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"""Constraints definition for minimize.""" 

from __future__ import division, print_function, absolute_import 

import numpy as np 

from ._hessian_update_strategy import BFGS 

from ._differentiable_functions import ( 

VectorFunction, LinearVectorFunction, IdentityVectorFunction) 

from .optimize import OptimizeWarning 

from warnings import warn 

from scipy._lib._numpy_compat import suppress_warnings 

from scipy.sparse import issparse 

 

class NonlinearConstraint(object): 

"""Nonlinear constraint on the variables. 

 

The constraint has the general inequality form:: 

 

lb <= fun(x) <= ub 

 

Here the vector of independent variables x is passed as ndarray of shape 

(n,) and ``fun`` returns a vector with m components. 

 

It is possible to use equal bounds to represent an equality constraint or 

infinite bounds to represent a one-sided constraint. 

 

Parameters 

---------- 

fun : callable 

The function defining the constraint. 

The signature is ``fun(x) -> array_like, shape (m,)``. 

lb, ub : array_like 

Lower and upper bounds on the constraint. Each array must have the 

shape (m,) or be a scalar, in the latter case a bound will be the same 

for all components of the constraint. Use ``np.inf`` with an 

appropriate sign to specify a one-sided constraint. 

Set components of `lb` and `ub` equal to represent an equality 

constraint. Note that you can mix constraints of different types: 

interval, one-sided or equality, by setting different components of 

`lb` and `ub` as necessary. 

jac : {callable, '2-point', '3-point', 'cs'}, optional 

Method of computing the Jacobian matrix (an m-by-n matrix, 

where element (i, j) is the partial derivative of f[i] with 

respect to x[j]). The keywords {'2-point', '3-point', 

'cs'} select a finite difference scheme for the numerical estimation. 

A callable must have the following signature: 

``jac(x) -> {ndarray, sparse matrix}, shape (m, n)``. 

Default is '2-point'. 

hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy, None}, optional 

Method for computing the Hessian matrix. The keywords 

{'2-point', '3-point', 'cs'} select a finite difference scheme for 

numerical estimation. Alternatively, objects implementing 

`HessianUpdateStrategy` interface can be used to approximate the 

Hessian. Currently available implementations are: 

 

- `BFGS` (default option) 

- `SR1` 

 

A callable must return the Hessian matrix of ``dot(fun, v)`` and 

must have the following signature: 

``hess(x, v) -> {LinearOperator, sparse matrix, array_like}, shape (n, n)``. 

Here ``v`` is ndarray with shape (m,) containing Lagrange multipliers. 

keep_feasible : array_like of bool, optional 

Whether to keep the constraint components feasible throughout 

iterations. A single value set this property for all components. 

Default is False. Has no effect for equality constraints. 

finite_diff_rel_step: None or array_like, optional 

Relative step size for the finite difference approximation. Default is 

None, which will select a reasonable value automatically depending 

on a finite difference scheme. 

finite_diff_jac_sparsity: {None, array_like, sparse matrix}, optional 

Defines the sparsity structure of the Jacobian matrix for finite 

difference estimation, its shape must be (m, n). If the Jacobian has 

only few non-zero elements in *each* row, providing the sparsity 

structure will greatly speed up the computations. A zero entry means 

that a corresponding element in the Jacobian is identically zero. 

If provided, forces the use of 'lsmr' trust-region solver. 

If None (default) then dense differencing will be used. 

 

Notes 

----- 

Finite difference schemes {'2-point', '3-point', 'cs'} may be used for 

approximating either the Jacobian or the Hessian. We, however, do not allow 

its use for approximating both simultaneously. Hence whenever the Jacobian 

is estimated via finite-differences, we require the Hessian to be estimated 

using one of the quasi-Newton strategies. 

 

The scheme 'cs' is potentially the most accurate, but requires the function 

to correctly handles complex inputs and be analytically continuable to the 

complex plane. The scheme '3-point' is more accurate than '2-point' but 

requires twice as many operations. 

""" 

def __init__(self, fun, lb, ub, jac='2-point', hess=BFGS(), 

keep_feasible=False, finite_diff_rel_step=None, 

finite_diff_jac_sparsity=None): 

self.fun = fun 

self.lb = lb 

self.ub = ub 

self.finite_diff_rel_step = finite_diff_rel_step 

self.finite_diff_jac_sparsity = finite_diff_jac_sparsity 

self.jac = jac 

self.hess = hess 

self.keep_feasible = keep_feasible 

 

 

class LinearConstraint(object): 

"""Linear constraint on the variables. 

 

The constraint has the general inequality form:: 

 

lb <= A.dot(x) <= ub 

 

Here the vector of independent variables x is passed as ndarray of shape 

(n,) and the matrix A has shape (m, n). 

 

It is possible to use equal bounds to represent an equality constraint or 

infinite bounds to represent a one-sided constraint. 

 

Parameters 

---------- 

A : {array_like, sparse matrix}, shape (m, n) 

Matrix defining the constraint. 

lb, ub : array_like 

Lower and upper bounds on the constraint. Each array must have the 

shape (m,) or be a scalar, in the latter case a bound will be the same 

for all components of the constraint. Use ``np.inf`` with an 

appropriate sign to specify a one-sided constraint. 

Set components of `lb` and `ub` equal to represent an equality 

constraint. Note that you can mix constraints of different types: 

interval, one-sided or equality, by setting different components of 

`lb` and `ub` as necessary. 

keep_feasible : array_like of bool, optional 

Whether to keep the constraint components feasible throughout 

iterations. A single value set this property for all components. 

Default is False. Has no effect for equality constraints. 

""" 

def __init__(self, A, lb, ub, keep_feasible=False): 

self.A = A 

self.lb = lb 

self.ub = ub 

self.keep_feasible = keep_feasible 

 

 

class Bounds(object): 

"""Bounds constraint on the variables. 

 

The constraint has the general inequality form:: 

 

lb <= x <= ub 

 

It is possible to use equal bounds to represent an equality constraint or 

infinite bounds to represent a one-sided constraint. 

 

Parameters 

---------- 

lb, ub : array_like, optional 

Lower and upper bounds on independent variables. Each array must 

have the same size as x or be a scalar, in which case a bound will be 

the same for all the variables. Set components of `lb` and `ub` equal 

to fix a variable. Use ``np.inf`` with an appropriate sign to disable 

bounds on all or some variables. Note that you can mix constraints of 

different types: interval, one-sided or equality, by setting different 

components of `lb` and `ub` as necessary. 

keep_feasible : array_like of bool, optional 

Whether to keep the constraint components feasible throughout 

iterations. A single value set this property for all components. 

Default is False. Has no effect for equality constraints. 

""" 

def __init__(self, lb, ub, keep_feasible=False): 

self.lb = lb 

self.ub = ub 

self.keep_feasible = keep_feasible 

 

def __repr__(self): 

if np.any(self.keep_feasible): 

return "{}({!r}, {!r}, keep_feasible={!r})".format(type(self).__name__, self.lb, self.ub, self.keep_feasible) 

else: 

return "{}({!r}, {!r})".format(type(self).__name__, self.lb, self.ub) 

 

 

class PreparedConstraint(object): 

"""Constraint prepared from a user defined constraint. 

 

On creation it will check whether a constraint definition is valid and 

the initial point is feasible. If created successfully, it will contain 

the attributes listed below. 

 

Parameters 

---------- 

constraint : {NonlinearConstraint, LinearConstraint`, Bounds} 

Constraint to check and prepare. 

x0 : array_like 

Initial vector of independent variables. 

sparse_jacobian : bool or None, optional 

If bool, then the Jacobian of the constraint will be converted 

to the corresponded format if necessary. If None (default), such 

conversion is not made. 

finite_diff_bounds : 2-tuple, optional 

Lower and upper bounds on the independent variables for the finite 

difference approximation, if applicable. Defaults to no bounds. 

 

Attributes 

---------- 

fun : {VectorFunction, LinearVectorFunction, IdentityVectorFunction} 

Function defining the constraint wrapped by one of the convenience 

classes. 

bounds : 2-tuple 

Contains lower and upper bounds for the constraints --- lb and ub. 

These are converted to ndarray and have a size equal to the number of 

the constraints. 

keep_feasible : ndarray 

Array indicating which components must be kept feasible with a size 

equal to the number of the constraints. 

""" 

def __init__(self, constraint, x0, sparse_jacobian=None, 

finite_diff_bounds=(-np.inf, np.inf)): 

if isinstance(constraint, NonlinearConstraint): 

fun = VectorFunction(constraint.fun, x0, 

constraint.jac, constraint.hess, 

constraint.finite_diff_rel_step, 

constraint.finite_diff_jac_sparsity, 

finite_diff_bounds, sparse_jacobian) 

elif isinstance(constraint, LinearConstraint): 

fun = LinearVectorFunction(constraint.A, x0, sparse_jacobian) 

elif isinstance(constraint, Bounds): 

fun = IdentityVectorFunction(x0, sparse_jacobian) 

else: 

raise ValueError("`constraint` of an unknown type is passed.") 

 

m = fun.m 

lb = np.asarray(constraint.lb, dtype=float) 

ub = np.asarray(constraint.ub, dtype=float) 

if lb.ndim == 0: 

lb = np.resize(lb, m) 

if ub.ndim == 0: 

ub = np.resize(ub, m) 

 

keep_feasible = np.asarray(constraint.keep_feasible, dtype=bool) 

if keep_feasible.ndim == 0: 

keep_feasible = np.resize(keep_feasible, m) 

if keep_feasible.shape != (m,): 

raise ValueError("`keep_feasible` has a wrong shape.") 

 

mask = keep_feasible & (lb != ub) 

f0 = fun.f 

if np.any(f0[mask] < lb[mask]) or np.any(f0[mask] > ub[mask]): 

raise ValueError("`x0` is infeasible with respect to some " 

"inequality constraint with `keep_feasible` " 

"set to True.") 

 

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, i-th containing lower and upper bound on a i-th 

variable. 

""" 

lb = np.asarray(lb) 

ub = np.asarray(ub) 

if lb.ndim == 0: 

lb = np.resize(lb, n) 

if ub.ndim == 0: 

ub = np.resize(ub, n) 

 

lb = [x if x > -np.inf else None for x in lb] 

ub = [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, i-th containing lower and upper bound on a i-th 

variable. 

""" 

lb, ub = zip(*bounds) 

lb = np.array([x if x is not None else -np.inf for x in lb]) 

ub = np.array([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 con.keep_feasible: 

warn("Constraint option `keep_feasible` is ignored by this " 

"method.", OptimizeWarning) 

 

A = con.A 

if issparse(A): 

A = A.todense() 

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.todense() 

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.todense() 

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: 

raise KeyError('Constraint %d has no type defined.' % ic) 

except TypeError: 

raise TypeError('Constraints must be a sequence of dictionaries.') 

except AttributeError: 

raise TypeError("Constraint's type must be a string.") 

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)