(maxima.info)Introduction to cobyla

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47.1 Introduction to cobyla
===========================

'fmin_cobyla' is a Common Lisp translation (via 'f2cl') of the Fortran
constrained optimization routine COBYLA by Powell[1][2][3].

   COBYLA minimizes an objective function F(X) subject to M inequality
constraints of the form g(X) >= 0 on X, where X is a vector of variables
that has N components.

   Equality constraints g(X)=0 can often be implemented by a pair of
inequality constraints g(X)>=0 and -g(X)>= 0.  Maxima's interface to
COBYLA allows equality constraints and internally converts the equality
constraints to a pair of inequality constraints.

   The algorithm employs linear approximations to the objective and
constraint functions, the approximations being formed by linear
interpolation at N+1 points in the space of the variables.  The
interpolation points are regarded as vertices of a simplex.  The
parameter RHO controls the size of the simplex and it is reduced
automatically from RHOBEG to RHOEND. For each RHO the subroutine tries
to achieve a good vector of variables for the current size, and then RHO
is reduced until the value RHOEND is reached.  Therefore RHOBEG and
RHOEND should be set to reasonable initial changes to and the required
accuracy in the variables respectively, but this accuracy should be
viewed as a subject for experimentation because it is not guaranteed.
The routine treats each constraint individually when calculating a
change to the variables, rather than lumping the constraints together
into a single penalty function.  The name of the subroutine is derived
from the phrase Constrained Optimization BY Linear Approximations.

   References:

   [1] Fortran Code is from
<http://plato.asu.edu/sub/nlores.html#general>

   [2] M. J. D. Powell, "A direct search optimization method that models
the objective and constraint functions by linear interpolation," in
Advances in Optimization and Numerical Analysis, eds.  S. Gomez and
J.-P. Hennart (Kluwer Academic: Dordrecht, 1994), p.  51-67.

   [3] M. J. D. Powell, "Direct search algorithms for optimization
calculations," Acta Numerica 7, 287-336 (1998).  Also available as
University of Cambridge, Department of Applied Mathematics and
Theoretical Physics, Numerical Analysis Group, Report NA1998/04 from
<http://www.damtp.cam.ac.uk/user/na/reports.html>