274 lines
9.2 KiB
Python
274 lines
9.2 KiB
Python
|
"""
|
||
|
|
Interface to Constrained Optimization By Linear Approximation
|
||
|
|
|
||
|
|
Functions
|
||
|
|
---------
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
fmin_cobyla
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
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
|
||
|
|
try:
|
||
|
|
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, 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.
|
||
|
|
|
||
|
|
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, 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.
|
||
|
|
disp : {0, 1, 2, 3}, optional
|
||
|
|
Controls the frequency of output; 0 implies no output.
|
||
|
|
maxfun : int, optional
|
||
|
|
Maximum number of function evaluations.
|
||
|
|
catol : float, optional
|
||
|
|
Absolute tolerance for constraint violations.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
x : ndarray
|
||
|
|
The argument that minimises `f`.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
minimize: Interface to minimization algorithms for multivariate
|
||
|
|
functions. See the 'COBYLA' `method` in particular.
|
||
|
|
|
||
|
|
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]
|
||
|
|
...
|
||
|
|
>>> 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."
|
||
|
|
try:
|
||
|
|
len(cons)
|
||
|
|
except TypeError:
|
||
|
|
if callable(cons):
|
||
|
|
cons = [cons]
|
||
|
|
else:
|
||
|
|
raise TypeError(err)
|
||
|
|
else:
|
||
|
|
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
|
||
|
|
opts = {'rhobeg': rhobeg,
|
||
|
|
'tol': rhoend,
|
||
|
|
'disp': disp,
|
||
|
|
'maxiter': maxfun,
|
||
|
|
'catol': catol}
|
||
|
|
|
||
|
|
sol = _minimize_cobyla(func, x0, args, constraints=con,
|
||
|
|
**opts)
|
||
|
|
if disp 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, 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.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
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
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
maxfun = maxiter
|
||
|
|
rhoend = tol
|
||
|
|
iprint = int(bool(disp))
|
||
|
|
|
||
|
|
# check constraints
|
||
|
|
if isinstance(constraints, dict):
|
||
|
|
constraints = (constraints, )
|
||
|
|
|
||
|
|
for ic, con in enumerate(constraints):
|
||
|
|
# check type
|
||
|
|
try:
|
||
|
|
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 '
|
||
|
|
'dictionary.')
|
||
|
|
except AttributeError:
|
||
|
|
raise TypeError("Constraint's type must be a string.")
|
||
|
|
else:
|
||
|
|
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'])
|
||
|
|
try:
|
||
|
|
cons_length = len(f)
|
||
|
|
except TypeError:
|
||
|
|
cons_length = 1
|
||
|
|
cons_lengths.append(cons_length)
|
||
|
|
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,
|
||
|
|
dinfo=info)
|
||
|
|
|
||
|
|
if info[3] > catol:
|
||
|
|
# Check constraint violation
|
||
|
|
info[0] = 4
|
||
|
|
|
||
|
|
return OptimizeResult(x=xopt,
|
||
|
|
status=int(info[0]),
|
||
|
|
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.',
|
||
|
|
5: 'NaN result encountered.'
|
||
|
|
}.get(info[0], 'Unknown exit status.'),
|
||
|
|
nfev=int(info[1]),
|
||
|
|
fun=info[2],
|
||
|
|
maxcv=info[3])
|