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42. augmented_lagrangian


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42.1 Functions and Variables for augmented_lagrangian

Function: augmented_lagrangian_method (FOM, xx, C, yy)
Function: augmented_lagrangian_method (FOM, xx, C, yy, optional_args)

Returns an approximate minimum of the expression FOM with respect to the variables xx, holding the constraints C equal to zero. yy is a list of initial guesses for xx. The method employed is the augmented Lagrangian method (see Refs [1] and [2]).

optional_args represents additional arguments, specified as symbol = value. The optional arguments recognized are:

niter

Number of iterations of the augmented Lagrangian algorithm

lbfgs_tolerance

Tolerance supplied to LBFGS

iprint

IPRINT parameter (a list of two integers which controls verbosity) supplied to LBFGS

%lambda

Initial value of %lambda to be used for calculating the augmented Lagrangian

This implementation minimizes the augmented Lagrangian by applying the limited-memory BFGS (LBFGS) algorithm, which is a quasi-Newton algorithm.

load(augmented_lagrangian) loads this function.

See also lbfgs.

References:

[1] http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html

[2] http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf

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]]

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