34. augmented_lagrangian

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34.1 Functions and Variables for augmented_lagrangian

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

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

Function: augmented_lagrangian_method ([FOM, grad], xx, C, yy)

Function: augmented_lagrangian_method ([FOM, grad], 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]).

grad, if present, is the gradient of FOM with respect to xx, represented as a list of expressions, one for each variable in xx. If not present, the gradient is constructed automatically.

FOM and each element of grad, if present, must be ordinary expressions, not names of functions or lambda expressions.

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

Examples:

(%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]]

Same example as before, but this time the gradient is supplied as an argument.

(%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]]