
(%o2)
/maxima/share/contrib/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 = 0.66665984108002, y = 0.33334027245545],
%lambda = [- 1.333337940892525]]
</pre>
<p>Same example as before, but this time the gradient is supplied as an argument.
</p>
<pre class="example">(%i1) load (lbfgs)$
(%i2) load (augmented_lagrangian)$
(%i3) FOM: x^2 + 2*y^2;
2 2
(%o3) 2 y + x
(%i4) FOM: x^2 + 2*y^2;
2 2
(%o4) 2 y + x
(%i5) xx: [x, y];
(%o5) [x, y]
(%i6) grad : [2*x, 4*y];
(%o6) [2 x, 4 y]
(%i7) C: [x + y - 1];
(%o7) [y + x - 1]
(%i8) yy: [1, 1];
(%o8) [1, 1]
(%i9) augmented_lagrangian_method ([FOM, grad], xx, C, yy, iprint = [-1, 0]);
(%o9) [[x = 0.666659841080025, y = .3333402724554462],
%lambda = [- 1.333337940892543]]
</pre>
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