571 lines
22 KiB
Python
571 lines
22 KiB
Python
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"""Constraints definition for minimize."""
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import numpy as np
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from ._hessian_update_strategy import BFGS
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from ._differentiable_functions import (
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VectorFunction, LinearVectorFunction, IdentityVectorFunction)
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from ._optimize import OptimizeWarning
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from warnings import warn, catch_warnings, simplefilter
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from numpy.testing import suppress_warnings
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from scipy.sparse import issparse
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def _arr_to_scalar(x):
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# If x is a numpy array, return x.item(). This will
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# fail if the array has more than one element.
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return x.item() if isinstance(x, np.ndarray) else x
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class NonlinearConstraint:
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"""Nonlinear constraint on the variables.
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The constraint has the general inequality form::
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lb <= fun(x) <= ub
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Here the vector of independent variables x is passed as ndarray of shape
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(n,) and ``fun`` returns a vector with m components.
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It is possible to use equal bounds to represent an equality constraint or
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infinite bounds to represent a one-sided constraint.
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Parameters
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----------
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fun : callable
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The function defining the constraint.
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The signature is ``fun(x) -> array_like, shape (m,)``.
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lb, ub : array_like
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Lower and upper bounds on the constraint. Each array must have the
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shape (m,) or be a scalar, in the latter case a bound will be the same
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for all components of the constraint. Use ``np.inf`` with an
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appropriate sign to specify a one-sided constraint.
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Set components of `lb` and `ub` equal to represent an equality
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constraint. Note that you can mix constraints of different types:
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interval, one-sided or equality, by setting different components of
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`lb` and `ub` as necessary.
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jac : {callable, '2-point', '3-point', 'cs'}, optional
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Method of computing the Jacobian matrix (an m-by-n matrix,
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where element (i, j) is the partial derivative of f[i] with
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respect to x[j]). The keywords {'2-point', '3-point',
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'cs'} select a finite difference scheme for the numerical estimation.
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A callable must have the following signature:
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``jac(x) -> {ndarray, sparse matrix}, shape (m, n)``.
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Default is '2-point'.
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hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy, None}, optional
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Method for computing the Hessian matrix. The keywords
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{'2-point', '3-point', 'cs'} select a finite difference scheme for
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numerical estimation. Alternatively, objects implementing
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`HessianUpdateStrategy` interface can be used to approximate the
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Hessian. Currently available implementations are:
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- `BFGS` (default option)
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- `SR1`
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A callable must return the Hessian matrix of ``dot(fun, v)`` and
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must have the following signature:
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``hess(x, v) -> {LinearOperator, sparse matrix, array_like}, shape (n, n)``.
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Here ``v`` is ndarray with shape (m,) containing Lagrange multipliers.
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keep_feasible : array_like of bool, optional
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Whether to keep the constraint components feasible throughout
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iterations. A single value set this property for all components.
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Default is False. Has no effect for equality constraints.
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finite_diff_rel_step: None or array_like, optional
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Relative step size for the finite difference approximation. Default is
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None, which will select a reasonable value automatically depending
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on a finite difference scheme.
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finite_diff_jac_sparsity: {None, array_like, sparse matrix}, optional
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Defines the sparsity structure of the Jacobian matrix for finite
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difference estimation, its shape must be (m, n). If the Jacobian has
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only few non-zero elements in *each* row, providing the sparsity
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structure will greatly speed up the computations. A zero entry means
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that a corresponding element in the Jacobian is identically zero.
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If provided, forces the use of 'lsmr' trust-region solver.
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If None (default) then dense differencing will be used.
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Notes
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-----
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Finite difference schemes {'2-point', '3-point', 'cs'} may be used for
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approximating either the Jacobian or the Hessian. We, however, do not allow
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its use for approximating both simultaneously. Hence whenever the Jacobian
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is estimated via finite-differences, we require the Hessian to be estimated
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using one of the quasi-Newton strategies.
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The scheme 'cs' is potentially the most accurate, but requires the function
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to correctly handles complex inputs and be analytically continuable to the
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complex plane. The scheme '3-point' is more accurate than '2-point' but
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requires twice as many operations.
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Examples
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--------
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Constrain ``x[0] < sin(x[1]) + 1.9``
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>>> from scipy.optimize import NonlinearConstraint
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>>> import numpy as np
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>>> con = lambda x: x[0] - np.sin(x[1])
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>>> nlc = NonlinearConstraint(con, -np.inf, 1.9)
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"""
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def __init__(self, fun, lb, ub, jac='2-point', hess=BFGS(),
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keep_feasible=False, finite_diff_rel_step=None,
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finite_diff_jac_sparsity=None):
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self.fun = fun
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self.lb = lb
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self.ub = ub
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self.finite_diff_rel_step = finite_diff_rel_step
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self.finite_diff_jac_sparsity = finite_diff_jac_sparsity
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self.jac = jac
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self.hess = hess
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self.keep_feasible = keep_feasible
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class LinearConstraint:
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"""Linear constraint on the variables.
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The constraint has the general inequality form::
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lb <= A.dot(x) <= ub
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Here the vector of independent variables x is passed as ndarray of shape
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(n,) and the matrix A has shape (m, n).
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It is possible to use equal bounds to represent an equality constraint or
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infinite bounds to represent a one-sided constraint.
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Parameters
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----------
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A : {array_like, sparse matrix}, shape (m, n)
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Matrix defining the constraint.
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lb, ub : array_like, optional
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Lower and upper limits on the constraint. Each array must have the
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shape (m,) or be a scalar, in the latter case a bound will be the same
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for all components of the constraint. Use ``np.inf`` with an
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appropriate sign to specify a one-sided constraint.
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Set components of `lb` and `ub` equal to represent an equality
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constraint. Note that you can mix constraints of different types:
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interval, one-sided or equality, by setting different components of
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`lb` and `ub` as necessary. Defaults to ``lb = -np.inf``
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and ``ub = np.inf`` (no limits).
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keep_feasible : array_like of bool, optional
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Whether to keep the constraint components feasible throughout
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iterations. A single value set this property for all components.
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Default is False. Has no effect for equality constraints.
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"""
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def _input_validation(self):
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if self.A.ndim != 2:
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message = "`A` must have exactly two dimensions."
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raise ValueError(message)
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try:
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shape = self.A.shape[0:1]
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self.lb = np.broadcast_to(self.lb, shape)
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self.ub = np.broadcast_to(self.ub, shape)
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self.keep_feasible = np.broadcast_to(self.keep_feasible, shape)
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except ValueError:
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message = ("`lb`, `ub`, and `keep_feasible` must be broadcastable "
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"to shape `A.shape[0:1]`")
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raise ValueError(message)
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def __init__(self, A, lb=-np.inf, ub=np.inf, keep_feasible=False):
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if not issparse(A):
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# In some cases, if the constraint is not valid, this emits a
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# VisibleDeprecationWarning about ragged nested sequences
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# before eventually causing an error. `scipy.optimize.milp` would
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# prefer that this just error out immediately so it can handle it
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# rather than concerning the user.
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with catch_warnings():
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simplefilter("error")
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self.A = np.atleast_2d(A).astype(np.float64)
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else:
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self.A = A
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self.lb = np.atleast_1d(lb).astype(np.float64)
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self.ub = np.atleast_1d(ub).astype(np.float64)
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self.keep_feasible = np.atleast_1d(keep_feasible).astype(bool)
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self._input_validation()
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def residual(self, x):
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"""
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Calculate the residual between the constraint function and the limits
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For a linear constraint of the form::
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lb <= A@x <= ub
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the lower and upper residuals between ``A@x`` and the limits are values
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``sl`` and ``sb`` such that::
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lb + sl == A@x == ub - sb
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When all elements of ``sl`` and ``sb`` are positive, all elements of
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the constraint are satisfied; a negative element in ``sl`` or ``sb``
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indicates that the corresponding element of the constraint is not
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satisfied.
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Parameters
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----------
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x: array_like
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Vector of independent variables
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Returns
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-------
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sl, sb : array-like
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The lower and upper residuals
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"""
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return self.A@x - self.lb, self.ub - self.A@x
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class Bounds:
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"""Bounds constraint on the variables.
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The constraint has the general inequality form::
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lb <= x <= ub
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It is possible to use equal bounds to represent an equality constraint or
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infinite bounds to represent a one-sided constraint.
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Parameters
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----------
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lb, ub : array_like, optional
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Lower and upper bounds on independent variables. `lb`, `ub`, and
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`keep_feasible` must be the same shape or broadcastable.
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Set components of `lb` and `ub` equal
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to fix a variable. Use ``np.inf`` with an appropriate sign to disable
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bounds on all or some variables. Note that you can mix constraints of
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different types: interval, one-sided or equality, by setting different
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components of `lb` and `ub` as necessary. Defaults to ``lb = -np.inf``
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and ``ub = np.inf`` (no bounds).
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keep_feasible : array_like of bool, optional
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Whether to keep the constraint components feasible throughout
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iterations. Must be broadcastable with `lb` and `ub`.
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Default is False. Has no effect for equality constraints.
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"""
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def _input_validation(self):
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try:
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res = np.broadcast_arrays(self.lb, self.ub, self.keep_feasible)
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self.lb, self.ub, self.keep_feasible = res
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except ValueError:
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message = "`lb`, `ub`, and `keep_feasible` must be broadcastable."
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raise ValueError(message)
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def __init__(self, lb=-np.inf, ub=np.inf, keep_feasible=False):
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self.lb = np.atleast_1d(lb)
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self.ub = np.atleast_1d(ub)
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self.keep_feasible = np.atleast_1d(keep_feasible).astype(bool)
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self._input_validation()
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def __repr__(self):
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start = f"{type(self).__name__}({self.lb!r}, {self.ub!r}"
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if np.any(self.keep_feasible):
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end = f", keep_feasible={self.keep_feasible!r})"
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else:
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end = ")"
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return start + end
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def residual(self, x):
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"""Calculate the residual (slack) between the input and the bounds
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For a bound constraint of the form::
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lb <= x <= ub
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the lower and upper residuals between `x` and the bounds are values
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``sl`` and ``sb`` such that::
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lb + sl == x == ub - sb
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When all elements of ``sl`` and ``sb`` are positive, all elements of
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``x`` lie within the bounds; a negative element in ``sl`` or ``sb``
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indicates that the corresponding element of ``x`` is out of bounds.
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Parameters
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----------
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x: array_like
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Vector of independent variables
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Returns
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-------
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sl, sb : array-like
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The lower and upper residuals
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"""
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return x - self.lb, self.ub - x
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class PreparedConstraint:
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"""Constraint prepared from a user defined constraint.
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On creation it will check whether a constraint definition is valid and
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the initial point is feasible. If created successfully, it will contain
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the attributes listed below.
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Parameters
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----------
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constraint : {NonlinearConstraint, LinearConstraint`, Bounds}
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Constraint to check and prepare.
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x0 : array_like
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Initial vector of independent variables.
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sparse_jacobian : bool or None, optional
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If bool, then the Jacobian of the constraint will be converted
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to the corresponded format if necessary. If None (default), such
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conversion is not made.
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finite_diff_bounds : 2-tuple, optional
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Lower and upper bounds on the independent variables for the finite
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difference approximation, if applicable. Defaults to no bounds.
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Attributes
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----------
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fun : {VectorFunction, LinearVectorFunction, IdentityVectorFunction}
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Function defining the constraint wrapped by one of the convenience
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classes.
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bounds : 2-tuple
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Contains lower and upper bounds for the constraints --- lb and ub.
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These are converted to ndarray and have a size equal to the number of
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the constraints.
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keep_feasible : ndarray
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Array indicating which components must be kept feasible with a size
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equal to the number of the constraints.
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"""
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def __init__(self, constraint, x0, sparse_jacobian=None,
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finite_diff_bounds=(-np.inf, np.inf)):
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if isinstance(constraint, NonlinearConstraint):
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fun = VectorFunction(constraint.fun, x0,
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constraint.jac, constraint.hess,
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constraint.finite_diff_rel_step,
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constraint.finite_diff_jac_sparsity,
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finite_diff_bounds, sparse_jacobian)
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elif isinstance(constraint, LinearConstraint):
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fun = LinearVectorFunction(constraint.A, x0, sparse_jacobian)
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elif isinstance(constraint, Bounds):
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fun = IdentityVectorFunction(x0, sparse_jacobian)
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else:
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raise ValueError("`constraint` of an unknown type is passed.")
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m = fun.m
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lb = np.asarray(constraint.lb, dtype=float)
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ub = np.asarray(constraint.ub, dtype=float)
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keep_feasible = np.asarray(constraint.keep_feasible, dtype=bool)
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lb = np.broadcast_to(lb, m)
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ub = np.broadcast_to(ub, m)
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keep_feasible = np.broadcast_to(keep_feasible, m)
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if keep_feasible.shape != (m,):
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raise ValueError("`keep_feasible` has a wrong shape.")
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mask = keep_feasible & (lb != ub)
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f0 = fun.f
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if np.any(f0[mask] < lb[mask]) or np.any(f0[mask] > ub[mask]):
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raise ValueError("`x0` is infeasible with respect to some "
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"inequality constraint with `keep_feasible` "
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"set to True.")
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|
|
|
||
|
|
self.fun = fun
|
||
|
|
self.bounds = (lb, ub)
|
||
|
|
self.keep_feasible = keep_feasible
|
||
|
|
|
||
|
|
def violation(self, x):
|
||
|
|
"""How much the constraint is exceeded by.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array-like
|
||
|
|
Vector of independent variables
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
excess : array-like
|
||
|
|
How much the constraint is exceeded by, for each of the
|
||
|
|
constraints specified by `PreparedConstraint.fun`.
|
||
|
|
"""
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(UserWarning)
|
||
|
|
ev = self.fun.fun(np.asarray(x))
|
||
|
|
|
||
|
|
excess_lb = np.maximum(self.bounds[0] - ev, 0)
|
||
|
|
excess_ub = np.maximum(ev - self.bounds[1], 0)
|
||
|
|
|
||
|
|
return excess_lb + excess_ub
|
||
|
|
|
||
|
|
|
||
|
|
def new_bounds_to_old(lb, ub, n):
|
||
|
|
"""Convert the new bounds representation to the old one.
|
||
|
|
|
||
|
|
The new representation is a tuple (lb, ub) and the old one is a list
|
||
|
|
containing n tuples, ith containing lower and upper bound on a ith
|
||
|
|
variable.
|
||
|
|
If any of the entries in lb/ub are -np.inf/np.inf they are replaced by
|
||
|
|
None.
|
||
|
|
"""
|
||
|
|
lb = np.broadcast_to(lb, n)
|
||
|
|
ub = np.broadcast_to(ub, n)
|
||
|
|
|
||
|
|
lb = [float(x) if x > -np.inf else None for x in lb]
|
||
|
|
ub = [float(x) if x < np.inf else None for x in ub]
|
||
|
|
|
||
|
|
return list(zip(lb, ub))
|
||
|
|
|
||
|
|
|
||
|
|
def old_bound_to_new(bounds):
|
||
|
|
"""Convert the old bounds representation to the new one.
|
||
|
|
|
||
|
|
The new representation is a tuple (lb, ub) and the old one is a list
|
||
|
|
containing n tuples, ith containing lower and upper bound on a ith
|
||
|
|
variable.
|
||
|
|
If any of the entries in lb/ub are None they are replaced by
|
||
|
|
-np.inf/np.inf.
|
||
|
|
"""
|
||
|
|
lb, ub = zip(*bounds)
|
||
|
|
|
||
|
|
# Convert occurrences of None to -inf or inf, and replace occurrences of
|
||
|
|
# any numpy array x with x.item(). Then wrap the results in numpy arrays.
|
||
|
|
lb = np.array([float(_arr_to_scalar(x)) if x is not None else -np.inf
|
||
|
|
for x in lb])
|
||
|
|
ub = np.array([float(_arr_to_scalar(x)) if x is not None else np.inf
|
||
|
|
for x in ub])
|
||
|
|
|
||
|
|
return lb, ub
|
||
|
|
|
||
|
|
|
||
|
|
def strict_bounds(lb, ub, keep_feasible, n_vars):
|
||
|
|
"""Remove bounds which are not asked to be kept feasible."""
|
||
|
|
strict_lb = np.resize(lb, n_vars).astype(float)
|
||
|
|
strict_ub = np.resize(ub, n_vars).astype(float)
|
||
|
|
keep_feasible = np.resize(keep_feasible, n_vars)
|
||
|
|
strict_lb[~keep_feasible] = -np.inf
|
||
|
|
strict_ub[~keep_feasible] = np.inf
|
||
|
|
return strict_lb, strict_ub
|
||
|
|
|
||
|
|
|
||
|
|
def new_constraint_to_old(con, x0):
|
||
|
|
"""
|
||
|
|
Converts new-style constraint objects to old-style constraint dictionaries.
|
||
|
|
"""
|
||
|
|
if isinstance(con, NonlinearConstraint):
|
||
|
|
if (con.finite_diff_jac_sparsity is not None or
|
||
|
|
con.finite_diff_rel_step is not None or
|
||
|
|
not isinstance(con.hess, BFGS) or # misses user specified BFGS
|
||
|
|
con.keep_feasible):
|
||
|
|
warn("Constraint options `finite_diff_jac_sparsity`, "
|
||
|
|
"`finite_diff_rel_step`, `keep_feasible`, and `hess`"
|
||
|
|
"are ignored by this method.", OptimizeWarning)
|
||
|
|
|
||
|
|
fun = con.fun
|
||
|
|
if callable(con.jac):
|
||
|
|
jac = con.jac
|
||
|
|
else:
|
||
|
|
jac = None
|
||
|
|
|
||
|
|
else: # LinearConstraint
|
||
|
|
if np.any(con.keep_feasible):
|
||
|
|
warn("Constraint option `keep_feasible` is ignored by this "
|
||
|
|
"method.", OptimizeWarning)
|
||
|
|
|
||
|
|
A = con.A
|
||
|
|
if issparse(A):
|
||
|
|
A = A.toarray()
|
||
|
|
fun = lambda x: np.dot(A, x)
|
||
|
|
jac = lambda x: A
|
||
|
|
|
||
|
|
# FIXME: when bugs in VectorFunction/LinearVectorFunction are worked out,
|
||
|
|
# use pcon.fun.fun and pcon.fun.jac. Until then, get fun/jac above.
|
||
|
|
pcon = PreparedConstraint(con, x0)
|
||
|
|
lb, ub = pcon.bounds
|
||
|
|
|
||
|
|
i_eq = lb == ub
|
||
|
|
i_bound_below = np.logical_xor(lb != -np.inf, i_eq)
|
||
|
|
i_bound_above = np.logical_xor(ub != np.inf, i_eq)
|
||
|
|
i_unbounded = np.logical_and(lb == -np.inf, ub == np.inf)
|
||
|
|
|
||
|
|
if np.any(i_unbounded):
|
||
|
|
warn("At least one constraint is unbounded above and below. Such "
|
||
|
|
"constraints are ignored.", OptimizeWarning)
|
||
|
|
|
||
|
|
ceq = []
|
||
|
|
if np.any(i_eq):
|
||
|
|
def f_eq(x):
|
||
|
|
y = np.array(fun(x)).flatten()
|
||
|
|
return y[i_eq] - lb[i_eq]
|
||
|
|
ceq = [{"type": "eq", "fun": f_eq}]
|
||
|
|
|
||
|
|
if jac is not None:
|
||
|
|
def j_eq(x):
|
||
|
|
dy = jac(x)
|
||
|
|
if issparse(dy):
|
||
|
|
dy = dy.toarray()
|
||
|
|
dy = np.atleast_2d(dy)
|
||
|
|
return dy[i_eq, :]
|
||
|
|
ceq[0]["jac"] = j_eq
|
||
|
|
|
||
|
|
cineq = []
|
||
|
|
n_bound_below = np.sum(i_bound_below)
|
||
|
|
n_bound_above = np.sum(i_bound_above)
|
||
|
|
if n_bound_below + n_bound_above:
|
||
|
|
def f_ineq(x):
|
||
|
|
y = np.zeros(n_bound_below + n_bound_above)
|
||
|
|
y_all = np.array(fun(x)).flatten()
|
||
|
|
y[:n_bound_below] = y_all[i_bound_below] - lb[i_bound_below]
|
||
|
|
y[n_bound_below:] = -(y_all[i_bound_above] - ub[i_bound_above])
|
||
|
|
return y
|
||
|
|
cineq = [{"type": "ineq", "fun": f_ineq}]
|
||
|
|
|
||
|
|
if jac is not None:
|
||
|
|
def j_ineq(x):
|
||
|
|
dy = np.zeros((n_bound_below + n_bound_above, len(x0)))
|
||
|
|
dy_all = jac(x)
|
||
|
|
if issparse(dy_all):
|
||
|
|
dy_all = dy_all.toarray()
|
||
|
|
dy_all = np.atleast_2d(dy_all)
|
||
|
|
dy[:n_bound_below, :] = dy_all[i_bound_below]
|
||
|
|
dy[n_bound_below:, :] = -dy_all[i_bound_above]
|
||
|
|
return dy
|
||
|
|
cineq[0]["jac"] = j_ineq
|
||
|
|
|
||
|
|
old_constraints = ceq + cineq
|
||
|
|
|
||
|
|
if len(old_constraints) > 1:
|
||
|
|
warn("Equality and inequality constraints are specified in the same "
|
||
|
|
"element of the constraint list. For efficient use with this "
|
||
|
|
"method, equality and inequality constraints should be specified "
|
||
|
|
"in separate elements of the constraint list. ", OptimizeWarning)
|
||
|
|
return old_constraints
|
||
|
|
|
||
|
|
|
||
|
|
def old_constraint_to_new(ic, con):
|
||
|
|
"""
|
||
|
|
Converts old-style constraint dictionaries to new-style constraint objects.
|
||
|
|
"""
|
||
|
|
# check type
|
||
|
|
try:
|
||
|
|
ctype = con['type'].lower()
|
||
|
|
except KeyError as e:
|
||
|
|
raise KeyError('Constraint %d has no type defined.' % ic) from e
|
||
|
|
except TypeError as e:
|
||
|
|
raise TypeError(
|
||
|
|
'Constraints must be a sequence of dictionaries.'
|
||
|
|
) from e
|
||
|
|
except AttributeError as e:
|
||
|
|
raise TypeError("Constraint's type must be a string.") from e
|
||
|
|
else:
|
||
|
|
if ctype not in ['eq', 'ineq']:
|
||
|
|
raise ValueError("Unknown constraint type '%s'." % con['type'])
|
||
|
|
if 'fun' not in con:
|
||
|
|
raise ValueError('Constraint %d has no function defined.' % ic)
|
||
|
|
|
||
|
|
lb = 0
|
||
|
|
if ctype == 'eq':
|
||
|
|
ub = 0
|
||
|
|
else:
|
||
|
|
ub = np.inf
|
||
|
|
|
||
|
|
jac = '2-point'
|
||
|
|
if 'args' in con:
|
||
|
|
args = con['args']
|
||
|
|
fun = lambda x: con['fun'](x, *args)
|
||
|
|
if 'jac' in con:
|
||
|
|
jac = lambda x: con['jac'](x, *args)
|
||
|
|
else:
|
||
|
|
fun = con['fun']
|
||
|
|
if 'jac' in con:
|
||
|
|
jac = con['jac']
|
||
|
|
|
||
|
|
return NonlinearConstraint(fun, lb, ub, jac)
|