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Inzynierka/Lib/site-packages/scipy/optimize/tests/test_optimize.py
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"""
Unit tests for optimization routines from optimize.py
Authors:
Ed Schofield, Nov 2005
Andrew Straw, April 2008
To run it in its simplest form::
nosetests test_optimize.py
"""
import itertools
import platform
import numpy as np
from numpy.testing import (assert_allclose, assert_equal,
assert_almost_equal,
assert_no_warnings, assert_warns,
assert_array_less, suppress_warnings)
import pytest
from pytest import raises as assert_raises
from scipy import optimize
from scipy.optimize._minimize import Bounds, NonlinearConstraint
from scipy.optimize._minimize import MINIMIZE_METHODS, MINIMIZE_SCALAR_METHODS
from scipy.optimize._linprog import LINPROG_METHODS
from scipy.optimize._root import ROOT_METHODS
from scipy.optimize._root_scalar import ROOT_SCALAR_METHODS
from scipy.optimize._qap import QUADRATIC_ASSIGNMENT_METHODS
from scipy.optimize._differentiable_functions import ScalarFunction, FD_METHODS
from scipy.optimize._optimize import MemoizeJac, show_options
def test_check_grad():
# Verify if check_grad is able to estimate the derivative of the
# expit (logistic sigmoid) function.
def expit(x):
return 1 / (1 + np.exp(-x))
def der_expit(x):
return np.exp(-x) / (1 + np.exp(-x))**2
x0 = np.array([1.5])
r = optimize.check_grad(expit, der_expit, x0)
assert_almost_equal(r, 0)
r = optimize.check_grad(expit, der_expit, x0,
direction='random', seed=1234)
assert_almost_equal(r, 0)
r = optimize.check_grad(expit, der_expit, x0, epsilon=1e-6)
assert_almost_equal(r, 0)
r = optimize.check_grad(expit, der_expit, x0, epsilon=1e-6,
direction='random', seed=1234)
assert_almost_equal(r, 0)
# Check if the epsilon parameter is being considered.
r = abs(optimize.check_grad(expit, der_expit, x0, epsilon=1e-1) - 0)
assert r > 1e-7
r = abs(optimize.check_grad(expit, der_expit, x0, epsilon=1e-1,
direction='random', seed=1234) - 0)
assert r > 1e-7
def x_sinx(x):
return (x*np.sin(x)).sum()
def der_x_sinx(x):
return np.sin(x) + x*np.cos(x)
x0 = np.arange(0, 2, 0.2)
r = optimize.check_grad(x_sinx, der_x_sinx, x0,
direction='random', seed=1234)
assert_almost_equal(r, 0)
assert_raises(ValueError, optimize.check_grad,
x_sinx, der_x_sinx, x0,
direction='random_projection', seed=1234)
# checking can be done for derivatives of vector valued functions
r = optimize.check_grad(himmelblau_grad, himmelblau_hess, himmelblau_x0,
direction='all', seed=1234)
assert r < 5e-7
class CheckOptimize:
""" Base test case for a simple constrained entropy maximization problem
(the machine translation example of Berger et al in
Computational Linguistics, vol 22, num 1, pp 39--72, 1996.)
"""
def setup_method(self):
self.F = np.array([[1, 1, 1],
[1, 1, 0],
[1, 0, 1],
[1, 0, 0],
[1, 0, 0]])
self.K = np.array([1., 0.3, 0.5])
self.startparams = np.zeros(3, np.float64)
self.solution = np.array([0., -0.524869316, 0.487525860])
self.maxiter = 1000
self.funccalls = 0
self.gradcalls = 0
self.trace = []
def func(self, x):
self.funccalls += 1
if self.funccalls > 6000:
raise RuntimeError("too many iterations in optimization routine")
log_pdot = np.dot(self.F, x)
logZ = np.log(sum(np.exp(log_pdot)))
f = logZ - np.dot(self.K, x)
self.trace.append(np.copy(x))
return f
def grad(self, x):
self.gradcalls += 1
log_pdot = np.dot(self.F, x)
logZ = np.log(sum(np.exp(log_pdot)))
p = np.exp(log_pdot - logZ)
return np.dot(self.F.transpose(), p) - self.K
def hess(self, x):
log_pdot = np.dot(self.F, x)
logZ = np.log(sum(np.exp(log_pdot)))
p = np.exp(log_pdot - logZ)
return np.dot(self.F.T,
np.dot(np.diag(p), self.F - np.dot(self.F.T, p)))
def hessp(self, x, p):
return np.dot(self.hess(x), p)
class CheckOptimizeParameterized(CheckOptimize):
def test_cg(self):
# conjugate gradient optimization routine
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
res = optimize.minimize(self.func, self.startparams, args=(),
method='CG', jac=self.grad,
options=opts)
params, fopt, func_calls, grad_calls, warnflag = \
res['x'], res['fun'], res['nfev'], res['njev'], res['status']
else:
retval = optimize.fmin_cg(self.func, self.startparams,
self.grad, (), maxiter=self.maxiter,
full_output=True, disp=self.disp,
retall=False)
(params, fopt, func_calls, grad_calls, warnflag) = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.7.0. Don't allow them to increase.
assert self.funccalls == 9, self.funccalls
assert self.gradcalls == 7, self.gradcalls
# Ensure that the function behaves the same; this is from SciPy 0.7.0
assert_allclose(self.trace[2:4],
[[0, -0.5, 0.5],
[0, -5.05700028e-01, 4.95985862e-01]],
atol=1e-14, rtol=1e-7)
def test_cg_cornercase(self):
def f(r):
return 2.5 * (1 - np.exp(-1.5*(r - 0.5)))**2
# Check several initial guesses. (Too far away from the
# minimum, the function ends up in the flat region of exp.)
for x0 in np.linspace(-0.75, 3, 71):
sol = optimize.minimize(f, [x0], method='CG')
assert sol.success
assert_allclose(sol.x, [0.5], rtol=1e-5)
def test_bfgs(self):
# Broyden-Fletcher-Goldfarb-Shanno optimization routine
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
res = optimize.minimize(self.func, self.startparams,
jac=self.grad, method='BFGS', args=(),
options=opts)
params, fopt, gopt, Hopt, func_calls, grad_calls, warnflag = (
res['x'], res['fun'], res['jac'], res['hess_inv'],
res['nfev'], res['njev'], res['status'])
else:
retval = optimize.fmin_bfgs(self.func, self.startparams, self.grad,
args=(), maxiter=self.maxiter,
full_output=True, disp=self.disp,
retall=False)
(params, fopt, gopt, Hopt,
func_calls, grad_calls, warnflag) = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.7.0. Don't allow them to increase.
assert self.funccalls == 10, self.funccalls
assert self.gradcalls == 8, self.gradcalls
# Ensure that the function behaves the same; this is from SciPy 0.7.0
assert_allclose(self.trace[6:8],
[[0, -5.25060743e-01, 4.87748473e-01],
[0, -5.24885582e-01, 4.87530347e-01]],
atol=1e-14, rtol=1e-7)
def test_bfgs_infinite(self):
# Test corner case where -Inf is the minimum. See gh-2019.
func = lambda x: -np.e**-x
fprime = lambda x: -func(x)
x0 = [0]
with np.errstate(over='ignore'):
if self.use_wrapper:
opts = {'disp': self.disp}
x = optimize.minimize(func, x0, jac=fprime, method='BFGS',
args=(), options=opts)['x']
else:
x = optimize.fmin_bfgs(func, x0, fprime, disp=self.disp)
assert not np.isfinite(func(x))
def test_bfgs_xrtol(self):
# test for #17345 to test xrtol parameter
x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
res = optimize.minimize(optimize.rosen,
x0, method='bfgs', options={'xrtol': 1e-3})
ref = optimize.minimize(optimize.rosen,
x0, method='bfgs', options={'gtol': 1e-3})
assert res.nit != ref.nit
def test_powell(self):
# Powell (direction set) optimization routine
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
res = optimize.minimize(self.func, self.startparams, args=(),
method='Powell', options=opts)
params, fopt, direc, numiter, func_calls, warnflag = (
res['x'], res['fun'], res['direc'], res['nit'],
res['nfev'], res['status'])
else:
retval = optimize.fmin_powell(self.func, self.startparams,
args=(), maxiter=self.maxiter,
full_output=True, disp=self.disp,
retall=False)
(params, fopt, direc, numiter, func_calls, warnflag) = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# params[0] does not affect the objective function
assert_allclose(params[1:], self.solution[1:], atol=5e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.7.0. Don't allow them to increase.
#
# However, some leeway must be added: the exact evaluation
# count is sensitive to numerical error, and floating-point
# computations are not bit-for-bit reproducible across
# machines, and when using e.g., MKL, data alignment
# etc., affect the rounding error.
#
assert self.funccalls <= 116 + 20, self.funccalls
assert self.gradcalls == 0, self.gradcalls
@pytest.mark.xfail(reason="This part of test_powell fails on some "
"platforms, but the solution returned by powell is "
"still valid.")
def test_powell_gh14014(self):
# This part of test_powell started failing on some CI platforms;
# see gh-14014. Since the solution is still correct and the comments
# in test_powell suggest that small differences in the bits are known
# to change the "trace" of the solution, seems safe to xfail to get CI
# green now and investigate later.
# Powell (direction set) optimization routine
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
res = optimize.minimize(self.func, self.startparams, args=(),
method='Powell', options=opts)
params, fopt, direc, numiter, func_calls, warnflag = (
res['x'], res['fun'], res['direc'], res['nit'],
res['nfev'], res['status'])
else:
retval = optimize.fmin_powell(self.func, self.startparams,
args=(), maxiter=self.maxiter,
full_output=True, disp=self.disp,
retall=False)
(params, fopt, direc, numiter, func_calls, warnflag) = retval
# Ensure that the function behaves the same; this is from SciPy 0.7.0
assert_allclose(self.trace[34:39],
[[0.72949016, -0.44156936, 0.47100962],
[0.72949016, -0.44156936, 0.48052496],
[1.45898031, -0.88313872, 0.95153458],
[0.72949016, -0.44156936, 0.47576729],
[1.72949016, -0.44156936, 0.47576729]],
atol=1e-14, rtol=1e-7)
def test_powell_bounded(self):
# Powell (direction set) optimization routine
# same as test_powell above, but with bounds
bounds = [(-np.pi, np.pi) for _ in self.startparams]
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
res = optimize.minimize(self.func, self.startparams, args=(),
bounds=bounds,
method='Powell', options=opts)
params, fopt, direc, numiter, func_calls, warnflag = (
res['x'], res['fun'], res['direc'], res['nit'],
res['nfev'], res['status'])
assert func_calls == self.funccalls
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6, rtol=1e-5)
# The exact evaluation count is sensitive to numerical error, and
# floating-point computations are not bit-for-bit reproducible
# across machines, and when using e.g. MKL, data alignment etc.
# affect the rounding error.
# It takes 155 calls on my machine, but we can add the same +20
# margin as is used in `test_powell`
assert self.funccalls <= 155 + 20
assert self.gradcalls == 0
def test_neldermead(self):
# Nelder-Mead simplex algorithm
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
res = optimize.minimize(self.func, self.startparams, args=(),
method='Nelder-mead', options=opts)
params, fopt, numiter, func_calls, warnflag = (
res['x'], res['fun'], res['nit'], res['nfev'],
res['status'])
else:
retval = optimize.fmin(self.func, self.startparams,
args=(), maxiter=self.maxiter,
full_output=True, disp=self.disp,
retall=False)
(params, fopt, numiter, func_calls, warnflag) = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.7.0. Don't allow them to increase.
assert self.funccalls == 167, self.funccalls
assert self.gradcalls == 0, self.gradcalls
# Ensure that the function behaves the same; this is from SciPy 0.7.0
assert_allclose(self.trace[76:78],
[[0.1928968, -0.62780447, 0.35166118],
[0.19572515, -0.63648426, 0.35838135]],
atol=1e-14, rtol=1e-7)
def test_neldermead_initial_simplex(self):
# Nelder-Mead simplex algorithm
simplex = np.zeros((4, 3))
simplex[...] = self.startparams
for j in range(3):
simplex[j+1, j] += 0.1
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': False,
'return_all': True, 'initial_simplex': simplex}
res = optimize.minimize(self.func, self.startparams, args=(),
method='Nelder-mead', options=opts)
params, fopt, numiter, func_calls, warnflag = (res['x'],
res['fun'],
res['nit'],
res['nfev'],
res['status'])
assert_allclose(res['allvecs'][0], simplex[0])
else:
retval = optimize.fmin(self.func, self.startparams,
args=(), maxiter=self.maxiter,
full_output=True, disp=False, retall=False,
initial_simplex=simplex)
(params, fopt, numiter, func_calls, warnflag) = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.17.0. Don't allow them to increase.
assert self.funccalls == 100, self.funccalls
assert self.gradcalls == 0, self.gradcalls
# Ensure that the function behaves the same; this is from SciPy 0.15.0
assert_allclose(self.trace[50:52],
[[0.14687474, -0.5103282, 0.48252111],
[0.14474003, -0.5282084, 0.48743951]],
atol=1e-14, rtol=1e-7)
def test_neldermead_initial_simplex_bad(self):
# Check it fails with a bad simplices
bad_simplices = []
simplex = np.zeros((3, 2))
simplex[...] = self.startparams[:2]
for j in range(2):
simplex[j+1, j] += 0.1
bad_simplices.append(simplex)
simplex = np.zeros((3, 3))
bad_simplices.append(simplex)
for simplex in bad_simplices:
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': False,
'return_all': False, 'initial_simplex': simplex}
assert_raises(ValueError,
optimize.minimize,
self.func,
self.startparams,
args=(),
method='Nelder-mead',
options=opts)
else:
assert_raises(ValueError, optimize.fmin,
self.func, self.startparams,
args=(), maxiter=self.maxiter,
full_output=True, disp=False, retall=False,
initial_simplex=simplex)
def test_ncg_negative_maxiter(self):
# Regression test for gh-8241
opts = {'maxiter': -1}
result = optimize.minimize(self.func, self.startparams,
method='Newton-CG', jac=self.grad,
args=(), options=opts)
assert result.status == 1
def test_ncg(self):
# line-search Newton conjugate gradient optimization routine
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
retval = optimize.minimize(self.func, self.startparams,
method='Newton-CG', jac=self.grad,
args=(), options=opts)['x']
else:
retval = optimize.fmin_ncg(self.func, self.startparams, self.grad,
args=(), maxiter=self.maxiter,
full_output=False, disp=self.disp,
retall=False)
params = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.7.0. Don't allow them to increase.
assert self.funccalls == 7, self.funccalls
assert self.gradcalls <= 22, self.gradcalls # 0.13.0
# assert self.gradcalls <= 18, self.gradcalls # 0.9.0
# assert self.gradcalls == 18, self.gradcalls # 0.8.0
# assert self.gradcalls == 22, self.gradcalls # 0.7.0
# Ensure that the function behaves the same; this is from SciPy 0.7.0
assert_allclose(self.trace[3:5],
[[-4.35700753e-07, -5.24869435e-01, 4.87527480e-01],
[-4.35700753e-07, -5.24869401e-01, 4.87527774e-01]],
atol=1e-6, rtol=1e-7)
def test_ncg_hess(self):
# Newton conjugate gradient with Hessian
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
retval = optimize.minimize(self.func, self.startparams,
method='Newton-CG', jac=self.grad,
hess=self.hess,
args=(), options=opts)['x']
else:
retval = optimize.fmin_ncg(self.func, self.startparams, self.grad,
fhess=self.hess,
args=(), maxiter=self.maxiter,
full_output=False, disp=self.disp,
retall=False)
params = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.7.0. Don't allow them to increase.
assert self.funccalls <= 7, self.funccalls # gh10673
assert self.gradcalls <= 18, self.gradcalls # 0.9.0
# assert self.gradcalls == 18, self.gradcalls # 0.8.0
# assert self.gradcalls == 22, self.gradcalls # 0.7.0
# Ensure that the function behaves the same; this is from SciPy 0.7.0
assert_allclose(self.trace[3:5],
[[-4.35700753e-07, -5.24869435e-01, 4.87527480e-01],
[-4.35700753e-07, -5.24869401e-01, 4.87527774e-01]],
atol=1e-6, rtol=1e-7)
def test_ncg_hessp(self):
# Newton conjugate gradient with Hessian times a vector p.
if self.use_wrapper:
opts = {'maxiter': self.maxiter, 'disp': self.disp,
'return_all': False}
retval = optimize.minimize(self.func, self.startparams,
method='Newton-CG', jac=self.grad,
hessp=self.hessp,
args=(), options=opts)['x']
else:
retval = optimize.fmin_ncg(self.func, self.startparams, self.grad,
fhess_p=self.hessp,
args=(), maxiter=self.maxiter,
full_output=False, disp=self.disp,
retall=False)
params = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.7.0. Don't allow them to increase.
assert self.funccalls <= 7, self.funccalls # gh10673
assert self.gradcalls <= 18, self.gradcalls # 0.9.0
# assert self.gradcalls == 18, self.gradcalls # 0.8.0
# assert self.gradcalls == 22, self.gradcalls # 0.7.0
# Ensure that the function behaves the same; this is from SciPy 0.7.0
assert_allclose(self.trace[3:5],
[[-4.35700753e-07, -5.24869435e-01, 4.87527480e-01],
[-4.35700753e-07, -5.24869401e-01, 4.87527774e-01]],
atol=1e-6, rtol=1e-7)
def test_maxfev_test():
rng = np.random.default_rng(271707100830272976862395227613146332411)
def cost(x):
return rng.random(1) * 1000 # never converged problem
for imaxfev in [1, 10, 50]:
# "TNC" and "L-BFGS-B" also supports max function evaluation, but
# these may violate the limit because of evaluating gradients
# by numerical differentiation. See the discussion in PR #14805.
for method in ['Powell', 'Nelder-Mead']:
result = optimize.minimize(cost, rng.random(10),
method=method,
options={'maxfev': imaxfev})
assert result["nfev"] == imaxfev
def test_wrap_scalar_function_with_validation():
def func_(x):
return x
fcalls, func = optimize._optimize.\
_wrap_scalar_function_maxfun_validation(func_, np.asarray(1), 5)
for i in range(5):
func(np.asarray(i))
assert fcalls[0] == i+1
msg = "Too many function calls"
with assert_raises(optimize._optimize._MaxFuncCallError, match=msg):
func(np.asarray(i)) # exceeded maximum function call
fcalls, func = optimize._optimize.\
_wrap_scalar_function_maxfun_validation(func_, np.asarray(1), 5)
msg = "The user-provided objective function must return a scalar value."
with assert_raises(ValueError, match=msg):
func(np.array([1, 1]))
def test_obj_func_returns_scalar():
match = ("The user-provided "
"objective function must "
"return a scalar value.")
with assert_raises(ValueError, match=match):
optimize.minimize(lambda x: x, np.array([1, 1]), method='BFGS')
def test_neldermead_iteration_num():
x0 = np.array([1.3, 0.7, 0.8, 1.9, 1.2])
res = optimize._minimize._minimize_neldermead(optimize.rosen, x0,
xatol=1e-8)
assert res.nit <= 339
def test_neldermead_xatol_fatol():
# gh4484
# test we can call with fatol, xatol specified
func = lambda x: x[0]**2 + x[1]**2
optimize._minimize._minimize_neldermead(func, [1, 1], maxiter=2,
xatol=1e-3, fatol=1e-3)
def test_neldermead_adaptive():
func = lambda x: np.sum(x**2)
p0 = [0.15746215, 0.48087031, 0.44519198, 0.4223638, 0.61505159,
0.32308456, 0.9692297, 0.4471682, 0.77411992, 0.80441652,
0.35994957, 0.75487856, 0.99973421, 0.65063887, 0.09626474]
res = optimize.minimize(func, p0, method='Nelder-Mead')
assert_equal(res.success, False)
res = optimize.minimize(func, p0, method='Nelder-Mead',
options={'adaptive': True})
assert_equal(res.success, True)
def test_bounded_powell_outsidebounds():
# With the bounded Powell method if you start outside the bounds the final
# should still be within the bounds (provided that the user doesn't make a
# bad choice for the `direc` argument).
func = lambda x: np.sum(x**2)
bounds = (-1, 1), (-1, 1), (-1, 1)
x0 = [-4, .5, -.8]
# we're starting outside the bounds, so we should get a warning
with assert_warns(optimize.OptimizeWarning):
res = optimize.minimize(func, x0, bounds=bounds, method="Powell")
assert_allclose(res.x, np.array([0.] * len(x0)), atol=1e-6)
assert_equal(res.success, True)
assert_equal(res.status, 0)
# However, now if we change the `direc` argument such that the
# set of vectors does not span the parameter space, then we may
# not end up back within the bounds. Here we see that the first
# parameter cannot be updated!
direc = [[0, 0, 0], [0, 1, 0], [0, 0, 1]]
# we're starting outside the bounds, so we should get a warning
with assert_warns(optimize.OptimizeWarning):
res = optimize.minimize(func, x0,
bounds=bounds, method="Powell",
options={'direc': direc})
assert_allclose(res.x, np.array([-4., 0, 0]), atol=1e-6)
assert_equal(res.success, False)
assert_equal(res.status, 4)
def test_bounded_powell_vs_powell():
# here we test an example where the bounded Powell method
# will return a different result than the standard Powell
# method.
# first we test a simple example where the minimum is at
# the origin and the minimum that is within the bounds is
# larger than the minimum at the origin.
func = lambda x: np.sum(x**2)
bounds = (-5, -1), (-10, -0.1), (1, 9.2), (-4, 7.6), (-15.9, -2)
x0 = [-2.1, -5.2, 1.9, 0, -2]
options = {'ftol': 1e-10, 'xtol': 1e-10}
res_powell = optimize.minimize(func, x0, method="Powell", options=options)
assert_allclose(res_powell.x, 0., atol=1e-6)
assert_allclose(res_powell.fun, 0., atol=1e-6)
res_bounded_powell = optimize.minimize(func, x0, options=options,
bounds=bounds,
method="Powell")
p = np.array([-1, -0.1, 1, 0, -2])
assert_allclose(res_bounded_powell.x, p, atol=1e-6)
assert_allclose(res_bounded_powell.fun, func(p), atol=1e-6)
# now we test bounded Powell but with a mix of inf bounds.
bounds = (None, -1), (-np.inf, -.1), (1, np.inf), (-4, None), (-15.9, -2)
res_bounded_powell = optimize.minimize(func, x0, options=options,
bounds=bounds,
method="Powell")
p = np.array([-1, -0.1, 1, 0, -2])
assert_allclose(res_bounded_powell.x, p, atol=1e-6)
assert_allclose(res_bounded_powell.fun, func(p), atol=1e-6)
# next we test an example where the global minimum is within
# the bounds, but the bounded Powell method performs better
# than the standard Powell method.
def func(x):
t = np.sin(-x[0]) * np.cos(x[1]) * np.sin(-x[0] * x[1]) * np.cos(x[1])
t -= np.cos(np.sin(x[1] * x[2]) * np.cos(x[2]))
return t**2
bounds = [(-2, 5)] * 3
x0 = [-0.5, -0.5, -0.5]
res_powell = optimize.minimize(func, x0, method="Powell")
res_bounded_powell = optimize.minimize(func, x0,
bounds=bounds,
method="Powell")
assert_allclose(res_powell.fun, 0.007136253919761627, atol=1e-6)
assert_allclose(res_bounded_powell.fun, 0, atol=1e-6)
# next we test the previous example where the we provide Powell
# with (-inf, inf) bounds, and compare it to providing Powell
# with no bounds. They should end up the same.
bounds = [(-np.inf, np.inf)] * 3
res_bounded_powell = optimize.minimize(func, x0,
bounds=bounds,
method="Powell")
assert_allclose(res_powell.fun, res_bounded_powell.fun, atol=1e-6)
assert_allclose(res_powell.nfev, res_bounded_powell.nfev, atol=1e-6)
assert_allclose(res_powell.x, res_bounded_powell.x, atol=1e-6)
# now test when x0 starts outside of the bounds.
x0 = [45.46254415, -26.52351498, 31.74830248]
bounds = [(-2, 5)] * 3
# we're starting outside the bounds, so we should get a warning
with assert_warns(optimize.OptimizeWarning):
res_bounded_powell = optimize.minimize(func, x0,
bounds=bounds,
method="Powell")
assert_allclose(res_bounded_powell.fun, 0, atol=1e-6)
def test_onesided_bounded_powell_stability():
# When the Powell method is bounded on only one side, a
# np.tan transform is done in order to convert it into a
# completely bounded problem. Here we do some simple tests
# of one-sided bounded Powell where the optimal solutions
# are large to test the stability of the transformation.
kwargs = {'method': 'Powell',
'bounds': [(-np.inf, 1e6)] * 3,
'options': {'ftol': 1e-8, 'xtol': 1e-8}}
x0 = [1, 1, 1]
# df/dx is constant.
f = lambda x: -np.sum(x)
res = optimize.minimize(f, x0, **kwargs)
assert_allclose(res.fun, -3e6, atol=1e-4)
# df/dx gets smaller and smaller.
def f(x):
return -np.abs(np.sum(x)) ** (0.1) * (1 if np.all(x > 0) else -1)
res = optimize.minimize(f, x0, **kwargs)
assert_allclose(res.fun, -(3e6) ** (0.1))
# df/dx gets larger and larger.
def f(x):
return -np.abs(np.sum(x)) ** 10 * (1 if np.all(x > 0) else -1)
res = optimize.minimize(f, x0, **kwargs)
assert_allclose(res.fun, -(3e6) ** 10, rtol=1e-7)
# df/dx gets larger for some of the variables and smaller for others.
def f(x):
t = -np.abs(np.sum(x[:2])) ** 5 - np.abs(np.sum(x[2:])) ** (0.1)
t *= (1 if np.all(x > 0) else -1)
return t
kwargs['bounds'] = [(-np.inf, 1e3)] * 3
res = optimize.minimize(f, x0, **kwargs)
assert_allclose(res.fun, -(2e3) ** 5 - (1e6) ** (0.1), rtol=1e-7)
class TestOptimizeWrapperDisp(CheckOptimizeParameterized):
use_wrapper = True
disp = True
class TestOptimizeWrapperNoDisp(CheckOptimizeParameterized):
use_wrapper = True
disp = False
class TestOptimizeNoWrapperDisp(CheckOptimizeParameterized):
use_wrapper = False
disp = True
class TestOptimizeNoWrapperNoDisp(CheckOptimizeParameterized):
use_wrapper = False
disp = False
class TestOptimizeSimple(CheckOptimize):
def test_bfgs_nan(self):
# Test corner case where nan is fed to optimizer. See gh-2067.
func = lambda x: x
fprime = lambda x: np.ones_like(x)
x0 = [np.nan]
with np.errstate(over='ignore', invalid='ignore'):
x = optimize.fmin_bfgs(func, x0, fprime, disp=False)
assert np.isnan(func(x))
def test_bfgs_nan_return(self):
# Test corner cases where fun returns NaN. See gh-4793.
# First case: NaN from first call.
func = lambda x: np.nan
with np.errstate(invalid='ignore'):
result = optimize.minimize(func, 0)
assert np.isnan(result['fun'])
assert result['success'] is False
# Second case: NaN from second call.
func = lambda x: 0 if x == 0 else np.nan
fprime = lambda x: np.ones_like(x) # Steer away from zero.
with np.errstate(invalid='ignore'):
result = optimize.minimize(func, 0, jac=fprime)
assert np.isnan(result['fun'])
assert result['success'] is False
def test_bfgs_numerical_jacobian(self):
# BFGS with numerical Jacobian and a vector epsilon parameter.
# define the epsilon parameter using a random vector
epsilon = np.sqrt(np.spacing(1.)) * np.random.rand(len(self.solution))
params = optimize.fmin_bfgs(self.func, self.startparams,
epsilon=epsilon, args=(),
maxiter=self.maxiter, disp=False)
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
def test_finite_differences_jac(self):
methods = ['BFGS', 'CG', 'TNC']
jacs = ['2-point', '3-point', None]
for method, jac in itertools.product(methods, jacs):
result = optimize.minimize(self.func, self.startparams,
method=method, jac=jac)
assert_allclose(self.func(result.x), self.func(self.solution),
atol=1e-6)
def test_finite_differences_hess(self):
# test that all the methods that require hess can use finite-difference
# For Newton-CG, trust-ncg, trust-krylov the FD estimated hessian is
# wrapped in a hessp function
# dogleg, trust-exact actually require true hessians at the moment, so
# they're excluded.
methods = ['trust-constr', 'Newton-CG', 'trust-ncg', 'trust-krylov']
hesses = FD_METHODS + (optimize.BFGS,)
for method, hess in itertools.product(methods, hesses):
if hess is optimize.BFGS:
hess = hess()
result = optimize.minimize(self.func, self.startparams,
method=method, jac=self.grad,
hess=hess)
assert result.success
# check that the methods demand some sort of Hessian specification
# Newton-CG creates its own hessp, and trust-constr doesn't need a hess
# specified either
methods = ['trust-ncg', 'trust-krylov', 'dogleg', 'trust-exact']
for method in methods:
with pytest.raises(ValueError):
optimize.minimize(self.func, self.startparams,
method=method, jac=self.grad,
hess=None)
def test_bfgs_gh_2169(self):
def f(x):
if x < 0:
return 1.79769313e+308
else:
return x + 1./x
xs = optimize.fmin_bfgs(f, [10.], disp=False)
assert_allclose(xs, 1.0, rtol=1e-4, atol=1e-4)
def test_bfgs_double_evaluations(self):
# check BFGS does not evaluate twice in a row at same point
def f(x):
xp = x[0]
assert xp not in seen
seen.add(xp)
return 10*x**2, 20*x
seen = set()
optimize.minimize(f, -100, method='bfgs', jac=True, tol=1e-7)
def test_l_bfgs_b(self):
# limited-memory bound-constrained BFGS algorithm
retval = optimize.fmin_l_bfgs_b(self.func, self.startparams,
self.grad, args=(),
maxiter=self.maxiter)
(params, fopt, d) = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
# Ensure that function call counts are 'known good'; these are from
# SciPy 0.7.0. Don't allow them to increase.
assert self.funccalls == 7, self.funccalls
assert self.gradcalls == 5, self.gradcalls
# Ensure that the function behaves the same; this is from SciPy 0.7.0
# test fixed in gh10673
assert_allclose(self.trace[3:5],
[[8.117083e-16, -5.196198e-01, 4.897617e-01],
[0., -0.52489628, 0.48753042]],
atol=1e-14, rtol=1e-7)
def test_l_bfgs_b_numjac(self):
# L-BFGS-B with numerical Jacobian
retval = optimize.fmin_l_bfgs_b(self.func, self.startparams,
approx_grad=True,
maxiter=self.maxiter)
(params, fopt, d) = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
def test_l_bfgs_b_funjac(self):
# L-BFGS-B with combined objective function and Jacobian
def fun(x):
return self.func(x), self.grad(x)
retval = optimize.fmin_l_bfgs_b(fun, self.startparams,
maxiter=self.maxiter)
(params, fopt, d) = retval
assert_allclose(self.func(params), self.func(self.solution),
atol=1e-6)
def test_l_bfgs_b_maxiter(self):
# gh7854
# Ensure that not more than maxiters are ever run.
class Callback:
def __init__(self):
self.nit = 0
self.fun = None
self.x = None
def __call__(self, x):
self.x = x
self.fun = optimize.rosen(x)
self.nit += 1
c = Callback()
res = optimize.minimize(optimize.rosen, [0., 0.], method='l-bfgs-b',
callback=c, options={'maxiter': 5})
assert_equal(res.nit, 5)
assert_almost_equal(res.x, c.x)
assert_almost_equal(res.fun, c.fun)
assert_equal(res.status, 1)
assert res.success is False
assert_equal(res.message,
'STOP: TOTAL NO. of ITERATIONS REACHED LIMIT')
def test_minimize_l_bfgs_b(self):
# Minimize with L-BFGS-B method
opts = {'disp': False, 'maxiter': self.maxiter}
r = optimize.minimize(self.func, self.startparams,
method='L-BFGS-B', jac=self.grad,
options=opts)
assert_allclose(self.func(r.x), self.func(self.solution),
atol=1e-6)
assert self.gradcalls == r.njev
self.funccalls = self.gradcalls = 0
# approximate jacobian
ra = optimize.minimize(self.func, self.startparams,
method='L-BFGS-B', options=opts)
# check that function evaluations in approximate jacobian are counted
# assert_(ra.nfev > r.nfev)
assert self.funccalls == ra.nfev
assert_allclose(self.func(ra.x), self.func(self.solution),
atol=1e-6)
self.funccalls = self.gradcalls = 0
# approximate jacobian
ra = optimize.minimize(self.func, self.startparams, jac='3-point',
method='L-BFGS-B', options=opts)
assert self.funccalls == ra.nfev
assert_allclose(self.func(ra.x), self.func(self.solution),
atol=1e-6)
def test_minimize_l_bfgs_b_ftol(self):
# Check that the `ftol` parameter in l_bfgs_b works as expected
v0 = None
for tol in [1e-1, 1e-4, 1e-7, 1e-10]:
opts = {'disp': False, 'maxiter': self.maxiter, 'ftol': tol}
sol = optimize.minimize(self.func, self.startparams,
method='L-BFGS-B', jac=self.grad,
options=opts)
v = self.func(sol.x)
if v0 is None:
v0 = v
else:
assert v < v0
assert_allclose(v, self.func(self.solution), rtol=tol)
def test_minimize_l_bfgs_maxls(self):
# check that the maxls is passed down to the Fortran routine
sol = optimize.minimize(optimize.rosen, np.array([-1.2, 1.0]),
method='L-BFGS-B', jac=optimize.rosen_der,
options={'disp': False, 'maxls': 1})
assert not sol.success
def test_minimize_l_bfgs_b_maxfun_interruption(self):
# gh-6162
f = optimize.rosen
g = optimize.rosen_der
values = []
x0 = np.full(7, 1000)
def objfun(x):
value = f(x)
values.append(value)
return value
# Look for an interesting test case.
# Request a maxfun that stops at a particularly bad function
# evaluation somewhere between 100 and 300 evaluations.
low, medium, high = 30, 100, 300
optimize.fmin_l_bfgs_b(objfun, x0, fprime=g, maxfun=high)
v, k = max((y, i) for i, y in enumerate(values[medium:]))
maxfun = medium + k
# If the minimization strategy is reasonable,
# the minimize() result should not be worse than the best
# of the first 30 function evaluations.
target = min(values[:low])
xmin, fmin, d = optimize.fmin_l_bfgs_b(f, x0, fprime=g, maxfun=maxfun)
assert_array_less(fmin, target)
def test_custom(self):
# This function comes from the documentation example.
def custmin(fun, x0, args=(), maxfev=None, stepsize=0.1,
maxiter=100, callback=None, **options):
bestx = x0
besty = fun(x0)
funcalls = 1
niter = 0
improved = True
stop = False
while improved and not stop and niter < maxiter:
improved = False
niter += 1
for dim in range(np.size(x0)):
for s in [bestx[dim] - stepsize, bestx[dim] + stepsize]:
testx = np.copy(bestx)
testx[dim] = s
testy = fun(testx, *args)
funcalls += 1
if testy < besty:
besty = testy
bestx = testx
improved = True
if callback is not None:
callback(bestx)
if maxfev is not None and funcalls >= maxfev:
stop = True
break
return optimize.OptimizeResult(fun=besty, x=bestx, nit=niter,
nfev=funcalls, success=(niter > 1))
x0 = [1.35, 0.9, 0.8, 1.1, 1.2]
res = optimize.minimize(optimize.rosen, x0, method=custmin,
options=dict(stepsize=0.05))
assert_allclose(res.x, 1.0, rtol=1e-4, atol=1e-4)
@pytest.mark.xfail(reason="output not reliable on all platforms")
def test_gh13321(self, capfd):
# gh-13321 reported issues with console output in fmin_l_bfgs_b;
# check that iprint=0 works.
kwargs = {'func': optimize.rosen, 'x0': [4, 3],
'fprime': optimize.rosen_der, 'bounds': ((3, 5), (3, 5))}
# "L-BFGS-B" is always in output; should show when iprint >= 0
# "At iterate" is iterate info; should show when iprint >= 1
optimize.fmin_l_bfgs_b(**kwargs, iprint=-1)
out, _ = capfd.readouterr()
assert "L-BFGS-B" not in out and "At iterate" not in out
optimize.fmin_l_bfgs_b(**kwargs, iprint=0)
out, _ = capfd.readouterr()
assert "L-BFGS-B" in out and "At iterate" not in out
optimize.fmin_l_bfgs_b(**kwargs, iprint=1)
out, _ = capfd.readouterr()
assert "L-BFGS-B" in out and "At iterate" in out
# `disp is not None` overrides `iprint` behavior
# `disp=0` should suppress all output
# `disp=1` should be the same as `iprint = 1`
optimize.fmin_l_bfgs_b(**kwargs, iprint=1, disp=False)
out, _ = capfd.readouterr()
assert "L-BFGS-B" not in out and "At iterate" not in out
optimize.fmin_l_bfgs_b(**kwargs, iprint=-1, disp=True)
out, _ = capfd.readouterr()
assert "L-BFGS-B" in out and "At iterate" in out
def test_gh10771(self):
# check that minimize passes bounds and constraints to a custom
# minimizer without altering them.
bounds = [(-2, 2), (0, 3)]
constraints = 'constraints'
def custmin(fun, x0, **options):
assert options['bounds'] is bounds
assert options['constraints'] is constraints
return optimize.OptimizeResult()
x0 = [1, 1]
optimize.minimize(optimize.rosen, x0, method=custmin,
bounds=bounds, constraints=constraints)
def test_minimize_tol_parameter(self):
# Check that the minimize() tol= argument does something
def func(z):
x, y = z
return x**2*y**2 + x**4 + 1
def dfunc(z):
x, y = z
return np.array([2*x*y**2 + 4*x**3, 2*x**2*y])
for method in ['nelder-mead', 'powell', 'cg', 'bfgs',
'newton-cg', 'l-bfgs-b', 'tnc',
'cobyla', 'slsqp']:
if method in ('nelder-mead', 'powell', 'cobyla'):
jac = None
else:
jac = dfunc
sol1 = optimize.minimize(func, [1, 1], jac=jac, tol=1e-10,
method=method)
sol2 = optimize.minimize(func, [1, 1], jac=jac, tol=1.0,
method=method)
assert func(sol1.x) < func(sol2.x), "%s: %s vs. %s" % (method, func(sol1.x), func(sol2.x))
@pytest.mark.parametrize('method',
['fmin', 'fmin_powell', 'fmin_cg', 'fmin_bfgs',
'fmin_ncg', 'fmin_l_bfgs_b', 'fmin_tnc',
'fmin_slsqp'] + MINIMIZE_METHODS)
def test_minimize_callback_copies_array(self, method):
# Check that arrays passed to callbacks are not modified
# inplace by the optimizer afterward
if method in ('fmin_tnc', 'fmin_l_bfgs_b'):
func = lambda x: (optimize.rosen(x), optimize.rosen_der(x))
else:
func = optimize.rosen
jac = optimize.rosen_der
hess = optimize.rosen_hess
x0 = np.zeros(10)
# Set options
kwargs = {}
if method.startswith('fmin'):
routine = getattr(optimize, method)
if method == 'fmin_slsqp':
kwargs['iter'] = 5
elif method == 'fmin_tnc':
kwargs['maxfun'] = 100
elif method in ('fmin', 'fmin_powell'):
kwargs['maxiter'] = 3500
else:
kwargs['maxiter'] = 5
else:
def routine(*a, **kw):
kw['method'] = method
return optimize.minimize(*a, **kw)
if method == 'tnc':
kwargs['options'] = dict(maxfun=100)
else:
kwargs['options'] = dict(maxiter=5)
if method in ('fmin_ncg',):
kwargs['fprime'] = jac
elif method in ('newton-cg',):
kwargs['jac'] = jac
elif method in ('trust-krylov', 'trust-exact', 'trust-ncg', 'dogleg',
'trust-constr'):
kwargs['jac'] = jac
kwargs['hess'] = hess
# Run with callback
results = []
def callback(x, *args, **kwargs):
results.append((x, np.copy(x)))
routine(func, x0, callback=callback, **kwargs)
# Check returned arrays coincide with their copies
# and have no memory overlap
assert len(results) > 2
assert all(np.all(x == y) for x, y in results)
assert not any(np.may_share_memory(x[0], y[0]) for x, y in itertools.combinations(results, 2))
@pytest.mark.parametrize('method', ['nelder-mead', 'powell', 'cg',
'bfgs', 'newton-cg', 'l-bfgs-b',
'tnc', 'cobyla', 'slsqp'])
def test_no_increase(self, method):
# Check that the solver doesn't return a value worse than the
# initial point.
def func(x):
return (x - 1)**2
def bad_grad(x):
# purposefully invalid gradient function, simulates a case
# where line searches start failing
return 2*(x - 1) * (-1) - 2
x0 = np.array([2.0])
f0 = func(x0)
jac = bad_grad
options = dict(maxfun=20) if method == 'tnc' else dict(maxiter=20)
if method in ['nelder-mead', 'powell', 'cobyla']:
jac = None
sol = optimize.minimize(func, x0, jac=jac, method=method,
options=options)
assert_equal(func(sol.x), sol.fun)
if method == 'slsqp':
pytest.xfail("SLSQP returns slightly worse")
assert func(sol.x) <= f0
def test_slsqp_respect_bounds(self):
# Regression test for gh-3108
def f(x):
return sum((x - np.array([1., 2., 3., 4.]))**2)
def cons(x):
a = np.array([[-1, -1, -1, -1], [-3, -3, -2, -1]])
return np.concatenate([np.dot(a, x) + np.array([5, 10]), x])
x0 = np.array([0.5, 1., 1.5, 2.])
res = optimize.minimize(f, x0, method='slsqp',
constraints={'type': 'ineq', 'fun': cons})
assert_allclose(res.x, np.array([0., 2, 5, 8])/3, atol=1e-12)
@pytest.mark.parametrize('method', ['Nelder-Mead', 'Powell', 'CG', 'BFGS',
'Newton-CG', 'L-BFGS-B', 'SLSQP',
'trust-constr', 'dogleg', 'trust-ncg',
'trust-exact', 'trust-krylov'])
def test_respect_maxiter(self, method):
# Check that the number of iterations equals max_iter, assuming
# convergence doesn't establish before
MAXITER = 4
x0 = np.zeros(10)
sf = ScalarFunction(optimize.rosen, x0, (), optimize.rosen_der,
optimize.rosen_hess, None, None)
# Set options
kwargs = {'method': method, 'options': dict(maxiter=MAXITER)}
if method in ('Newton-CG',):
kwargs['jac'] = sf.grad
elif method in ('trust-krylov', 'trust-exact', 'trust-ncg', 'dogleg',
'trust-constr'):
kwargs['jac'] = sf.grad
kwargs['hess'] = sf.hess
sol = optimize.minimize(sf.fun, x0, **kwargs)
assert sol.nit == MAXITER
assert sol.nfev >= sf.nfev
if hasattr(sol, 'njev'):
assert sol.njev >= sf.ngev
# method specific tests
if method == 'SLSQP':
assert sol.status == 9 # Iteration limit reached
@pytest.mark.parametrize('method', ['Nelder-Mead', 'Powell',
'fmin', 'fmin_powell'])
def test_runtime_warning(self, method):
x0 = np.zeros(10)
sf = ScalarFunction(optimize.rosen, x0, (), optimize.rosen_der,
optimize.rosen_hess, None, None)
options = {"maxiter": 1, "disp": True}
with pytest.warns(RuntimeWarning,
match=r'Maximum number of iterations'):
if method.startswith('fmin'):
routine = getattr(optimize, method)
routine(sf.fun, x0, **options)
else:
optimize.minimize(sf.fun, x0, method=method, options=options)
def test_respect_maxiter_trust_constr_ineq_constraints(self):
# special case of minimization with trust-constr and inequality
# constraints to check maxiter limit is obeyed when using internal
# method 'tr_interior_point'
MAXITER = 4
f = optimize.rosen
jac = optimize.rosen_der
hess = optimize.rosen_hess
fun = lambda x: np.array([0.2 * x[0] - 0.4 * x[1] - 0.33 * x[2]])
cons = ({'type': 'ineq',
'fun': fun},)
x0 = np.zeros(10)
sol = optimize.minimize(f, x0, constraints=cons, jac=jac, hess=hess,
method='trust-constr',
options=dict(maxiter=MAXITER))
assert sol.nit == MAXITER
def test_minimize_automethod(self):
def f(x):
return x**2
def cons(x):
return x - 2
x0 = np.array([10.])
sol_0 = optimize.minimize(f, x0)
sol_1 = optimize.minimize(f, x0, constraints=[{'type': 'ineq',
'fun': cons}])
sol_2 = optimize.minimize(f, x0, bounds=[(5, 10)])
sol_3 = optimize.minimize(f, x0,
constraints=[{'type': 'ineq', 'fun': cons}],
bounds=[(5, 10)])
sol_4 = optimize.minimize(f, x0,
constraints=[{'type': 'ineq', 'fun': cons}],
bounds=[(1, 10)])
for sol in [sol_0, sol_1, sol_2, sol_3, sol_4]:
assert sol.success
assert_allclose(sol_0.x, 0, atol=1e-7)
assert_allclose(sol_1.x, 2, atol=1e-7)
assert_allclose(sol_2.x, 5, atol=1e-7)
assert_allclose(sol_3.x, 5, atol=1e-7)
assert_allclose(sol_4.x, 2, atol=1e-7)
def test_minimize_coerce_args_param(self):
# Regression test for gh-3503
def Y(x, c):
return np.sum((x-c)**2)
def dY_dx(x, c=None):
return 2*(x-c)
c = np.array([3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5])
xinit = np.random.randn(len(c))
optimize.minimize(Y, xinit, jac=dY_dx, args=(c), method="BFGS")
def test_initial_step_scaling(self):
# Check that optimizer initial step is not huge even if the
# function and gradients are
scales = [1e-50, 1, 1e50]
methods = ['CG', 'BFGS', 'L-BFGS-B', 'Newton-CG']
def f(x):
if first_step_size[0] is None and x[0] != x0[0]:
first_step_size[0] = abs(x[0] - x0[0])
if abs(x).max() > 1e4:
raise AssertionError("Optimization stepped far away!")
return scale*(x[0] - 1)**2
def g(x):
return np.array([scale*(x[0] - 1)])
for scale, method in itertools.product(scales, methods):
if method in ('CG', 'BFGS'):
options = dict(gtol=scale*1e-8)
else:
options = dict()
if scale < 1e-10 and method in ('L-BFGS-B', 'Newton-CG'):
# XXX: return initial point if they see small gradient
continue
x0 = [-1.0]
first_step_size = [None]
res = optimize.minimize(f, x0, jac=g, method=method,
options=options)
err_msg = "{0} {1}: {2}: {3}".format(method, scale,
first_step_size,
res)
assert res.success, err_msg
assert_allclose(res.x, [1.0], err_msg=err_msg)
assert res.nit <= 3, err_msg
if scale > 1e-10:
if method in ('CG', 'BFGS'):
assert_allclose(first_step_size[0], 1.01, err_msg=err_msg)
else:
# Newton-CG and L-BFGS-B use different logic for the first
# step, but are both scaling invariant with step sizes ~ 1
assert first_step_size[0] > 0.5 and first_step_size[0] < 3, err_msg
else:
# step size has upper bound of ||grad||, so line
# search makes many small steps
pass
@pytest.mark.parametrize('method', ['nelder-mead', 'powell', 'cg', 'bfgs',
'newton-cg', 'l-bfgs-b', 'tnc',
'cobyla', 'slsqp', 'trust-constr',
'dogleg', 'trust-ncg', 'trust-exact',
'trust-krylov'])
def test_nan_values(self, method):
# Check nan values result to failed exit status
np.random.seed(1234)
count = [0]
def func(x):
return np.nan
def func2(x):
count[0] += 1
if count[0] > 2:
return np.nan
else:
return np.random.rand()
def grad(x):
return np.array([1.0])
def hess(x):
return np.array([[1.0]])
x0 = np.array([1.0])
needs_grad = method in ('newton-cg', 'trust-krylov', 'trust-exact',
'trust-ncg', 'dogleg')
needs_hess = method in ('trust-krylov', 'trust-exact', 'trust-ncg',
'dogleg')
funcs = [func, func2]
grads = [grad] if needs_grad else [grad, None]
hesss = [hess] if needs_hess else [hess, None]
options = dict(maxfun=20) if method == 'tnc' else dict(maxiter=20)
with np.errstate(invalid='ignore'), suppress_warnings() as sup:
sup.filter(UserWarning, "delta_grad == 0.*")
sup.filter(RuntimeWarning, ".*does not use Hessian.*")
sup.filter(RuntimeWarning, ".*does not use gradient.*")
for f, g, h in itertools.product(funcs, grads, hesss):
count = [0]
sol = optimize.minimize(f, x0, jac=g, hess=h, method=method,
options=options)
assert_equal(sol.success, False)
@pytest.mark.parametrize('method', ['nelder-mead', 'cg', 'bfgs',
'l-bfgs-b', 'tnc',
'cobyla', 'slsqp', 'trust-constr',
'dogleg', 'trust-ncg', 'trust-exact',
'trust-krylov'])
def test_duplicate_evaluations(self, method):
# check that there are no duplicate evaluations for any methods
jac = hess = None
if method in ('newton-cg', 'trust-krylov', 'trust-exact',
'trust-ncg', 'dogleg'):
jac = self.grad
if method in ('trust-krylov', 'trust-exact', 'trust-ncg',
'dogleg'):
hess = self.hess
with np.errstate(invalid='ignore'), suppress_warnings() as sup:
# for trust-constr
sup.filter(UserWarning, "delta_grad == 0.*")
optimize.minimize(self.func, self.startparams,
method=method, jac=jac, hess=hess)
for i in range(1, len(self.trace)):
if np.array_equal(self.trace[i - 1], self.trace[i]):
raise RuntimeError(
"Duplicate evaluations made by {}".format(method))
@pytest.mark.parametrize(
'method',
['l-bfgs-b', 'tnc', 'Powell', 'Nelder-Mead']
)
def test_minimize_with_scalar(method):
# checks that minimize works with a scalar being provided to it.
def f(x):
return np.sum(x ** 2)
res = optimize.minimize(f, 17, bounds=[(-100, 100)], method=method)
assert res.success
assert_allclose(res.x, [0.0], atol=1e-5)
class TestLBFGSBBounds:
def setup_method(self):
self.bounds = ((1, None), (None, None))
self.solution = (1, 0)
def fun(self, x, p=2.0):
return 1.0 / p * (x[0]**p + x[1]**p)
def jac(self, x, p=2.0):
return x**(p - 1)
def fj(self, x, p=2.0):
return self.fun(x, p), self.jac(x, p)
def test_l_bfgs_b_bounds(self):
x, f, d = optimize.fmin_l_bfgs_b(self.fun, [0, -1],
fprime=self.jac,
bounds=self.bounds)
assert d['warnflag'] == 0, d['task']
assert_allclose(x, self.solution, atol=1e-6)
def test_l_bfgs_b_funjac(self):
# L-BFGS-B with fun and jac combined and extra arguments
x, f, d = optimize.fmin_l_bfgs_b(self.fj, [0, -1], args=(2.0, ),
bounds=self.bounds)
assert d['warnflag'] == 0, d['task']
assert_allclose(x, self.solution, atol=1e-6)
def test_minimize_l_bfgs_b_bounds(self):
# Minimize with method='L-BFGS-B' with bounds
res = optimize.minimize(self.fun, [0, -1], method='L-BFGS-B',
jac=self.jac, bounds=self.bounds)
assert res['success'], res['message']
assert_allclose(res.x, self.solution, atol=1e-6)
@pytest.mark.parametrize('bounds', [
([(10, 1), (1, 10)]),
([(1, 10), (10, 1)]),
([(10, 1), (10, 1)])
])
def test_minimize_l_bfgs_b_incorrect_bounds(self, bounds):
with pytest.raises(ValueError, match='.*bounds.*'):
optimize.minimize(self.fun, [0, -1], method='L-BFGS-B',
jac=self.jac, bounds=bounds)
def test_minimize_l_bfgs_b_bounds_FD(self):
# test that initial starting value outside bounds doesn't raise
# an error (done with clipping).
# test all different finite differences combos, with and without args
jacs = ['2-point', '3-point', None]
argss = [(2.,), ()]
for jac, args in itertools.product(jacs, argss):
res = optimize.minimize(self.fun, [0, -1], args=args,
method='L-BFGS-B',
jac=jac, bounds=self.bounds,
options={'finite_diff_rel_step': None})
assert res['success'], res['message']
assert_allclose(res.x, self.solution, atol=1e-6)
class TestOptimizeScalar:
def setup_method(self):
self.solution = 1.5
def fun(self, x, a=1.5):
"""Objective function"""
return (x - a)**2 - 0.8
def test_brent(self):
x = optimize.brent(self.fun)
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.brent(self.fun, brack=(-3, -2))
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.brent(self.fun, full_output=True)
assert_allclose(x[0], self.solution, atol=1e-6)
x = optimize.brent(self.fun, brack=(-15, -1, 15))
assert_allclose(x, self.solution, atol=1e-6)
message = r"\(f\(xb\) < f\(xa\)\) and \(f\(xb\) < f\(xc\)\)"
with pytest.raises(ValueError, match=message):
optimize.brent(self.fun, brack=(-1, 0, 1))
message = r"\(xa < xb\) and \(xb < xc\)"
with pytest.raises(ValueError, match=message):
optimize.brent(self.fun, brack=(0, -1, 1))
def test_golden(self):
x = optimize.golden(self.fun)
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.golden(self.fun, brack=(-3, -2))
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.golden(self.fun, full_output=True)
assert_allclose(x[0], self.solution, atol=1e-6)
x = optimize.golden(self.fun, brack=(-15, -1, 15))
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.golden(self.fun, tol=0)
assert_allclose(x, self.solution)
maxiter_test_cases = [0, 1, 5]
for maxiter in maxiter_test_cases:
x0 = optimize.golden(self.fun, maxiter=0, full_output=True)
x = optimize.golden(self.fun, maxiter=maxiter, full_output=True)
nfev0, nfev = x0[2], x[2]
assert_equal(nfev - nfev0, maxiter)
message = r"\(f\(xb\) < f\(xa\)\) and \(f\(xb\) < f\(xc\)\)"
with pytest.raises(ValueError, match=message):
optimize.golden(self.fun, brack=(-1, 0, 1))
message = r"\(xa < xb\) and \(xb < xc\)"
with pytest.raises(ValueError, match=message):
optimize.golden(self.fun, brack=(0, -1, 1))
def test_fminbound(self):
x = optimize.fminbound(self.fun, 0, 1)
assert_allclose(x, 1, atol=1e-4)
x = optimize.fminbound(self.fun, 1, 5)
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.fminbound(self.fun, np.array([1]), np.array([5]))
assert_allclose(x, self.solution, atol=1e-6)
assert_raises(ValueError, optimize.fminbound, self.fun, 5, 1)
def test_fminbound_scalar(self):
with pytest.raises(ValueError, match='.*must be finite scalars.*'):
optimize.fminbound(self.fun, np.zeros((1, 2)), 1)
x = optimize.fminbound(self.fun, 1, np.array(5))
assert_allclose(x, self.solution, atol=1e-6)
def test_gh11207(self):
def fun(x):
return x**2
optimize.fminbound(fun, 0, 0)
def test_minimize_scalar(self):
# combine all tests above for the minimize_scalar wrapper
x = optimize.minimize_scalar(self.fun).x
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.minimize_scalar(self.fun, method='Brent')
assert x.success
x = optimize.minimize_scalar(self.fun, method='Brent',
options=dict(maxiter=3))
assert not x.success
x = optimize.minimize_scalar(self.fun, bracket=(-3, -2),
args=(1.5, ), method='Brent').x
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.minimize_scalar(self.fun, method='Brent',
args=(1.5,)).x
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.minimize_scalar(self.fun, bracket=(-15, -1, 15),
args=(1.5, ), method='Brent').x
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.minimize_scalar(self.fun, bracket=(-3, -2),
args=(1.5, ), method='golden').x
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.minimize_scalar(self.fun, method='golden',
args=(1.5,)).x
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.minimize_scalar(self.fun, bracket=(-15, -1, 15),
args=(1.5, ), method='golden').x
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.minimize_scalar(self.fun, bounds=(0, 1), args=(1.5,),
method='Bounded').x
assert_allclose(x, 1, atol=1e-4)
x = optimize.minimize_scalar(self.fun, bounds=(1, 5), args=(1.5, ),
method='bounded').x
assert_allclose(x, self.solution, atol=1e-6)
x = optimize.minimize_scalar(self.fun, bounds=(np.array([1]),
np.array([5])),
args=(np.array([1.5]), ),
method='bounded').x
assert_allclose(x, self.solution, atol=1e-6)
assert_raises(ValueError, optimize.minimize_scalar, self.fun,
bounds=(5, 1), method='bounded', args=(1.5, ))
assert_raises(ValueError, optimize.minimize_scalar, self.fun,
bounds=(np.zeros(2), 1), method='bounded', args=(1.5, ))
x = optimize.minimize_scalar(self.fun, bounds=(1, np.array(5)),
method='bounded').x
assert_allclose(x, self.solution, atol=1e-6)
def test_minimize_scalar_custom(self):
# This function comes from the documentation example.
def custmin(fun, bracket, args=(), maxfev=None, stepsize=0.1,
maxiter=100, callback=None, **options):
bestx = (bracket[1] + bracket[0]) / 2.0
besty = fun(bestx)
funcalls = 1
niter = 0
improved = True
stop = False
while improved and not stop and niter < maxiter:
improved = False
niter += 1
for testx in [bestx - stepsize, bestx + stepsize]:
testy = fun(testx, *args)
funcalls += 1
if testy < besty:
besty = testy
bestx = testx
improved = True
if callback is not None:
callback(bestx)
if maxfev is not None and funcalls >= maxfev:
stop = True
break
return optimize.OptimizeResult(fun=besty, x=bestx, nit=niter,
nfev=funcalls, success=(niter > 1))
res = optimize.minimize_scalar(self.fun, bracket=(0, 4),
method=custmin,
options=dict(stepsize=0.05))
assert_allclose(res.x, self.solution, atol=1e-6)
def test_minimize_scalar_coerce_args_param(self):
# Regression test for gh-3503
optimize.minimize_scalar(self.fun, args=1.5)
@pytest.mark.parametrize('method', ['brent', 'bounded', 'golden'])
def test_disp(self, method):
# test that all minimize_scalar methods accept a disp option.
for disp in [0, 1, 2, 3]:
optimize.minimize_scalar(self.fun, options={"disp": disp})
@pytest.mark.parametrize('method', ['brent', 'bounded', 'golden'])
def test_result_attributes(self, method):
kwargs = {"bounds": [-10, 10]} if method == 'bounded' else {}
result = optimize.minimize_scalar(self.fun, method=method, **kwargs)
assert hasattr(result, "x")
assert hasattr(result, "success")
assert hasattr(result, "message")
assert hasattr(result, "fun")
assert hasattr(result, "nfev")
assert hasattr(result, "nit")
@pytest.mark.parametrize('method', ['brent', 'bounded', 'golden'])
def test_nan_values(self, method):
# Check nan values result to failed exit status
np.random.seed(1234)
count = [0]
def func(x):
count[0] += 1
if count[0] > 4:
return np.nan
else:
return x**2 + 0.1 * np.sin(x)
bracket = (-1, 0, 1)
bounds = (-1, 1)
with np.errstate(invalid='ignore'), suppress_warnings() as sup:
sup.filter(UserWarning, "delta_grad == 0.*")
sup.filter(RuntimeWarning, ".*does not use Hessian.*")
sup.filter(RuntimeWarning, ".*does not use gradient.*")
count = [0]
kwargs = {"bounds": bounds} if method == 'bounded' else {}
sol = optimize.minimize_scalar(func, bracket=bracket,
**kwargs, method=method,
options=dict(maxiter=20))
assert_equal(sol.success, False)
def test_minimize_scalar_defaults_gh10911(self):
# Previously, bounds were silently ignored unless `method='bounds'`
# was chosen. See gh-10911. Check that this is no longer the case.
def f(x):
return x**2
res = optimize.minimize_scalar(f)
assert_allclose(res.x, 0, atol=1e-8)
res = optimize.minimize_scalar(f, bounds=(1, 100),
options={'xatol': 1e-10})
assert_allclose(res.x, 1)
def test_minimize_non_finite_bounds_gh10911(self):
# Previously, minimize_scalar misbehaved with infinite bounds.
# See gh-10911. Check that it now raises an error, instead.
msg = "Optimization bounds must be finite scalars."
with pytest.raises(ValueError, match=msg):
optimize.minimize_scalar(np.sin, bounds=(1, np.inf))
with pytest.raises(ValueError, match=msg):
optimize.minimize_scalar(np.sin, bounds=(np.nan, 1))
@pytest.mark.parametrize("method", ['brent', 'golden'])
def test_minimize_unbounded_method_with_bounds_gh10911(self, method):
# Previously, `bounds` were silently ignored when `method='brent'` or
# `method='golden'`. See gh-10911. Check that error is now raised.
msg = "Use of `bounds` is incompatible with..."
with pytest.raises(ValueError, match=msg):
optimize.minimize_scalar(np.sin, method=method, bounds=(1, 2))
def test_brent_negative_tolerance():
assert_raises(ValueError, optimize.brent, np.cos, tol=-.01)
class TestNewtonCg:
def test_rosenbrock(self):
x0 = np.array([-1.2, 1.0])
sol = optimize.minimize(optimize.rosen, x0,
jac=optimize.rosen_der,
hess=optimize.rosen_hess,
tol=1e-5,
method='Newton-CG')
assert sol.success, sol.message
assert_allclose(sol.x, np.array([1, 1]), rtol=1e-4)
def test_himmelblau(self):
x0 = np.array(himmelblau_x0)
sol = optimize.minimize(himmelblau,
x0,
jac=himmelblau_grad,
hess=himmelblau_hess,
method='Newton-CG',
tol=1e-6)
assert sol.success, sol.message
assert_allclose(sol.x, himmelblau_xopt, rtol=1e-4)
assert_allclose(sol.fun, himmelblau_min, atol=1e-4)
def test_finite_difference(self):
x0 = np.array([-1.2, 1.0])
sol = optimize.minimize(optimize.rosen, x0,
jac=optimize.rosen_der,
hess='2-point',
tol=1e-5,
method='Newton-CG')
assert sol.success, sol.message
assert_allclose(sol.x, np.array([1, 1]), rtol=1e-4)
def test_hessian_update_strategy(self):
x0 = np.array([-1.2, 1.0])
sol = optimize.minimize(optimize.rosen, x0,
jac=optimize.rosen_der,
hess=optimize.BFGS(),
tol=1e-5,
method='Newton-CG')
assert sol.success, sol.message
assert_allclose(sol.x, np.array([1, 1]), rtol=1e-4)
def test_line_for_search():
# _line_for_search is only used in _linesearch_powell, which is also
# tested below. Thus there are more tests of _line_for_search in the
# test_linesearch_powell_bounded function.
line_for_search = optimize._optimize._line_for_search
# args are x0, alpha, lower_bound, upper_bound
# returns lmin, lmax
lower_bound = np.array([-5.3, -1, -1.5, -3])
upper_bound = np.array([1.9, 1, 2.8, 3])
# test when starting in the bounds
x0 = np.array([0., 0, 0, 0])
# and when starting outside of the bounds
x1 = np.array([0., 2, -3, 0])
all_tests = (
(x0, np.array([1., 0, 0, 0]), -5.3, 1.9),
(x0, np.array([0., 1, 0, 0]), -1, 1),
(x0, np.array([0., 0, 1, 0]), -1.5, 2.8),
(x0, np.array([0., 0, 0, 1]), -3, 3),
(x0, np.array([1., 1, 0, 0]), -1, 1),
(x0, np.array([1., 0, -1, 2]), -1.5, 1.5),
(x0, np.array([2., 0, -1, 2]), -1.5, 0.95),
(x1, np.array([1., 0, 0, 0]), -5.3, 1.9),
(x1, np.array([0., 1, 0, 0]), -3, -1),
(x1, np.array([0., 0, 1, 0]), 1.5, 5.8),
(x1, np.array([0., 0, 0, 1]), -3, 3),
(x1, np.array([1., 1, 0, 0]), -3, -1),
(x1, np.array([1., 0, -1, 0]), -5.3, -1.5),
)
for x, alpha, lmin, lmax in all_tests:
mi, ma = line_for_search(x, alpha, lower_bound, upper_bound)
assert_allclose(mi, lmin, atol=1e-6)
assert_allclose(ma, lmax, atol=1e-6)
# now with infinite bounds
lower_bound = np.array([-np.inf, -1, -np.inf, -3])
upper_bound = np.array([np.inf, 1, 2.8, np.inf])
all_tests = (
(x0, np.array([1., 0, 0, 0]), -np.inf, np.inf),
(x0, np.array([0., 1, 0, 0]), -1, 1),
(x0, np.array([0., 0, 1, 0]), -np.inf, 2.8),
(x0, np.array([0., 0, 0, 1]), -3, np.inf),
(x0, np.array([1., 1, 0, 0]), -1, 1),
(x0, np.array([1., 0, -1, 2]), -1.5, np.inf),
(x1, np.array([1., 0, 0, 0]), -np.inf, np.inf),
(x1, np.array([0., 1, 0, 0]), -3, -1),
(x1, np.array([0., 0, 1, 0]), -np.inf, 5.8),
(x1, np.array([0., 0, 0, 1]), -3, np.inf),
(x1, np.array([1., 1, 0, 0]), -3, -1),
(x1, np.array([1., 0, -1, 0]), -5.8, np.inf),
)
for x, alpha, lmin, lmax in all_tests:
mi, ma = line_for_search(x, alpha, lower_bound, upper_bound)
assert_allclose(mi, lmin, atol=1e-6)
assert_allclose(ma, lmax, atol=1e-6)
def test_linesearch_powell():
# helper function in optimize.py, not a public function.
linesearch_powell = optimize._optimize._linesearch_powell
# args are func, p, xi, fval, lower_bound=None, upper_bound=None, tol=1e-3
# returns new_fval, p + direction, direction
func = lambda x: np.sum((x - np.array([-1., 2., 1.5, -.4]))**2)
p0 = np.array([0., 0, 0, 0])
fval = func(p0)
lower_bound = np.array([-np.inf] * 4)
upper_bound = np.array([np.inf] * 4)
all_tests = (
(np.array([1., 0, 0, 0]), -1),
(np.array([0., 1, 0, 0]), 2),
(np.array([0., 0, 1, 0]), 1.5),
(np.array([0., 0, 0, 1]), -.4),
(np.array([-1., 0, 1, 0]), 1.25),
(np.array([0., 0, 1, 1]), .55),
(np.array([2., 0, -1, 1]), -.65),
)
for xi, l in all_tests:
f, p, direction = linesearch_powell(func, p0, xi,
fval=fval, tol=1e-5)
assert_allclose(f, func(l * xi), atol=1e-6)
assert_allclose(p, l * xi, atol=1e-6)
assert_allclose(direction, l * xi, atol=1e-6)
f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5,
lower_bound=lower_bound,
upper_bound=upper_bound,
fval=fval)
assert_allclose(f, func(l * xi), atol=1e-6)
assert_allclose(p, l * xi, atol=1e-6)
assert_allclose(direction, l * xi, atol=1e-6)
def test_linesearch_powell_bounded():
# helper function in optimize.py, not a public function.
linesearch_powell = optimize._optimize._linesearch_powell
# args are func, p, xi, fval, lower_bound=None, upper_bound=None, tol=1e-3
# returns new_fval, p+direction, direction
func = lambda x: np.sum((x-np.array([-1., 2., 1.5, -.4]))**2)
p0 = np.array([0., 0, 0, 0])
fval = func(p0)
# first choose bounds such that the same tests from
# test_linesearch_powell should pass.
lower_bound = np.array([-2.]*4)
upper_bound = np.array([2.]*4)
all_tests = (
(np.array([1., 0, 0, 0]), -1),
(np.array([0., 1, 0, 0]), 2),
(np.array([0., 0, 1, 0]), 1.5),
(np.array([0., 0, 0, 1]), -.4),
(np.array([-1., 0, 1, 0]), 1.25),
(np.array([0., 0, 1, 1]), .55),
(np.array([2., 0, -1, 1]), -.65),
)
for xi, l in all_tests:
f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5,
lower_bound=lower_bound,
upper_bound=upper_bound,
fval=fval)
assert_allclose(f, func(l * xi), atol=1e-6)
assert_allclose(p, l * xi, atol=1e-6)
assert_allclose(direction, l * xi, atol=1e-6)
# now choose bounds such that unbounded vs bounded gives different results
lower_bound = np.array([-.3]*3 + [-1])
upper_bound = np.array([.45]*3 + [.9])
all_tests = (
(np.array([1., 0, 0, 0]), -.3),
(np.array([0., 1, 0, 0]), .45),
(np.array([0., 0, 1, 0]), .45),
(np.array([0., 0, 0, 1]), -.4),
(np.array([-1., 0, 1, 0]), .3),
(np.array([0., 0, 1, 1]), .45),
(np.array([2., 0, -1, 1]), -.15),
)
for xi, l in all_tests:
f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5,
lower_bound=lower_bound,
upper_bound=upper_bound,
fval=fval)
assert_allclose(f, func(l * xi), atol=1e-6)
assert_allclose(p, l * xi, atol=1e-6)
assert_allclose(direction, l * xi, atol=1e-6)
# now choose as above but start outside the bounds
p0 = np.array([-1., 0, 0, 2])
fval = func(p0)
all_tests = (
(np.array([1., 0, 0, 0]), .7),
(np.array([0., 1, 0, 0]), .45),
(np.array([0., 0, 1, 0]), .45),
(np.array([0., 0, 0, 1]), -2.4),
)
for xi, l in all_tests:
f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5,
lower_bound=lower_bound,
upper_bound=upper_bound,
fval=fval)
assert_allclose(f, func(p0 + l * xi), atol=1e-6)
assert_allclose(p, p0 + l * xi, atol=1e-6)
assert_allclose(direction, l * xi, atol=1e-6)
# now mix in inf
p0 = np.array([0., 0, 0, 0])
fval = func(p0)
# now choose bounds that mix inf
lower_bound = np.array([-.3, -np.inf, -np.inf, -1])
upper_bound = np.array([np.inf, .45, np.inf, .9])
all_tests = (
(np.array([1., 0, 0, 0]), -.3),
(np.array([0., 1, 0, 0]), .45),
(np.array([0., 0, 1, 0]), 1.5),
(np.array([0., 0, 0, 1]), -.4),
(np.array([-1., 0, 1, 0]), .3),
(np.array([0., 0, 1, 1]), .55),
(np.array([2., 0, -1, 1]), -.15),
)
for xi, l in all_tests:
f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5,
lower_bound=lower_bound,
upper_bound=upper_bound,
fval=fval)
assert_allclose(f, func(l * xi), atol=1e-6)
assert_allclose(p, l * xi, atol=1e-6)
assert_allclose(direction, l * xi, atol=1e-6)
# now choose as above but start outside the bounds
p0 = np.array([-1., 0, 0, 2])
fval = func(p0)
all_tests = (
(np.array([1., 0, 0, 0]), .7),
(np.array([0., 1, 0, 0]), .45),
(np.array([0., 0, 1, 0]), 1.5),
(np.array([0., 0, 0, 1]), -2.4),
)
for xi, l in all_tests:
f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5,
lower_bound=lower_bound,
upper_bound=upper_bound,
fval=fval)
assert_allclose(f, func(p0 + l * xi), atol=1e-6)
assert_allclose(p, p0 + l * xi, atol=1e-6)
assert_allclose(direction, l * xi, atol=1e-6)
def test_powell_limits():
# gh15342 - powell was going outside bounds for some function evaluations.
bounds = optimize.Bounds([0, 0], [0.6, 20])
def fun(x):
a, b = x
assert (x >= bounds.lb).all() and (x <= bounds.ub).all()
return a ** 2 + b ** 2
optimize.minimize(fun, x0=[0.6, 20], method='Powell', bounds=bounds)
# Another test from the original report - gh-13411
bounds = optimize.Bounds(lb=[0,], ub=[1,], keep_feasible=[True,])
def func(x):
assert x >= 0 and x <= 1
return np.exp(x)
optimize.minimize(fun=func, x0=[0.5], method='powell', bounds=bounds)
class TestRosen:
def test_hess(self):
# Compare rosen_hess(x) times p with rosen_hess_prod(x,p). See gh-1775.
x = np.array([3, 4, 5])
p = np.array([2, 2, 2])
hp = optimize.rosen_hess_prod(x, p)
dothp = np.dot(optimize.rosen_hess(x), p)
assert_equal(hp, dothp)
def himmelblau(p):
"""
R^2 -> R^1 test function for optimization. The function has four local
minima where himmelblau(xopt) == 0.
"""
x, y = p
a = x*x + y - 11
b = x + y*y - 7
return a*a + b*b
def himmelblau_grad(p):
x, y = p
return np.array([4*x**3 + 4*x*y - 42*x + 2*y**2 - 14,
2*x**2 + 4*x*y + 4*y**3 - 26*y - 22])
def himmelblau_hess(p):
x, y = p
return np.array([[12*x**2 + 4*y - 42, 4*x + 4*y],
[4*x + 4*y, 4*x + 12*y**2 - 26]])
himmelblau_x0 = [-0.27, -0.9]
himmelblau_xopt = [3, 2]
himmelblau_min = 0.0
def test_minimize_multiple_constraints():
# Regression test for gh-4240.
def func(x):
return np.array([25 - 0.2 * x[0] - 0.4 * x[1] - 0.33 * x[2]])
def func1(x):
return np.array([x[1]])
def func2(x):
return np.array([x[2]])
cons = ({'type': 'ineq', 'fun': func},
{'type': 'ineq', 'fun': func1},
{'type': 'ineq', 'fun': func2})
f = lambda x: -1 * (x[0] + x[1] + x[2])
res = optimize.minimize(f, [0, 0, 0], method='SLSQP', constraints=cons)
assert_allclose(res.x, [125, 0, 0], atol=1e-10)
class TestOptimizeResultAttributes:
# Test that all minimizers return an OptimizeResult containing
# all the OptimizeResult attributes
def setup_method(self):
self.x0 = [5, 5]
self.func = optimize.rosen
self.jac = optimize.rosen_der
self.hess = optimize.rosen_hess
self.hessp = optimize.rosen_hess_prod
self.bounds = [(0., 10.), (0., 10.)]
def test_attributes_present(self):
attributes = ['nit', 'nfev', 'x', 'success', 'status', 'fun',
'message']
skip = {'cobyla': ['nit']}
for method in MINIMIZE_METHODS:
with suppress_warnings() as sup:
sup.filter(RuntimeWarning,
("Method .+ does not use (gradient|Hessian.*)"
" information"))
res = optimize.minimize(self.func, self.x0, method=method,
jac=self.jac, hess=self.hess,
hessp=self.hessp)
for attribute in attributes:
if method in skip and attribute in skip[method]:
continue
assert hasattr(res, attribute)
assert attribute in dir(res)
# gh13001, OptimizeResult.message should be a str
assert isinstance(res.message, str)
def f1(z, *params):
x, y = z
a, b, c, d, e, f, g, h, i, j, k, l, scale = params
return (a * x**2 + b * x * y + c * y**2 + d*x + e*y + f)
def f2(z, *params):
x, y = z
a, b, c, d, e, f, g, h, i, j, k, l, scale = params
return (-g*np.exp(-((x-h)**2 + (y-i)**2) / scale))
def f3(z, *params):
x, y = z
a, b, c, d, e, f, g, h, i, j, k, l, scale = params
return (-j*np.exp(-((x-k)**2 + (y-l)**2) / scale))
def brute_func(z, *params):
return f1(z, *params) + f2(z, *params) + f3(z, *params)
class TestBrute:
# Test the "brute force" method
def setup_method(self):
self.params = (2, 3, 7, 8, 9, 10, 44, -1, 2, 26, 1, -2, 0.5)
self.rranges = (slice(-4, 4, 0.25), slice(-4, 4, 0.25))
self.solution = np.array([-1.05665192, 1.80834843])
def brute_func(self, z, *params):
# an instance method optimizing
return brute_func(z, *params)
def test_brute(self):
# test fmin
resbrute = optimize.brute(brute_func, self.rranges, args=self.params,
full_output=True, finish=optimize.fmin)
assert_allclose(resbrute[0], self.solution, atol=1e-3)
assert_allclose(resbrute[1], brute_func(self.solution, *self.params),
atol=1e-3)
# test minimize
resbrute = optimize.brute(brute_func, self.rranges, args=self.params,
full_output=True,
finish=optimize.minimize)
assert_allclose(resbrute[0], self.solution, atol=1e-3)
assert_allclose(resbrute[1], brute_func(self.solution, *self.params),
atol=1e-3)
# test that brute can optimize an instance method (the other tests use
# a non-class based function
resbrute = optimize.brute(self.brute_func, self.rranges,
args=self.params, full_output=True,
finish=optimize.minimize)
assert_allclose(resbrute[0], self.solution, atol=1e-3)
def test_1D(self):
# test that for a 1-D problem the test function is passed an array,
# not a scalar.
def f(x):
assert len(x.shape) == 1
assert x.shape[0] == 1
return x ** 2
optimize.brute(f, [(-1, 1)], Ns=3, finish=None)
def test_workers(self):
# check that parallel evaluation works
resbrute = optimize.brute(brute_func, self.rranges, args=self.params,
full_output=True, finish=None)
resbrute1 = optimize.brute(brute_func, self.rranges, args=self.params,
full_output=True, finish=None, workers=2)
assert_allclose(resbrute1[-1], resbrute[-1])
assert_allclose(resbrute1[0], resbrute[0])
def test_runtime_warning(self):
rng = np.random.default_rng(1234)
def func(z, *params):
return rng.random(1) * 1000 # never converged problem
with pytest.warns(RuntimeWarning,
match=r'Either final optimization did not succeed'):
optimize.brute(func, self.rranges, args=self.params, disp=True)
def test_coerce_args_param(self):
# optimize.brute should coerce non-iterable args to a tuple.
def f(x, *args):
return x ** args[0]
resbrute = optimize.brute(f, (slice(-4, 4, .25),), args=2)
assert_allclose(resbrute, 0)
def test_cobyla_threadsafe():
# Verify that cobyla is threadsafe. Will segfault if it is not.
import concurrent.futures
import time
def objective1(x):
time.sleep(0.1)
return x[0]**2
def objective2(x):
time.sleep(0.1)
return (x[0]-1)**2
min_method = "COBYLA"
def minimizer1():
return optimize.minimize(objective1,
[0.0],
method=min_method)
def minimizer2():
return optimize.minimize(objective2,
[0.0],
method=min_method)
with concurrent.futures.ThreadPoolExecutor() as pool:
tasks = []
tasks.append(pool.submit(minimizer1))
tasks.append(pool.submit(minimizer2))
for t in tasks:
res = t.result()
class TestIterationLimits:
# Tests that optimisation does not give up before trying requested
# number of iterations or evaluations. And that it does not succeed
# by exceeding the limits.
def setup_method(self):
self.funcalls = 0
def slow_func(self, v):
self.funcalls += 1
r, t = np.sqrt(v[0]**2+v[1]**2), np.arctan2(v[0], v[1])
return np.sin(r*20 + t)+r*0.5
def test_neldermead_limit(self):
self.check_limits("Nelder-Mead", 200)
def test_powell_limit(self):
self.check_limits("powell", 1000)
def check_limits(self, method, default_iters):
for start_v in [[0.1, 0.1], [1, 1], [2, 2]]:
for mfev in [50, 500, 5000]:
self.funcalls = 0
res = optimize.minimize(self.slow_func, start_v,
method=method,
options={"maxfev": mfev})
assert self.funcalls == res["nfev"]
if res["success"]:
assert res["nfev"] < mfev
else:
assert res["nfev"] >= mfev
for mit in [50, 500, 5000]:
res = optimize.minimize(self.slow_func, start_v,
method=method,
options={"maxiter": mit})
if res["success"]:
assert res["nit"] <= mit
else:
assert res["nit"] >= mit
for mfev, mit in [[50, 50], [5000, 5000], [5000, np.inf]]:
self.funcalls = 0
res = optimize.minimize(self.slow_func, start_v,
method=method,
options={"maxiter": mit,
"maxfev": mfev})
assert self.funcalls == res["nfev"]
if res["success"]:
assert res["nfev"] < mfev and res["nit"] <= mit
else:
assert res["nfev"] >= mfev or res["nit"] >= mit
for mfev, mit in [[np.inf, None], [None, np.inf]]:
self.funcalls = 0
res = optimize.minimize(self.slow_func, start_v,
method=method,
options={"maxiter": mit,
"maxfev": mfev})
assert self.funcalls == res["nfev"]
if res["success"]:
if mfev is None:
assert res["nfev"] < default_iters*2
else:
assert res["nit"] <= default_iters*2
else:
assert res["nfev"] >= default_iters*2 or res["nit"] >= default_iters*2
def test_result_x_shape_when_len_x_is_one():
def fun(x):
return x * x
def jac(x):
return 2. * x
def hess(x):
return np.array([[2.]])
methods = ['Nelder-Mead', 'Powell', 'CG', 'BFGS', 'L-BFGS-B', 'TNC',
'COBYLA', 'SLSQP']
for method in methods:
res = optimize.minimize(fun, np.array([0.1]), method=method)
assert res.x.shape == (1,)
# use jac + hess
methods = ['trust-constr', 'dogleg', 'trust-ncg', 'trust-exact',
'trust-krylov', 'Newton-CG']
for method in methods:
res = optimize.minimize(fun, np.array([0.1]), method=method, jac=jac,
hess=hess)
assert res.x.shape == (1,)
class FunctionWithGradient:
def __init__(self):
self.number_of_calls = 0
def __call__(self, x):
self.number_of_calls += 1
return np.sum(x**2), 2 * x
@pytest.fixture
def function_with_gradient():
return FunctionWithGradient()
def test_memoize_jac_function_before_gradient(function_with_gradient):
memoized_function = MemoizeJac(function_with_gradient)
x0 = np.array([1.0, 2.0])
assert_allclose(memoized_function(x0), 5.0)
assert function_with_gradient.number_of_calls == 1
assert_allclose(memoized_function.derivative(x0), 2 * x0)
assert function_with_gradient.number_of_calls == 1, \
"function is not recomputed " \
"if gradient is requested after function value"
assert_allclose(
memoized_function(2 * x0), 20.0,
err_msg="different input triggers new computation")
assert function_with_gradient.number_of_calls == 2, \
"different input triggers new computation"
def test_memoize_jac_gradient_before_function(function_with_gradient):
memoized_function = MemoizeJac(function_with_gradient)
x0 = np.array([1.0, 2.0])
assert_allclose(memoized_function.derivative(x0), 2 * x0)
assert function_with_gradient.number_of_calls == 1
assert_allclose(memoized_function(x0), 5.0)
assert function_with_gradient.number_of_calls == 1, \
"function is not recomputed " \
"if function value is requested after gradient"
assert_allclose(
memoized_function.derivative(2 * x0), 4 * x0,
err_msg="different input triggers new computation")
assert function_with_gradient.number_of_calls == 2, \
"different input triggers new computation"
def test_memoize_jac_with_bfgs(function_with_gradient):
""" Tests that using MemoizedJac in combination with ScalarFunction
and BFGS does not lead to repeated function evaluations.
Tests changes made in response to GH11868.
"""
memoized_function = MemoizeJac(function_with_gradient)
jac = memoized_function.derivative
hess = optimize.BFGS()
x0 = np.array([1.0, 0.5])
scalar_function = ScalarFunction(
memoized_function, x0, (), jac, hess, None, None)
assert function_with_gradient.number_of_calls == 1
scalar_function.fun(x0 + 0.1)
assert function_with_gradient.number_of_calls == 2
scalar_function.fun(x0 + 0.2)
assert function_with_gradient.number_of_calls == 3
def test_gh12696():
# Test that optimize doesn't throw warning gh-12696
with assert_no_warnings():
optimize.fminbound(
lambda x: np.array([x**2]), -np.pi, np.pi, disp=False)
# --- Test minimize with equal upper and lower bounds --- #
def setup_test_equal_bounds():
np.random.seed(0)
x0 = np.random.rand(4)
lb = np.array([0, 2, -1, -1.0])
ub = np.array([3, 2, 2, -1.0])
i_eb = (lb == ub)
def check_x(x, check_size=True, check_values=True):
if check_size:
assert x.size == 4
if check_values:
assert_allclose(x[i_eb], lb[i_eb])
def func(x):
check_x(x)
return optimize.rosen(x)
def grad(x):
check_x(x)
return optimize.rosen_der(x)
def callback(x, *args):
check_x(x)
def constraint1(x):
check_x(x, check_values=False)
return x[0:1] - 1
def jacobian1(x):
check_x(x, check_values=False)
dc = np.zeros_like(x)
dc[0] = 1
return dc
def constraint2(x):
check_x(x, check_values=False)
return x[2:3] - 0.5
def jacobian2(x):
check_x(x, check_values=False)
dc = np.zeros_like(x)
dc[2] = 1
return dc
c1a = NonlinearConstraint(constraint1, -np.inf, 0)
c1b = NonlinearConstraint(constraint1, -np.inf, 0, jacobian1)
c2a = NonlinearConstraint(constraint2, -np.inf, 0)
c2b = NonlinearConstraint(constraint2, -np.inf, 0, jacobian2)
# test using the three methods that accept bounds, use derivatives, and
# have some trouble when bounds fix variables
methods = ('L-BFGS-B', 'SLSQP', 'TNC')
# test w/out gradient, w/ gradient, and w/ combined objective/gradient
kwds = ({"fun": func, "jac": False},
{"fun": func, "jac": grad},
{"fun": (lambda x: (func(x), grad(x))),
"jac": True})
# test with both old- and new-style bounds
bound_types = (lambda lb, ub: list(zip(lb, ub)),
Bounds)
# Test for many combinations of constraints w/ and w/out jacobian
# Pairs in format: (test constraints, reference constraints)
# (always use analytical jacobian in reference)
constraints = ((None, None), ([], []),
(c1a, c1b), (c2b, c2b),
([c1b], [c1b]), ([c2a], [c2b]),
([c1a, c2a], [c1b, c2b]),
([c1a, c2b], [c1b, c2b]),
([c1b, c2b], [c1b, c2b]))
# test with and without callback function
callbacks = (None, callback)
data = {"methods": methods, "kwds": kwds, "bound_types": bound_types,
"constraints": constraints, "callbacks": callbacks,
"lb": lb, "ub": ub, "x0": x0, "i_eb": i_eb}
return data
eb_data = setup_test_equal_bounds()
# This test is about handling fixed variables, not the accuracy of the solvers
@pytest.mark.xfail_on_32bit("Failures due to floating point issues, not logic")
@pytest.mark.parametrize('method', eb_data["methods"])
@pytest.mark.parametrize('kwds', eb_data["kwds"])
@pytest.mark.parametrize('bound_type', eb_data["bound_types"])
@pytest.mark.parametrize('constraints', eb_data["constraints"])
@pytest.mark.parametrize('callback', eb_data["callbacks"])
def test_equal_bounds(method, kwds, bound_type, constraints, callback):
"""
Tests that minimizers still work if (bounds.lb == bounds.ub).any()
gh12502 - Divide by zero in Jacobian numerical differentiation when
equality bounds constraints are used
"""
# GH-15051; slightly more skips than necessary; hopefully fixed by GH-14882
if (platform.machine() == 'aarch64' and method == "TNC"
and kwds["jac"] is False and callback is not None):
pytest.skip('Tolerance violation on aarch')
lb, ub = eb_data["lb"], eb_data["ub"]
x0, i_eb = eb_data["x0"], eb_data["i_eb"]
test_constraints, reference_constraints = constraints
if test_constraints and not method == 'SLSQP':
pytest.skip('Only SLSQP supports nonlinear constraints')
# reference constraints always have analytical jacobian
# if test constraints are not the same, we'll need finite differences
fd_needed = (test_constraints != reference_constraints)
bounds = bound_type(lb, ub) # old- or new-style
kwds.update({"x0": x0, "method": method, "bounds": bounds,
"constraints": test_constraints, "callback": callback})
res = optimize.minimize(**kwds)
expected = optimize.minimize(optimize.rosen, x0, method=method,
jac=optimize.rosen_der, bounds=bounds,
constraints=reference_constraints)
# compare the output of a solution with FD vs that of an analytic grad
assert res.success
assert_allclose(res.fun, expected.fun, rtol=1e-6)
assert_allclose(res.x, expected.x, rtol=5e-4)
if fd_needed or kwds['jac'] is False:
expected.jac[i_eb] = np.nan
assert res.jac.shape[0] == 4
assert_allclose(res.jac[i_eb], expected.jac[i_eb], rtol=1e-6)
if not (kwds['jac'] or test_constraints or isinstance(bounds, Bounds)):
# compare the output to an equivalent FD minimization that doesn't
# need factorization
def fun(x):
new_x = np.array([np.nan, 2, np.nan, -1])
new_x[[0, 2]] = x
return optimize.rosen(new_x)
fd_res = optimize.minimize(fun,
x0[[0, 2]],
method=method,
bounds=bounds[::2])
assert_allclose(res.fun, fd_res.fun)
# TODO this test should really be equivalent to factorized version
# above, down to res.nfev. However, testing found that when TNC is
# called with or without a callback the output is different. The two
# should be the same! This indicates that the TNC callback may be
# mutating something when it should't.
assert_allclose(res.x[[0, 2]], fd_res.x, rtol=2e-6)
@pytest.mark.parametrize('method', eb_data["methods"])
def test_all_bounds_equal(method):
# this only tests methods that have parameters factored out when lb==ub
# it does not test other methods that work with bounds
def f(x, p1=1):
return np.linalg.norm(x) + p1
bounds = [(1, 1), (2, 2)]
x0 = (1.0, 3.0)
res = optimize.minimize(f, x0, bounds=bounds, method=method)
assert res.success
assert_allclose(res.fun, f([1.0, 2.0]))
assert res.nfev == 1
assert res.message == 'All independent variables were fixed by bounds.'
args = (2,)
res = optimize.minimize(f, x0, bounds=bounds, method=method, args=args)
assert res.success
assert_allclose(res.fun, f([1.0, 2.0], 2))
if method.upper() == 'SLSQP':
def con(x):
return np.sum(x)
nlc = NonlinearConstraint(con, -np.inf, 0.0)
res = optimize.minimize(
f, x0, bounds=bounds, method=method, constraints=[nlc]
)
assert res.success is False
assert_allclose(res.fun, f([1.0, 2.0]))
assert res.nfev == 1
message = "All independent variables were fixed by bounds, but"
assert res.message.startswith(message)
nlc = NonlinearConstraint(con, -np.inf, 4)
res = optimize.minimize(
f, x0, bounds=bounds, method=method, constraints=[nlc]
)
assert res.success is True
assert_allclose(res.fun, f([1.0, 2.0]))
assert res.nfev == 1
message = "All independent variables were fixed by bounds at values"
assert res.message.startswith(message)
def test_eb_constraints():
# make sure constraint functions aren't overwritten when equal bounds
# are employed, and a parameter is factored out. GH14859
def f(x):
return x[0]**3 + x[1]**2 + x[2]*x[3]
def cfun(x):
return x[0] + x[1] + x[2] + x[3] - 40
constraints = [{'type': 'ineq', 'fun': cfun}]
bounds = [(0, 20)] * 4
bounds[1] = (5, 5)
optimize.minimize(
f,
x0=[1, 2, 3, 4],
method='SLSQP',
bounds=bounds,
constraints=constraints,
)
assert constraints[0]['fun'] == cfun
def test_show_options():
solver_methods = {
'minimize': MINIMIZE_METHODS,
'minimize_scalar': MINIMIZE_SCALAR_METHODS,
'root': ROOT_METHODS,
'root_scalar': ROOT_SCALAR_METHODS,
'linprog': LINPROG_METHODS,
'quadratic_assignment': QUADRATIC_ASSIGNMENT_METHODS,
}
for solver, methods in solver_methods.items():
for method in methods:
# testing that `show_options` works without error
show_options(solver, method)
unknown_solver_method = {
'minimize': "ekki", # unknown method
'maximize': "cg", # unknown solver
'maximize_scalar': "ekki", # unknown solver and method
}
for solver, method in unknown_solver_method.items():
# testing that `show_options` raises ValueError
assert_raises(ValueError, show_options, solver, method)
def test_bounds_with_list():
# gh13501. Bounds created with lists weren't working for Powell.
bounds = optimize.Bounds(lb=[5., 5.], ub=[10., 10.])
optimize.minimize(
optimize.rosen, x0=np.array([9, 9]), method='Powell', bounds=bounds
)
def test_x_overwritten_user_function():
# if the user overwrites the x-array in the user function it's likely
# that the minimizer stops working properly.
# gh13740
def fquad(x):
a = np.arange(np.size(x))
x -= a
x *= x
return np.sum(x)
def fquad_jac(x):
a = np.arange(np.size(x))
x *= 2
x -= 2 * a
return x
fquad_hess = lambda x: np.eye(np.size(x)) * 2.0
meth_jac = [
'newton-cg', 'dogleg', 'trust-ncg', 'trust-exact',
'trust-krylov', 'trust-constr'
]
meth_hess = [
'dogleg', 'trust-ncg', 'trust-exact', 'trust-krylov', 'trust-constr'
]
x0 = np.ones(5) * 1.5
for meth in MINIMIZE_METHODS:
jac = None
hess = None
if meth in meth_jac:
jac = fquad_jac
if meth in meth_hess:
hess = fquad_hess
res = optimize.minimize(fquad, x0, method=meth, jac=jac, hess=hess)
assert_allclose(res.x, np.arange(np.size(x0)), atol=2e-4)
class TestGlobalOptimization:
def test_optimize_result_attributes(self):
def func(x):
return x ** 2
# Note that `brute` solver does not return `OptimizeResult`
results = [optimize.basinhopping(func, x0=1),
optimize.differential_evolution(func, [(-4, 4)]),
optimize.shgo(func, [(-4, 4)]),
optimize.dual_annealing(func, [(-4, 4)]),
optimize.direct(func, [(-4, 4)]),
]
for result in results:
assert isinstance(result, optimize.OptimizeResult)
assert hasattr(result, "x")
assert hasattr(result, "success")
assert hasattr(result, "message")
assert hasattr(result, "fun")
assert hasattr(result, "nfev")
assert hasattr(result, "nit")
def test_approx_fprime():
# check that approx_fprime (serviced by approx_derivative) works for
# jac and hess
g = optimize.approx_fprime(himmelblau_x0, himmelblau)
assert_allclose(g, himmelblau_grad(himmelblau_x0), rtol=5e-6)
h = optimize.approx_fprime(himmelblau_x0, himmelblau_grad)
assert_allclose(h, himmelblau_hess(himmelblau_x0), rtol=5e-6)
def test_gh12594():
# gh-12594 reported an error in `_linesearch_powell` and
# `_line_for_search` when `Bounds` was passed lists instead of arrays.
# Check that results are the same whether the inputs are lists or arrays.
def f(x):
return x[0]**2 + (x[1] - 1)**2
bounds = Bounds(lb=[-10, -10], ub=[10, 10])
res = optimize.minimize(f, x0=(0, 0), method='Powell', bounds=bounds)
bounds = Bounds(lb=np.array([-10, -10]), ub=np.array([10, 10]))
ref = optimize.minimize(f, x0=(0, 0), method='Powell', bounds=bounds)
assert_allclose(res.fun, ref.fun)
assert_allclose(res.x, ref.x)
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