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"""Interface to Constrained Optimization By Linear Approximation
Functions:
fmin_coblya(func, x0, cons, args=(), consargs=None, rhobeg=1.0, rhoend=1e-4,
iprint=1, maxfun=1000)
Minimize a function using the Constrained Optimization BY Linear
Approximation (COBYLA) method
"""
import _cobyla
from numpy import copy
__all__ = ['fmin_cobyla']
def fmin_cobyla(func, x0, cons, args=(), consargs=None, rhobeg=1.0, rhoend=1e-4,
iprint=1, maxfun=1000, disp=None):
"""
Minimize a function using the Constrained Optimization BY Linear
Approximation (COBYLA) method. This method wraps a FORTRAN
implentation of the algorithm.
Parameters
----------
func : callable
Function to minimize. In the form func(x, \\*args).
x0 : ndarray
Initial guess.
cons : sequence
Constraint functions; must all be ``>=0`` (a single function
if only 1 constraint). Each function takes the parameters `x`
as its first argument.
args : tuple
Extra arguments to pass to function.
consargs : tuple
Extra arguments to pass to constraint functions (default of None means
use same extra arguments as those passed to func).
Use ``()`` for no extra arguments.
rhobeg :
Reasonable initial changes to the variables.
rhoend :
Final accuracy in the optimization (not precisely guaranteed). This
is a lower bound on the size of the trust region.
iprint : {0, 1, 2, 3}
Controls the frequency of output; 0 implies no output. Deprecated.
disp : {0, 1, 2, 3}
Over-rides the iprint interface. Preferred.
maxfun : int
Maximum number of function evaluations.
Returns
-------
x : ndarray
The argument that minimises `f`.
Notes
-----
This algorithm is based on linear approximations to the objective
function and each constraint. We briefly describe the algorithm.
Suppose the function is being minimized over k variables. At the
jth iteration the algorithm has k+1 points v_1, ..., v_(k+1),
an approximate solution x_j, and a radius RHO_j.
(i.e. linear plus a constant) approximations to the objective
function and constraint functions such that their function values
agree with the linear approximation on the k+1 points v_1,.., v_(k+1).
This gives a linear program to solve (where the linear approximations
of the constraint functions are constrained to be non-negative).
However the linear approximations are likely only good
approximations near the current simplex, so the linear program is
given the further requirement that the solution, which
will become x_(j+1), must be within RHO_j from x_j. RHO_j only
decreases, never increases. The initial RHO_j is rhobeg and the
final RHO_j is rhoend. In this way COBYLA's iterations behave
like a trust region algorithm.
Additionally, the linear program may be inconsistent, or the
approximation may give poor improvement. For details about
how these issues are resolved, as well as how the points v_i are
updated, refer to the source code or the references below.
References
----------
Powell M.J.D. (1994), "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), pp. 51-67
Powell M.J.D. (1998), "Direct search algorithms for optimization
calculations", Acta Numerica 7, 287-336
Powell M.J.D. (2007), "A view of algorithms for optimization without
derivatives", Cambridge University Technical Report DAMTP 2007/NA03
Examples
--------
Minimize the objective function f(x,y) = x*y subject
to the constraints x**2 + y**2 < 1 and y > 0::
>>> def objective(x):
... return x[0]*x[1]
...
>>> def constr1(x):
... return 1 - (x[0]**2 + x[1]**2)
...
>>> def constr2(x):
... return x[1]
...
>>> fmin_cobyla(objective, [0.0, 0.1], [constr1, constr2], rhoend=1e-7)
Normal return from subroutine COBYLA
NFVALS = 64 F =-5.000000E-01 MAXCV = 1.998401E-14
X =-7.071069E-01 7.071067E-01
array([-0.70710685, 0.70710671])
The exact solution is (-sqrt(2)/2, sqrt(2)/2).
"""
err = "cons must be a sequence of callable functions or a single"\
" callable function."
try:
m = len(cons)
except TypeError:
if callable(cons):
m = 1
cons = [cons]
else:
raise TypeError(err)
else:
for thisfunc in cons:
if not callable(thisfunc):
raise TypeError(err)
if consargs is None:
consargs = args
if disp is not None:
iprint = disp
def calcfc(x, con):
f = func(x, *args)
k = 0
for constraints in cons:
con[k] = constraints(x, *consargs)
k += 1
return f
xopt = _cobyla.minimize(calcfc, m=m, x=copy(x0), rhobeg=rhobeg,
rhoend=rhoend, iprint=iprint, maxfun=maxfun)
return xopt
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