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Interface to Constrained Optimization By Linear Approximation
.. autosummary::
:toctree: generated/
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy._lib.six import callable
from scipy.optimize import _cobyla
from .optimize import OptimizeResult, _check_unknown_options
from itertools import izip
except ImportError:
izip = zip
__all__ = ['fmin_cobyla']
def fmin_cobyla(func, x0, cons, args=(), consargs=None, rhobeg=1.0,
rhoend=1e-4, iprint=1, maxfun=1000, disp=None, catol=2e-4):
Minimize a function using the Constrained Optimization BY Linear
Approximation (COBYLA) method. This method wraps a FORTRAN
implementation of the algorithm.
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, and it can return either a single number or
an array or list of numbers.
args : tuple, optional
Extra arguments to pass to function.
consargs : tuple, optional
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 : float, optional
Reasonable initial changes to the variables.
rhoend : float, optional
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}, optional
Controls the frequency of output; 0 implies no output. Deprecated.
disp : {0, 1, 2, 3}, optional
Over-rides the iprint interface. Preferred.
maxfun : int, optional
Maximum number of function evaluations.
catol : float, optional
Absolute tolerance for constraint violations.
x : ndarray
The argument that minimises `f`.
See also
minimize: Interface to minimization algorithms for multivariate
functions. See the 'COBYLA' `method` in particular.
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.
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
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]
>>> from scipy.optimize import fmin_cobyla
>>> fmin_cobyla(objective, [0.0, 0.1], [constr1, constr2], rhoend=1e-7)
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."
except TypeError:
if callable(cons):
cons = [cons]
raise TypeError(err)
for thisfunc in cons:
if not callable(thisfunc):
raise TypeError(err)
if consargs is None:
consargs = args
# build constraints
con = tuple({'type': 'ineq', 'fun': c, 'args': consargs} for c in cons)
# options
if disp is not None:
iprint = disp
opts = {'rhobeg': rhobeg,
'tol': rhoend,
'iprint': iprint,
'disp': iprint != 0,
'maxiter': maxfun,
'catol': catol}
sol = _minimize_cobyla(func, x0, args, constraints=con,
if iprint > 0 and not sol['success']:
print("COBYLA failed to find a solution: %s" % (sol.message,))
return sol['x']
def _minimize_cobyla(fun, x0, args=(), constraints=(),
rhobeg=1.0, tol=1e-4, iprint=1, maxiter=1000,
disp=False, catol=2e-4, **unknown_options):
Minimize a scalar function of one or more variables using the
Constrained Optimization BY Linear Approximation (COBYLA) algorithm.
rhobeg : float
Reasonable initial changes to the variables.
tol : float
Final accuracy in the optimization (not precisely guaranteed).
This is a lower bound on the size of the trust region.
disp : bool
Set to True to print convergence messages. If False,
`verbosity` is ignored as set to 0.
maxiter : int
Maximum number of function evaluations.
catol : float
Tolerance (absolute) for constraint violations
maxfun = maxiter
rhoend = tol
if not disp:
iprint = 0
# check constraints
if isinstance(constraints, dict):
constraints = (constraints, )
for ic, con in enumerate(constraints):
# check type
ctype = con['type'].lower()
except KeyError:
raise KeyError('Constraint %d has no type defined.' % ic)
except TypeError:
raise TypeError('Constraints must be defined using a '
except AttributeError:
raise TypeError("Constraint's type must be a string.")
if ctype != 'ineq':
raise ValueError("Constraints of type '%s' not handled by "
"COBYLA." % con['type'])
# check function
if 'fun' not in con:
raise KeyError('Constraint %d has no function defined.' % ic)
# check extra arguments
if 'args' not in con:
con['args'] = ()
# m is the total number of constraint values
# it takes into account that some constraints may be vector-valued
cons_lengths = []
for c in constraints:
f = c['fun'](x0, *c['args'])
cons_length = len(f)
except TypeError:
cons_length = 1
m = sum(cons_lengths)
def calcfc(x, con):
f = fun(x, *args)
i = 0
for size, c in izip(cons_lengths, constraints):
con[i: i + size] = c['fun'](x, *c['args'])
i += size
return f
info = np.zeros(4, np.float64)
xopt, info = _cobyla.minimize(calcfc, m=m, x=np.copy(x0), rhobeg=rhobeg,
rhoend=rhoend, iprint=iprint, maxfun=maxfun,
if info[3] > catol:
# Check constraint violation
info[0] = 4
return OptimizeResult(x=xopt,
success=info[0] == 1,
message={1: 'Optimization terminated successfully.',
2: 'Maximum number of function evaluations has '
'been exceeded.',
3: 'Rounding errors are becoming damaging in '
'COBYLA subroutine.',
4: 'Did not converge to a solution satisfying '
'the constraints. See `maxcv` for magnitude '
'of violation.'
}.get(info[0], 'Unknown exit status.'),
if __name__ == '__main__':
from math import sqrt
def fun(x):
return x[0] * x[1]
def cons(x):
return 1 - x[0]**2 - x[1]**2
x = fmin_cobyla(fun, [1., 1.], cons, iprint=3, disp=1)
print('\nTheoretical solution: %e, %e' % (1. / sqrt(2.), -1. / sqrt(2.)))