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37.1 Introduction to cobyla | ||
37.2 Functions and Variables for cobyla | ||
37.3 Examples for cobyla |
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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
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Returns an approximate minimum of the expression F with respect to the variables X, subject to an optional set of constraints. Y is a list of initial guesses for X.
F must be an ordinary expressions, not names of functions or lambda expressions.
optional_args
represents additional arguments, specified as
symbol = value
. The optional arguments recognized are:
constraints
List of inequality and equality constraints that must be satisfied by X.
The inequality constraints must be actual inequalities of the form
g(X) >= h(X)
or g(X) <= h(X)
. The
equality constraints must be of the form g(X) = h(X)
.
rhobeg
Initial value of the internal RHO variable which controls the size of simplex. (Defaults to 1.0)
rhoend
The desired final value rho parameter. It is approximately the accuracy in the variables. (Defaults to 1d-6.)
iprint
Verbose output level. (Defaults to 0)
maxfun
The maximum number of function evaluations. (Defaults to 1000).
On return, a vector is given:
var = value
for each of the
variables listed in X.
load(fmin_cobyla)
loads this function.
This function is identical to fmin_cobyla
, except that bigfloat
operations are used, and the default value for rhoend is
10^(fpprec/2)
.
See fmin_cobyla
for more information.
load(fmin_cobyla)
loads this function.
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Minimize x1*x2 with 1-x1^2-x2^2 >= 0. The theoretical solution is x1 = 1/sqrt(2), x2 = -1/sqrt(2).
(%i1) load(fmin_cobyla)$ (%i2) fmin_cobyla(x1*x2, [x1, x2], [1,1], constraints = [x1^2+x2^2<=1], iprint=1); Normal return from subroutine COBYLA NFVALS = 66 F =-5.000000E-01 MAXCV = 1.999956E-12 X = 7.071058E-01 -7.071077E-01 (%o2) [[x1 = .7071058493484819, x2 = - .7071077130247994], - .499999999999263
There are additional examples in the share/cobyla/ex directory.
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