274 lines
9.2 KiB
Python
274 lines
9.2 KiB
Python
"""
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Interface to Constrained Optimization By Linear Approximation
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Functions
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---------
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.. autosummary::
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:toctree: generated/
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fmin_cobyla
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"""
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from __future__ import division, print_function, absolute_import
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import numpy as np
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from scipy._lib.six import callable
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from scipy.optimize import _cobyla
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from .optimize import OptimizeResult, _check_unknown_options
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try:
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from itertools import izip
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except ImportError:
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izip = zip
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__all__ = ['fmin_cobyla']
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def fmin_cobyla(func, x0, cons, args=(), consargs=None, rhobeg=1.0,
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rhoend=1e-4, maxfun=1000, disp=None, catol=2e-4):
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"""
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Minimize a function using the Constrained Optimization BY Linear
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Approximation (COBYLA) method. This method wraps a FORTRAN
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implementation of the algorithm.
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Parameters
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----------
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func : callable
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Function to minimize. In the form func(x, \\*args).
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x0 : ndarray
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Initial guess.
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cons : sequence
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Constraint functions; must all be ``>=0`` (a single function
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if only 1 constraint). Each function takes the parameters `x`
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as its first argument, and it can return either a single number or
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an array or list of numbers.
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args : tuple, optional
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Extra arguments to pass to function.
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consargs : tuple, optional
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Extra arguments to pass to constraint functions (default of None means
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use same extra arguments as those passed to func).
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Use ``()`` for no extra arguments.
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rhobeg : float, optional
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Reasonable initial changes to the variables.
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rhoend : float, optional
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Final accuracy in the optimization (not precisely guaranteed). This
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is a lower bound on the size of the trust region.
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disp : {0, 1, 2, 3}, optional
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Controls the frequency of output; 0 implies no output.
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maxfun : int, optional
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Maximum number of function evaluations.
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catol : float, optional
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Absolute tolerance for constraint violations.
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Returns
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-------
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x : ndarray
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The argument that minimises `f`.
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See also
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--------
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minimize: Interface to minimization algorithms for multivariate
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functions. See the 'COBYLA' `method` in particular.
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Notes
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-----
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This algorithm is based on linear approximations to the objective
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function and each constraint. We briefly describe the algorithm.
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Suppose the function is being minimized over k variables. At the
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jth iteration the algorithm has k+1 points v_1, ..., v_(k+1),
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an approximate solution x_j, and a radius RHO_j.
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(i.e. linear plus a constant) approximations to the objective
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function and constraint functions such that their function values
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agree with the linear approximation on the k+1 points v_1,.., v_(k+1).
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This gives a linear program to solve (where the linear approximations
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of the constraint functions are constrained to be non-negative).
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However the linear approximations are likely only good
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approximations near the current simplex, so the linear program is
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given the further requirement that the solution, which
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will become x_(j+1), must be within RHO_j from x_j. RHO_j only
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decreases, never increases. The initial RHO_j is rhobeg and the
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final RHO_j is rhoend. In this way COBYLA's iterations behave
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like a trust region algorithm.
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Additionally, the linear program may be inconsistent, or the
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approximation may give poor improvement. For details about
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how these issues are resolved, as well as how the points v_i are
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updated, refer to the source code or the references below.
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References
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----------
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Powell M.J.D. (1994), "A direct search optimization method that models
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the objective and constraint functions by linear interpolation.", in
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Advances in Optimization and Numerical Analysis, eds. S. Gomez and
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J-P Hennart, Kluwer Academic (Dordrecht), pp. 51-67
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Powell M.J.D. (1998), "Direct search algorithms for optimization
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calculations", Acta Numerica 7, 287-336
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Powell M.J.D. (2007), "A view of algorithms for optimization without
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derivatives", Cambridge University Technical Report DAMTP 2007/NA03
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Examples
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--------
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Minimize the objective function f(x,y) = x*y subject
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to the constraints x**2 + y**2 < 1 and y > 0::
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>>> def objective(x):
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... return x[0]*x[1]
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...
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>>> def constr1(x):
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... return 1 - (x[0]**2 + x[1]**2)
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...
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>>> def constr2(x):
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... return x[1]
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...
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>>> from scipy.optimize import fmin_cobyla
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>>> fmin_cobyla(objective, [0.0, 0.1], [constr1, constr2], rhoend=1e-7)
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array([-0.70710685, 0.70710671])
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The exact solution is (-sqrt(2)/2, sqrt(2)/2).
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"""
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err = "cons must be a sequence of callable functions or a single"\
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" callable function."
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try:
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len(cons)
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except TypeError:
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if callable(cons):
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cons = [cons]
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else:
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raise TypeError(err)
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else:
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for thisfunc in cons:
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if not callable(thisfunc):
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raise TypeError(err)
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if consargs is None:
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consargs = args
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# build constraints
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con = tuple({'type': 'ineq', 'fun': c, 'args': consargs} for c in cons)
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# options
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opts = {'rhobeg': rhobeg,
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'tol': rhoend,
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'disp': disp,
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'maxiter': maxfun,
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'catol': catol}
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sol = _minimize_cobyla(func, x0, args, constraints=con,
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**opts)
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if disp and not sol['success']:
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print("COBYLA failed to find a solution: %s" % (sol.message,))
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return sol['x']
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def _minimize_cobyla(fun, x0, args=(), constraints=(),
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rhobeg=1.0, tol=1e-4, maxiter=1000,
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disp=False, catol=2e-4, **unknown_options):
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"""
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Minimize a scalar function of one or more variables using the
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Constrained Optimization BY Linear Approximation (COBYLA) algorithm.
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Options
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-------
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rhobeg : float
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Reasonable initial changes to the variables.
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tol : float
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Final accuracy in the optimization (not precisely guaranteed).
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This is a lower bound on the size of the trust region.
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disp : bool
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Set to True to print convergence messages. If False,
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`verbosity` is ignored as set to 0.
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maxiter : int
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Maximum number of function evaluations.
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catol : float
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Tolerance (absolute) for constraint violations
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"""
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_check_unknown_options(unknown_options)
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maxfun = maxiter
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rhoend = tol
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iprint = int(bool(disp))
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# check constraints
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if isinstance(constraints, dict):
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constraints = (constraints, )
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for ic, con in enumerate(constraints):
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# check type
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try:
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ctype = con['type'].lower()
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except KeyError:
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raise KeyError('Constraint %d has no type defined.' % ic)
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except TypeError:
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raise TypeError('Constraints must be defined using a '
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'dictionary.')
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except AttributeError:
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raise TypeError("Constraint's type must be a string.")
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else:
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if ctype != 'ineq':
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raise ValueError("Constraints of type '%s' not handled by "
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"COBYLA." % con['type'])
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# check function
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if 'fun' not in con:
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raise KeyError('Constraint %d has no function defined.' % ic)
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# check extra arguments
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if 'args' not in con:
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con['args'] = ()
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# m is the total number of constraint values
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# it takes into account that some constraints may be vector-valued
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cons_lengths = []
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for c in constraints:
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f = c['fun'](x0, *c['args'])
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try:
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cons_length = len(f)
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except TypeError:
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cons_length = 1
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cons_lengths.append(cons_length)
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m = sum(cons_lengths)
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def calcfc(x, con):
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f = fun(x, *args)
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i = 0
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for size, c in izip(cons_lengths, constraints):
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con[i: i + size] = c['fun'](x, *c['args'])
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i += size
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return f
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info = np.zeros(4, np.float64)
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xopt, info = _cobyla.minimize(calcfc, m=m, x=np.copy(x0), rhobeg=rhobeg,
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rhoend=rhoend, iprint=iprint, maxfun=maxfun,
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dinfo=info)
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if info[3] > catol:
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# Check constraint violation
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info[0] = 4
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return OptimizeResult(x=xopt,
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status=int(info[0]),
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success=info[0] == 1,
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message={1: 'Optimization terminated successfully.',
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2: 'Maximum number of function evaluations '
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'has been exceeded.',
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3: 'Rounding errors are becoming damaging '
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'in COBYLA subroutine.',
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4: 'Did not converge to a solution '
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'satisfying the constraints. See '
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'`maxcv` for magnitude of violation.',
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5: 'NaN result encountered.'
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}.get(info[0], 'Unknown exit status.'),
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nfev=int(info[1]),
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fun=info[2],
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maxcv=info[3])
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