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test_optimize.py
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test_optimize.py
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"""
Unit tests for optimization routines from optimize.py and tnc.py
Authors:
Ed Schofield, Nov 2005
Andrew Straw, April 2008
To run it in its simplest form::
nosetests test_optimize.py
"""
from numpy.testing import assert_raises, assert_almost_equal, \
assert_equal, assert_, TestCase, run_module_suite
from scipy import optimize
import numpy as np
from scipy.optimize.tnc import RCSTRINGS, MSG_NONE
from math import pow
class TestOptimize(TestCase):
""" 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(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(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)
def test_cg(self, use_wrapper=False):
""" conjugate gradient optimization routine """
if use_wrapper:
opts = {'maxit': self.maxiter, 'disp': False}
params, info = optimize.minimize(self.func, self.startparams,
args=(), method='CG',
jac=self.grad, options=opts,
full_output=True,
retall=False)
fopt, func_calls, grad_calls, warnflag = \
info['fun'], info['nfev'], info['njev'], info['status']
else:
retval = optimize.fmin_cg(self.func, self.startparams, self.grad, (),
maxiter=self.maxiter,
full_output=True, disp=False, retall=False)
(params, fopt, func_calls, grad_calls, warnflag) = retval
err = abs(self.func(params) - self.func(self.solution))
#print "CG: Difference is: " + str(err)
assert_(err < 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_(np.allclose(self.trace[2:4],
[[0, -0.5, 0.5],
[0, -5.05700028e-01, 4.95985862e-01]],
atol=1e-14, rtol=1e-7), self.trace[2:4])
def test_bfgs(self, use_wrapper=False):
""" Broyden-Fletcher-Goldfarb-Shanno optimization routine """
if use_wrapper:
opts = {'maxit': self.maxiter, 'disp': False}
params, info = optimize.minimize(self.func, self.startparams,
jac=self.grad, method='BFGS',
args=(), options=opts,
full_output=True,
retall=False)
fopt, gopt, Hopt, func_calls, grad_calls, warnflag = \
info['fun'], info['jac'], info['hess'], info['nfev'], \
info['njev'], info['status']
else:
retval = optimize.fmin_bfgs(self.func, self.startparams, self.grad,
args=(), maxiter=self.maxiter,
full_output=True, disp=False, retall=False)
(params, fopt, gopt, Hopt, func_calls, grad_calls, warnflag) = retval
err = abs(self.func(params) - self.func(self.solution))
#print "BFGS: Difference is: " + str(err)
assert_(err < 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_(np.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), self.trace[6:8])
def test_bfgs_nan(self):
"""Test corner case where nan is fed to optimizer. See #1542."""
func = lambda x: x
fprime = lambda x: np.ones_like(x)
x0 = [np.nan]
olderr = np.seterr(over='ignore')
try:
x = optimize.fmin_bfgs(func, x0, fprime, disp=False)
assert_(np.isnan(func(x)))
finally:
np.seterr(**olderr)
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.finfo(float).eps) * np.random.rand(len(self.solution))
params = optimize.fmin_bfgs(self.func, self.startparams,
epsilon=epsilon, args=(),
maxiter=self.maxiter, disp=False)
err = abs(self.func(params) - self.func(self.solution))
assert_(err < 1e-6, err)
def test_bfgs_infinite(self, use_wrapper=False):
"""Test corner case where -Inf is the minimum. See #1494."""
func = lambda x: -np.e**-x
fprime = lambda x: -func(x)
x0 = [0]
olderr = np.seterr(over='ignore')
try:
if use_wrapper:
opts = {'disp': False}
x = optimize.minimize(func, x0, jac=fprime, method='BFGS',
args=(), options=opts)
else:
x = optimize.fmin_bfgs(func, x0, fprime, disp=False)
assert_(not np.isfinite(func(x)))
finally:
np.seterr(**olderr)
def test_powell(self, use_wrapper=False):
""" Powell (direction set) optimization routine
"""
if use_wrapper:
opts = {'maxit': self.maxiter, 'disp': False}
params, info = optimize.minimize(self.func, self.startparams,
args=(), method='Powell',
options=opts,
full_output=True,
retall=False)
fopt, direc, numiter, func_calls, warnflag = \
info['fun'], info['direc'], info['nit'], info['nfev'], \
info['status']
else:
retval = optimize.fmin_powell(self.func, self.startparams,
args=(), maxiter=self.maxiter,
full_output=True, disp=False, retall=False)
(params, fopt, direc, numiter, func_calls, warnflag) = retval
err = abs(self.func(params) - self.func(self.solution))
#print "Powell: Difference is: " + str(err)
assert_(err < 1e-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)
# Ensure that the function behaves the same; this is from Scipy 0.7.0
assert_(np.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), self.trace[34:39])
def test_neldermead(self, use_wrapper=False):
""" Nelder-Mead simplex algorithm
"""
if use_wrapper:
opts = {'maxit': self.maxiter, 'disp': False}
params, info = optimize.minimize(self.func, self.startparams,
args=(), method='Nelder-mead',
options=opts,
full_output=True,
retall=False)
fopt, numiter, func_calls, warnflag = \
info['fun'], info['nit'], info['nfev'], info['status']
else:
retval = optimize.fmin(self.func, self.startparams,
args=(), maxiter=self.maxiter,
full_output=True, disp=False, retall=False)
(params, fopt, numiter, func_calls, warnflag) = retval
err = abs(self.func(params) - self.func(self.solution))
#print "Nelder-Mead: Difference is: " + str(err)
assert_(err < 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_(np.allclose(self.trace[76:78],
[[0.1928968 , -0.62780447, 0.35166118],
[0.19572515, -0.63648426, 0.35838135]],
atol=1e-14, rtol=1e-7), self.trace[76:78])
def test_ncg(self, use_wrapper=False):
""" line-search Newton conjugate gradient optimization routine
"""
if use_wrapper:
opts = {'maxit': self.maxiter, 'disp': False}
retval = optimize.minimize(self.func, self.startparams,
method='Newton-CG', jac=self.grad,
args=(), options=opts,
full_output=False, retall=False)
else:
retval = optimize.fmin_ncg(self.func, self.startparams, self.grad,
args=(), maxiter=self.maxiter,
full_output=False, disp=False,
retall=False)
params = retval
err = abs(self.func(params) - self.func(self.solution))
#print "NCG: Difference is: " + str(err)
assert_(err < 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 <= 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_(np.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), self.trace[:5])
def test_ncg_hess(self, use_wrapper=False):
""" Newton conjugate gradient with Hessian """
if use_wrapper:
opts = {'maxit': self.maxiter, 'disp': False}
retval = optimize.minimize(self.func, self.startparams,
method='Newton-CG', jac=self.grad,
hess = self.hess,
args=(), options=opts,
full_output=False, retall=False)
else:
retval = optimize.fmin_ncg(self.func, self.startparams, self.grad,
fhess = self.hess,
args=(), maxiter=self.maxiter,
full_output=False, disp=False,
retall=False)
params = retval
err = abs(self.func(params) - self.func(self.solution))
#print "NCG: Difference is: " + str(err)
assert_(err < 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 <= 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_(np.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), self.trace[:5])
def test_ncg_hessp(self, use_wrapper=False):
""" Newton conjugate gradient with Hessian times a vector p """
if use_wrapper:
opts = {'maxit': self.maxiter, 'disp': False}
retval = optimize.minimize(self.func, self.startparams,
method='Newton-CG', jac=self.grad,
hess = self.hessp,
args=(), options=opts,
full_output=False, retall=False)
else:
retval = optimize.fmin_ncg(self.func, self.startparams, self.grad,
fhess_p = self.hessp,
args=(), maxiter=self.maxiter,
full_output=False, disp=False,
retall=False)
params = retval
err = abs(self.func(params) - self.func(self.solution))
#print "NCG: Difference is: " + str(err)
assert_(err < 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 <= 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_(np.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), self.trace[:5])
def test_l_bfgs_b(self, use_wrapper=False):
""" limited-memory bound-constrained BFGS algorithm
"""
if use_wrapper:
opts = {'maxfev': self.maxiter}
retval = optimize.fmincon(self.func, self.startparams,
method='l-BFGS-B', args=(),
jac=self.grad, options=opts,
full_output=True)
params = retval[0]
else:
retval = optimize.fmin_l_bfgs_b(self.func, self.startparams,
self.grad, args=(),
maxfun=self.maxiter)
(params, fopt, d) = retval
err = abs(self.func(params) - self.func(self.solution))
#print "LBFGSB: Difference is: " + str(err)
assert_(err < 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
assert_(np.allclose(self.trace[3:5],
[[0. , -0.52489628, 0.48753042],
[0. , -0.52489628, 0.48753042]],
atol=1e-14, rtol=1e-7), self.trace[3:5])
def test_minimize(self):
"""Tests for the minimize wrapper."""
self.setUp()
self.test_bfgs(True)
self.setUp()
self.test_bfgs_infinite(True)
self.setUp()
self.test_cg(True)
self.setUp()
self.test_ncg(True)
self.setUp()
self.test_ncg_hess(True)
self.setUp()
self.test_ncg_hessp(True)
self.setUp()
self.test_neldermead(True)
self.setUp()
self.test_powell(True)
def test_brent(self):
""" brent algorithm
"""
x = optimize.brent(lambda x: (x-1.5)**2-0.8)
err1 = abs(x - 1.5)
x = optimize.brent(lambda x: (x-1.5)**2-0.8, brack = (-3,-2))
err2 = abs(x - 1.5)
x = optimize.brent(lambda x: (x-1.5)**2-0.8, full_output=True)
err3 = abs(x[0] - 1.5)
x = optimize.brent(lambda x: (x-1.5)**2-0.8, brack = (-15,-1,15))
err4 = abs(x - 1.5)
assert_(max((err1,err2,err3,err4)) < 1e-6)
def test_fminbound(self):
"""Test fminbound
"""
x = optimize.fminbound(lambda x: (x - 1.5)**2 - 0.8, 0, 1)
assert_(abs(x - 1) < 1e-5)
x = optimize.fminbound(lambda x: (x - 1.5)**2 - 0.8, 1, 5)
assert_(abs(x - 1.5) < 1e-6)
x = optimize.fminbound(lambda x: (x - 1.5)**2 - 0.8,
np.array([1]), np.array([5]))
assert_(abs(x - 1.5) < 1e-6)
assert_raises(ValueError,
optimize.fminbound, lambda x: (x - 1.5)**2 - 0.8, 5, 1)
def test_fminbound_scalar(self):
assert_raises(ValueError,
optimize.fminbound, lambda x: (x - 1.5)**2 - 0.8,
np.zeros(2), 1)
assert_almost_equal(
optimize.fminbound(lambda x: (x - 1.5)**2 - 0.8, 1, np.array(5)),
1.5)
class TestTnc(TestCase):
"""TNC non-linear optimization.
These tests are taken from Prof. K. Schittkowski's test examples
for constrained non-linear programming.
http://www.uni-bayreuth.de/departments/math/~kschittkowski/home.htm
"""
tests = []
def setUp(self):
def test1fg(x):
f = 100.0*pow((x[1]-pow(x[0],2)),2)+pow(1.0-x[0],2)
dif = [0,0]
dif[1] = 200.0*(x[1]-pow(x[0],2))
dif[0] = -2.0*(x[0]*(dif[1]-1.0)+1.0)
return f, dif
self.tests.append((test1fg, [-2,1], ([-np.inf,None],[-1.5,None]),
[1,1]))
def test2fg(x):
f = 100.0*pow((x[1]-pow(x[0],2)),2)+pow(1.0-x[0],2)
dif = [0,0]
dif[1] = 200.0*(x[1]-pow(x[0],2))
dif[0] = -2.0*(x[0]*(dif[1]-1.0)+1.0)
return f, dif
self.tests.append((test2fg, [-2,1], [(-np.inf,None),(1.5,None)],
[-1.2210262419616387,1.5]))
def test3fg(x):
f = x[1]+pow(x[1]-x[0],2)*1.0e-5
dif = [0,0]
dif[0] = -2.0*(x[1]-x[0])*1.0e-5
dif[1] = 1.0-dif[0]
return f, dif
self.tests.append((test3fg, [10,1], [(-np.inf,None),(0.0, None)],
[0,0]))
def test4fg(x):
f = pow(x[0]+1.0,3)/3.0+x[1]
dif = [0,0]
dif[0] = pow(x[0]+1.0,2)
dif[1] = 1.0
return f, dif
self.tests.append((test4fg, [1.125,0.125], [(1, None),(0, None)],
[1,0]))
def test5fg(x):
f = np.sin(x[0]+x[1])+pow(x[0]-x[1],2)-1.5*x[0]+2.5*x[1]+1.0
dif = [0,0]
v1 = np.cos(x[0]+x[1])
v2 = 2.0*(x[0]-x[1])
dif[0] = v1+v2-1.5
dif[1] = v1-v2+2.5
return f, dif
self.tests.append((test5fg, [0,0], [(-1.5, 4),(-3,3)],
[-0.54719755119659763, -1.5471975511965976]))
def test38fg(x):
f = (100.0*pow(x[1]-pow(x[0],2),2) + \
pow(1.0-x[0],2)+90.0*pow(x[3]-pow(x[2],2),2) + \
pow(1.0-x[2],2)+10.1*(pow(x[1]-1.0,2)+pow(x[3]-1.0,2)) + \
19.8*(x[1]-1.0)*(x[3]-1.0))*1.0e-5
dif = [0,0,0,0]
dif[0] = (-400.0*x[0]*(x[1]-pow(x[0],2))-2.0*(1.0-x[0]))*1.0e-5
dif[1] = (200.0*(x[1]-pow(x[0],2))+20.2 \
*(x[1]-1.0)+19.8*(x[3]-1.0))*1.0e-5
dif[2] = (-360.0*x[2]*(x[3]-pow(x[2],2))-2.0\
*(1.0-x[2]))*1.0e-5
dif[3] = (180.0*(x[3]-pow(x[2],2))+20.2\
*(x[3]-1.0)+19.8*(x[1]-1.0))*1.0e-5
return f, dif
self.tests.append((test38fg, np.array([-3,-1,-3,-1]), [(-10,10)]*4, [1]*4))
def test45fg(x):
f = 2.0-x[0]*x[1]*x[2]*x[3]*x[4]/120.0
dif = [0]*5
dif[0] = -x[1]*x[2]*x[3]*x[4]/120.0
dif[1] = -x[0]*x[2]*x[3]*x[4]/120.0
dif[2] = -x[0]*x[1]*x[3]*x[4]/120.0
dif[3] = -x[0]*x[1]*x[2]*x[4]/120.0
dif[4] = -x[0]*x[1]*x[2]*x[3]/120.0
return f, dif
self.tests.append((test45fg, [2]*5, [(0,1),(0,2),(0,3),(0,4),(0,5)],
[1,2,3,4,5]))
def test_tnc(self):
for fg, x, bounds, xopt in self.tests:
x, nf, rc = optimize.fmin_tnc(fg, x, bounds=bounds,
messages=MSG_NONE, maxfun=200)
err = "Failed optimization of %s.\n" \
"After %d function evaluations, TNC returned: %s.""" % \
(fg.__name__, nf, RCSTRINGS[rc])
ef = abs(fg(xopt)[0] - fg(x)[0])
if ef > 1e-8:
raise err
class TestRosen(TestCase):
def test_hess(self):
"""Compare rosen_hess(x) times p with rosen_hess_prod(x,p) (ticket #1248)"""
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)
if __name__ == "__main__":
run_module_suite()