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from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import assert_array_equal, assert_equal
from scipy.optimize._constraints import (NonlinearConstraint, Bounds,
PreparedConstraint)
from scipy.optimize._trustregion_constr.canonical_constraint \
import CanonicalConstraint, initial_constraints_as_canonical
def create_quadratic_function(n, m, rng):
a = rng.rand(m)
A = rng.rand(m, n)
H = rng.rand(m, n, n)
HT = np.transpose(H, (1, 2, 0))
def fun(x):
return a + A.dot(x) + 0.5 * H.dot(x).dot(x)
def jac(x):
return A + H.dot(x)
def hess(x, v):
return HT.dot(v)
return fun, jac, hess
def test_bounds_cases():
# Test 1: no constraints.
user_constraint = Bounds(-np.inf, np.inf)
x0 = np.array([-1, 2])
prepared_constraint = PreparedConstraint(user_constraint, x0, False)
c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint)
assert_equal(c.n_eq, 0)
assert_equal(c.n_ineq, 0)
c_eq, c_ineq = c.fun(x0)
assert_array_equal(c_eq, [])
assert_array_equal(c_ineq, [])
J_eq, J_ineq = c.jac(x0)
assert_array_equal(J_eq, np.empty((0, 2)))
assert_array_equal(J_ineq, np.empty((0, 2)))
assert_array_equal(c.keep_feasible, [])
# Test 2: infinite lower bound.
user_constraint = Bounds(-np.inf, [0, np.inf, 1], [False, True, True])
x0 = np.array([-1, -2, -3], dtype=float)
prepared_constraint = PreparedConstraint(user_constraint, x0, False)
c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint)
assert_equal(c.n_eq, 0)
assert_equal(c.n_ineq, 2)
c_eq, c_ineq = c.fun(x0)
assert_array_equal(c_eq, [])
assert_array_equal(c_ineq, [-1, -4])
J_eq, J_ineq = c.jac(x0)
assert_array_equal(J_eq, np.empty((0, 3)))
assert_array_equal(J_ineq, np.array([[1, 0, 0], [0, 0, 1]]))
assert_array_equal(c.keep_feasible, [False, True])
# Test 3: infinite upper bound.
user_constraint = Bounds([0, 1, -np.inf], np.inf, [True, False, True])
x0 = np.array([1, 2, 3], dtype=float)
prepared_constraint = PreparedConstraint(user_constraint, x0, False)
c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint)
assert_equal(c.n_eq, 0)
assert_equal(c.n_ineq, 2)
c_eq, c_ineq = c.fun(x0)
assert_array_equal(c_eq, [])
assert_array_equal(c_ineq, [-1, -1])
J_eq, J_ineq = c.jac(x0)
assert_array_equal(J_eq, np.empty((0, 3)))
assert_array_equal(J_ineq, np.array([[-1, 0, 0], [0, -1, 0]]))
assert_array_equal(c.keep_feasible, [True, False])
# Test 4: interval constraint.
user_constraint = Bounds([-1, -np.inf, 2, 3], [1, np.inf, 10, 3],
[False, True, True, True])
x0 = np.array([0, 10, 8, 5])
prepared_constraint = PreparedConstraint(user_constraint, x0, False)
c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint)
assert_equal(c.n_eq, 1)
assert_equal(c.n_ineq, 4)
c_eq, c_ineq = c.fun(x0)
assert_array_equal(c_eq, [2])
assert_array_equal(c_ineq, [-1, -2, -1, -6])
J_eq, J_ineq = c.jac(x0)
assert_array_equal(J_eq, [[0, 0, 0, 1]])
assert_array_equal(J_ineq, [[1, 0, 0, 0],
[0, 0, 1, 0],
[-1, 0, 0, 0],
[0, 0, -1, 0]])
assert_array_equal(c.keep_feasible, [False, True, False, True])
def test_nonlinear_constraint():
n = 3
m = 5
rng = np.random.RandomState(0)
x0 = rng.rand(n)
fun, jac, hess = create_quadratic_function(n, m, rng)
f = fun(x0)
J = jac(x0)
lb = [-10, 3, -np.inf, -np.inf, -5]
ub = [10, 3, np.inf, 3, np.inf]
user_constraint = NonlinearConstraint(
fun, lb, ub, jac, hess, [True, False, False, True, False])
for sparse_jacobian in [False, True]:
prepared_constraint = PreparedConstraint(user_constraint, x0,
sparse_jacobian)
c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint)
assert_array_equal(c.n_eq, 1)
assert_array_equal(c.n_ineq, 4)
c_eq, c_ineq = c.fun(x0)
assert_array_equal(c_eq, [f[1] - lb[1]])
assert_array_equal(c_ineq, [f[3] - ub[3], lb[4] - f[4],
f[0] - ub[0], lb[0] - f[0]])
J_eq, J_ineq = c.jac(x0)
if sparse_jacobian:
J_eq = J_eq.toarray()
J_ineq = J_ineq.toarray()
assert_array_equal(J_eq, J[1, None])
assert_array_equal(J_ineq, np.vstack((J[3], -J[4], J[0], -J[0])))
v_eq = rng.rand(c.n_eq)
v_ineq = rng.rand(c.n_ineq)
v = np.zeros(m)
v[1] = v_eq[0]
v[3] = v_ineq[0]
v[4] = -v_ineq[1]
v[0] = v_ineq[2] - v_ineq[3]
assert_array_equal(c.hess(x0, v_eq, v_ineq), hess(x0, v))
assert_array_equal(c.keep_feasible, [True, False, True, True])
def test_concatenation():
rng = np.random.RandomState(0)
n = 4
x0 = np.random.rand(n)
f1 = x0
J1 = np.eye(n)
lb1 = [-1, -np.inf, -2, 3]
ub1 = [1, np.inf, np.inf, 3]
bounds = Bounds(lb1, ub1, [False, False, True, False])
fun, jac, hess = create_quadratic_function(n, 5, rng)
f2 = fun(x0)
J2 = jac(x0)
lb2 = [-10, 3, -np.inf, -np.inf, -5]
ub2 = [10, 3, np.inf, 5, np.inf]
nonlinear = NonlinearConstraint(
fun, lb2, ub2, jac, hess, [True, False, False, True, False])
for sparse_jacobian in [False, True]:
bounds_prepared = PreparedConstraint(bounds, x0, sparse_jacobian)
nonlinear_prepared = PreparedConstraint(nonlinear, x0, sparse_jacobian)
c1 = CanonicalConstraint.from_PreparedConstraint(bounds_prepared)
c2 = CanonicalConstraint.from_PreparedConstraint(nonlinear_prepared)
c = CanonicalConstraint.concatenate([c1, c2], sparse_jacobian)
assert_equal(c.n_eq, 2)
assert_equal(c.n_ineq, 7)
c_eq, c_ineq = c.fun(x0)
assert_array_equal(c_eq, [f1[3] - lb1[3], f2[1] - lb2[1]])
assert_array_equal(c_ineq, [lb1[2] - f1[2], f1[0] - ub1[0],
lb1[0] - f1[0], f2[3] - ub2[3],
lb2[4] - f2[4], f2[0] - ub2[0],
lb2[0] - f2[0]])
J_eq, J_ineq = c.jac(x0)
if sparse_jacobian:
J_eq = J_eq.toarray()
J_ineq = J_ineq.toarray()
assert_array_equal(J_eq, np.vstack((J1[3], J2[1])))
assert_array_equal(J_ineq, np.vstack((-J1[2], J1[0], -J1[0], J2[3],
-J2[4], J2[0], -J2[0])))
v_eq = rng.rand(c.n_eq)
v_ineq = rng.rand(c.n_ineq)
v = np.zeros(5)
v[1] = v_eq[1]
v[3] = v_ineq[3]
v[4] = -v_ineq[4]
v[0] = v_ineq[5] - v_ineq[6]
H = c.hess(x0, v_eq, v_ineq).dot(np.eye(n))
assert_array_equal(H, hess(x0, v))
assert_array_equal(c.keep_feasible,
[True, False, False, True, False, True, True])
def test_empty():
x = np.array([1, 2, 3])
c = CanonicalConstraint.empty(3)
assert_equal(c.n_eq, 0)
assert_equal(c.n_ineq, 0)
c_eq, c_ineq = c.fun(x)
assert_array_equal(c_eq, [])
assert_array_equal(c_ineq, [])
J_eq, J_ineq = c.jac(x)
assert_array_equal(J_eq, np.empty((0, 3)))
assert_array_equal(J_ineq, np.empty((0, 3)))
H = c.hess(x, None, None).toarray()
assert_array_equal(H, np.zeros((3, 3)))
def test_initial_constraints_as_canonical():
# rng is only used to generate the coefficients of the quadratic
# function that is used by the nonlinear constraint.
rng = np.random.RandomState(0)
x0 = np.array([0.5, 0.4, 0.3, 0.2])
n = len(x0)
lb1 = [-1, -np.inf, -2, 3]
ub1 = [1, np.inf, np.inf, 3]
bounds = Bounds(lb1, ub1, [False, False, True, False])
fun, jac, hess = create_quadratic_function(n, 5, rng)
lb2 = [-10, 3, -np.inf, -np.inf, -5]
ub2 = [10, 3, np.inf, 5, np.inf]
nonlinear = NonlinearConstraint(
fun, lb2, ub2, jac, hess, [True, False, False, True, False])
for sparse_jacobian in [False, True]:
bounds_prepared = PreparedConstraint(bounds, x0, sparse_jacobian)
nonlinear_prepared = PreparedConstraint(nonlinear, x0, sparse_jacobian)
f1 = bounds_prepared.fun.f
J1 = bounds_prepared.fun.J
f2 = nonlinear_prepared.fun.f
J2 = nonlinear_prepared.fun.J
c_eq, c_ineq, J_eq, J_ineq = initial_constraints_as_canonical(
n, [bounds_prepared, nonlinear_prepared], sparse_jacobian)
assert_array_equal(c_eq, [f1[3] - lb1[3], f2[1] - lb2[1]])
assert_array_equal(c_ineq, [lb1[2] - f1[2], f1[0] - ub1[0],
lb1[0] - f1[0], f2[3] - ub2[3],
lb2[4] - f2[4], f2[0] - ub2[0],
lb2[0] - f2[0]])
if sparse_jacobian:
J1 = J1.toarray()
J2 = J2.toarray()
J_eq = J_eq.toarray()
J_ineq = J_ineq.toarray()
assert_array_equal(J_eq, np.vstack((J1[3], J2[1])))
assert_array_equal(J_ineq, np.vstack((-J1[2], J1[0], -J1[0], J2[3],
-J2[4], J2[0], -J2[0])))
def test_initial_constraints_as_canonical_empty():
n = 3
for sparse_jacobian in [False, True]:
c_eq, c_ineq, J_eq, J_ineq = initial_constraints_as_canonical(
n, [], sparse_jacobian)
assert_array_equal(c_eq, [])
assert_array_equal(c_ineq, [])
if sparse_jacobian:
J_eq = J_eq.toarray()
J_ineq = J_ineq.toarray()
assert_array_equal(J_eq, np.empty((0, n)))
assert_array_equal(J_ineq, np.empty((0, n)))
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