(maxima.info)Functions and Variables for augmented_lagrangian

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41.1 Functions and Variables for augmented_lagrangian
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 -- Function: augmented_lagrangian_method
          augmented_lagrangian_method (<FOM>, <xx>, <C>, <yy>)
          augmented_lagrangian_method (<FOM>, <xx>, <C>, <yy>,
          optional_args)
          augmented_lagrangian_method ([<FOM>, <grad>], <xx>, <C>, <yy>)

          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 Note: lbfgs-pkg

     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)  /home/gunter/src/maxima-code/share/lbfgs/lbfgs.mac
          (%i2) load ("augmented_lagrangian");
          (%o2) /home/gunter/src/maxima-code/share/contrib/augmented_lagra\
          ngian.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 = 0.666659841080023, y = 0.333340272455448],
                                           %lambda = [- 1.333337940892518]]

     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) xx: [x, y];
          (%o4)                        [x, y]
          (%i5) grad : [2*x, 4*y];
          (%o5)                      [2 x, 4 y]
          (%i6) C: [x + y - 1];
          (%o6)                      [y + x - 1]
          (%i7) yy: [1, 1];
          (%o7)                        [1, 1]
          (%i8) augmented_lagrangian_method ([FOM, grad], xx, C, yy,
                                       iprint = [-1, 0]);
          (%o8) [[x = 0.6666598410800247, y = 0.3333402724554464],
                                           %lambda = [- 1.333337940892525]]