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  • Committer: Bazaar Package Importer
  • Author(s): Camm Maguire
  • Date: 2006-10-18 14:52:42 UTC
  • mto: (1.1.5 upstream)
  • mto: This revision was merged to the branch mainline in revision 4.
  • Revision ID: james.westby@ubuntu.com-20061018145242-vzyrm5hmxr8kiosf
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doc/info/augmented_lagrangian.texi
 
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@menu
 
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* Definitions for augmented_lagrangian::
 
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@end menu
 
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@node Definitions for augmented_lagrangian,  , augmented_lagrangian, augmented_lagrangian
 
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@section Definitions for augmented_lagrangian
 
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@deffn {Function} augmented_lagrangian_method (@var{FOM}, @var{xx}, @var{C}, @var{yy})
 
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@deffnx {Function} augmented_lagrangian_method (@var{FOM}, @var{xx}, @var{C}, @var{yy}, optional_args)
 
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Returns an approximate minimum of the expression @var{FOM}
 
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with respect to the variables @var{xx},
 
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holding the constraints @var{C} equal to zero.
 
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@var{yy} is a list of initial guesses for @var{xx}.
 
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The method employed is the augmented Lagrangian method (see Refs [1] and [2]).
 
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@code{optional_args} represents additional arguments,
 
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specified as @code{@var{symbol} = @var{value}}.
 
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The optional arguments recognized are:
 
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@table @code
 
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@item niter
 
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Number of iterations of the augmented Lagrangian algorithm
 
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@item lbfgs_tolerance
 
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Tolerance supplied to LBFGS
 
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@item iprint
 
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IPRINT parameter (a list of two integers which controls verbosity) supplied to LBFGS
 
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@item %lambda
 
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Initial value of @code{%lambda} to be used for calculating the augmented Lagrangian
 
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@end table
 
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This implementation minimizes the augmented Lagrangian by
 
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applying the limited-memory BFGS (LBFGS) algorithm,
 
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which is a quasi-Newton algorithm.
 
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@code{load(augmented_lagrangian)} loads this function.
 
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See also @code{lbfgs}.
 
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References:
 
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[1] http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html
 
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[2] http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf
 
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Example:
 
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@c ===beg===
 
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@c load (lbfgs);
 
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@c load (augmented_lagrangian);
 
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@c FOM: x^2 + 2*y^2;
 
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@c xx: [x, y];
 
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@c C: [x + y - 1];
 
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@c yy: [1, 1];
 
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@c augmented_lagrangian_method (FOM, xx, C, yy, iprint = [-1, 0]);
 
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@c ===end===
 
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@example
 
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(%i1) load (lbfgs);
 
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(%o1) /home/robert/tmp/maxima-release-branch/maxima/share/lbfgs/\
 
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lbfgs.mac
 
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(%i2) load (augmented_lagrangian);
 
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(%o2) /home/robert/tmp/maxima-release-branch/maxima/share/contri\
 
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b/augmented_lagrangian.mac
 
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(%i3) FOM: x^2 + 2*y^2;
 
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                               2    2
 
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(%o3)                       2 y  + x
 
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(%i4) xx: [x, y];
 
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(%o4)                        [x, y]
 
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(%i5) C: [x + y - 1];
 
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(%o5)                      [y + x - 1]
 
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(%i6) yy: [1, 1];
 
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(%o6)                        [1, 1]
 
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(%i7) augmented_lagrangian_method (FOM, xx, C, yy, iprint = [-1, 0]);
 
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(%o7) [[x = 0.6478349888525, y = 0.32391749442625], 
 
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                                 %lambda = [- 1.267422460983745]]
 
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@end example
 
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@end deffn
 
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