"""
Unit test for SLSQP optimization.
"""
from numpy.testing import (assert_, assert_array_almost_equal,
assert_allclose, assert_equal)
from pytest import raises as assert_raises
import numpy as np
from scipy.optimize import fmin_slsqp, minimize, Bounds
class MyCallBack(object):
"""pass a custom callback function
This makes sure it's being used.
"""
def __init__(self):
self.been_called = False
self.ncalls = 0
def __call__(self, x):
self.been_called = True
self.ncalls += 1
class TestSLSQP(object):
"""
Test SLSQP algorithm using Example 14.4 from Numerical Methods for
Engineers by Steven Chapra and Raymond Canale.
This example maximizes the function f(x) = 2*x*y + 2*x - x**2 - 2*y**2,
which has a maximum at x=2, y=1.
"""
def setup_method(self):
self.opts = {'disp': False}
def fun(self, d, sign=1.0):
"""
Arguments:
d - A list of two elements, where d[0] represents x and d[1] represents y
in the following equation.
sign - A multiplier for f. Since we want to optimize it, and the SciPy
optimizers can only minimize functions, we need to multiply it by
-1 to achieve the desired solution
Returns:
2*x*y + 2*x - x**2 - 2*y**2
"""
x = d[0]
y = d[1]
return sign*(2*x*y + 2*x - x**2 - 2*y**2)
def jac(self, d, sign=1.0):
"""
This is the derivative of fun, returning a NumPy array
representing df/dx and df/dy.
"""
x = d[0]
y = d[1]
dfdx = sign*(-2*x + 2*y + 2)
dfdy = sign*(2*x - 4*y)
return np.array([dfdx, dfdy], float)
def fun_and_jac(self, d, sign=1.0):
return self.fun(d, sign), self.jac(d, sign)
def f_eqcon(self, x, sign=1.0):
""" Equality constraint """
return np.array([x[0] - x[1]])
def fprime_eqcon(self, x, sign=1.0):
""" Equality constraint, derivative """
return np.array([[1, -1]])
def f_eqcon_scalar(self, x, sign=1.0):
""" Scalar equality constraint """
return self.f_eqcon(x, sign)[0]
def fprime_eqcon_scalar(self, x, sign=1.0):
""" Scalar equality constraint, derivative """
return self.fprime_eqcon(x, sign)[0].tolist()
def f_ieqcon(self, x, sign=1.0):
""" Inequality constraint """
return np.array([x[0] - x[1] - 1.0])
def fprime_ieqcon(self, x, sign=1.0):
""" Inequality constraint, derivative """
return np.array([[1, -1]])
def f_ieqcon2(self, x):
""" Vector inequality constraint """
return np.asarray(x)
def fprime_ieqcon2(self, x):
""" Vector inequality constraint, derivative """
return np.identity(x.shape[0])
# minimize
def test_minimize_unbounded_approximated(self):
# Minimize, method='SLSQP': unbounded, approximated jacobian.
jacs = [None, False, '2-point', '3-point']
for jac in jacs:
res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ),
jac=jac, method='SLSQP',
options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [2, 1])
def test_minimize_unbounded_given(self):
# Minimize, method='SLSQP': unbounded, given Jacobian.
res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ),
jac=self.jac, method='SLSQP', options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [2, 1])
def test_minimize_bounded_approximated(self):
# Minimize, method='SLSQP': bounded, approximated jacobian.
jacs = [None, False, '2-point', '3-point']
for jac in jacs:
with np.errstate(invalid='ignore'):
res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ),
jac=jac,
bounds=((2.5, None), (None, 0.5)),
method='SLSQP', options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [2.5, 0.5])
assert_(2.5 <= res.x[0])
assert_(res.x[1] <= 0.5)
def test_minimize_unbounded_combined(self):
# Minimize, method='SLSQP': unbounded, combined function and Jacobian.
res = minimize(self.fun_and_jac, [-1.0, 1.0], args=(-1.0, ),
jac=True, method='SLSQP', options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [2, 1])
def test_minimize_equality_approximated(self):
# Minimize with method='SLSQP': equality constraint, approx. jacobian.
jacs = [None, False, '2-point', '3-point']
for jac in jacs:
res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ),
jac=jac,
constraints={'type': 'eq',
'fun': self.f_eqcon,
'args': (-1.0, )},
method='SLSQP', options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [1, 1])
def test_minimize_equality_given(self):
# Minimize with method='SLSQP': equality constraint, given Jacobian.
res = minimize(self.fun, [-1.0, 1.0], jac=self.jac,
method='SLSQP', args=(-1.0,),
constraints={'type': 'eq', 'fun':self.f_eqcon,
'args': (-1.0, )},
options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [1, 1])
def test_minimize_equality_given2(self):
# Minimize with method='SLSQP': equality constraint, given Jacobian
# for fun and const.
res = minimize(self.fun, [-1.0, 1.0], method='SLSQP',
jac=self.jac, args=(-1.0,),
constraints={'type': 'eq',
'fun': self.f_eqcon,
'args': (-1.0, ),
'jac': self.fprime_eqcon},
options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [1, 1])
def test_minimize_equality_given_cons_scalar(self):
# Minimize with method='SLSQP': scalar equality constraint, given
# Jacobian for fun and const.
res = minimize(self.fun, [-1.0, 1.0], method='SLSQP',
jac=self.jac, args=(-1.0,),
constraints={'type': 'eq',
'fun': self.f_eqcon_scalar,
'args': (-1.0, ),
'jac': self.fprime_eqcon_scalar},
options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [1, 1])
def test_minimize_inequality_given(self):
# Minimize with method='SLSQP': inequality constraint, given Jacobian.
res = minimize(self.fun, [-1.0, 1.0], method='SLSQP',
jac=self.jac, args=(-1.0, ),
constraints={'type': 'ineq',
'fun': self.f_ieqcon,
'args': (-1.0, )},
options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [2, 1], atol=1e-3)
def test_minimize_inequality_given_vector_constraints(self):
# Minimize with method='SLSQP': vector inequality constraint, given
# Jacobian.
res = minimize(self.fun, [-1.0, 1.0], jac=self.jac,
method='SLSQP', args=(-1.0,),
constraints={'type': 'ineq',
'fun': self.f_ieqcon2,
'jac': self.fprime_ieqcon2},
options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [2, 1])
def test_minimize_bound_equality_given2(self):
# Minimize with method='SLSQP': bounds, eq. const., given jac. for
# fun. and const.
res = minimize(self.fun, [-1.0, 1.0], method='SLSQP',
jac=self.jac, args=(-1.0, ),
bounds=[(-0.8, 1.), (-1, 0.8)],
constraints={'type': 'eq',
'fun': self.f_eqcon,
'args': (-1.0, ),
'jac': self.fprime_eqcon},
options=self.opts)
assert_(res['success'], res['message'])
assert_allclose(res.x, [0.8, 0.8], atol=1e-3)
assert_(-0.8 <= res.x[0] <= 1)
assert_(-1 <= res.x[1] <= 0.8)
# fmin_slsqp
def test_unbounded_approximated(self):
# SLSQP: unbounded, approximated Jacobian.
res = fmin_slsqp(self.fun, [-1.0, 1.0], args=(-1.0, ),
iprint = 0, full_output = 1)
x, fx, its, imode, smode = res
assert_(imode == 0, imode)
assert_array_almost_equal(x, [2, 1])
def test_unbounded_given(self):
# SLSQP: unbounded, given Jacobian.
res = fmin_slsqp(self.fun, [-1.0, 1.0], args=(-1.0, ),
fprime = self.jac, iprint = 0,
full_output = 1)
x, fx, its, imode, smode = res
assert_(imode == 0, imode)
assert_array_almost_equal(x, [2, 1])
def test_equality_approximated(self):
# SLSQP: equality constraint, approximated Jacobian.
res = fmin_slsqp(self.fun,[-1.0,1.0], args=(-1.0,),
eqcons = [self.f_eqcon],
iprint = 0, full_output = 1)
x, fx, its, imode, smode = res
assert_(imode == 0, imode)
assert_array_almost_equal(x, [1, 1])
def test_equality_given(self):
# SLSQP: equality constraint, given Jacobian.
res = fmin_slsqp(self.fun, [-1.0, 1.0],
fprime=self.jac, args=(-1.0,),
eqcons = [self.f_eqcon], iprint = 0,
full_output = 1)
x, fx, its, imode, smode = res
assert_(imode == 0, imode)
assert_array_almost_equal(x, [1, 1])
def test_equality_given2(self):
# SLSQP: equality constraint, given Jacobian for fun and const.
res = fmin_slsqp(self.fun, [-1.0, 1.0],
fprime=self.jac, args=(-1.0,),
f_eqcons = self.f_eqcon,
fprime_eqcons = self.fprime_eqcon,
iprint = 0,
full_output = 1)
x, fx, its, imode, smode = res
assert_(imode == 0, imode)
assert_array_almost_equal(x, [1, 1])
def test_inequality_given(self):
# SLSQP: inequality constraint, given Jacobian.
res = fmin_slsqp(self.fun, [-1.0, 1.0],
fprime=self.jac, args=(-1.0, ),
ieqcons = [self.f_ieqcon],
iprint = 0, full_output = 1)
x, fx, its, imode, smode = res
assert_(imode == 0, imode)
assert_array_almost_equal(x, [2, 1], decimal=3)
def test_bound_equality_given2(self):
# SLSQP: bounds, eq. const., given jac. for fun. and const.
res = fmin_slsqp(self.fun, [-1.0, 1.0],
fprime=self.jac, args=(-1.0, ),
bounds = [(-0.8, 1.), (-1, 0.8)],
f_eqcons = self.f_eqcon,
fprime_eqcons = self.fprime_eqcon,
iprint = 0, full_output = 1)
x, fx, its, imode, smode = res
assert_(imode == 0, imode)
assert_array_almost_equal(x, [0.8, 0.8], decimal=3)
assert_(-0.8 <= x[0] <= 1)
assert_(-1 <= x[1] <= 0.8)
def test_scalar_constraints(self):
# Regression test for gh-2182
x = fmin_slsqp(lambda z: z**2, [3.],
ieqcons=[lambda z: z[0] - 1],
iprint=0)
assert_array_almost_equal(x, [1.])
x = fmin_slsqp(lambda z: z**2, [3.],
f_ieqcons=lambda z: [z[0] - 1],
iprint=0)
assert_array_almost_equal(x, [1.])
def test_integer_bounds(self):
# This should not raise an exception
fmin_slsqp(lambda z: z**2 - 1, [0], bounds=[[0, 1]], iprint=0)
def test_array_bounds(self):
# This should pass (1-element arrays are castable to floats in numpy)
bounds = [(-np.inf, np.inf), (np.array([2]), np.array([3]))]
x = fmin_slsqp(lambda z: np.sum(z**2 - 1), [2.5, 2.5], bounds=bounds, iprint=0)
assert_array_almost_equal(x, [0, 2])
def test_obj_must_return_scalar(self):
# Regression test for Github Issue #5433
# If objective function does not return a scalar, raises ValueError
with assert_raises(ValueError):
fmin_slsqp(lambda x: [0, 1], [1, 2, 3])
def test_obj_returns_scalar_in_list(self):
# Test for Github Issue #5433 and PR #6691
# Objective function should be able to return length-1 Python list
# containing the scalar
fmin_slsqp(lambda x: [0], [1, 2, 3], iprint=0)
def test_callback(self):
# Minimize, method='SLSQP': unbounded, approximated jacobian. Check for callback
callback = MyCallBack()
res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ),
method='SLSQP', callback=callback, options=self.opts)
assert_(res['success'], res['message'])
assert_(callback.been_called)
assert_equal(callback.ncalls, res['nit'])
def test_inconsistent_linearization(self):
# SLSQP must be able to solve this problem, even if the
# linearized problem at the starting point is infeasible.
# Linearized constraints are
#
# 2*x0[0]*x[0] >= 1
#
# At x0 = [0, 1], the second constraint is clearly infeasible.
# This triggers a call with n2==1 in the LSQ subroutine.
x = [0, 1]
f1 = lambda x: x[0] + x[1] - 2
f2 = lambda x: x[0]**2 - 1
sol = minimize(
lambda x: x[0]**2 + x[1]**2,
x,
constraints=({'type':'eq','fun': f1},
{'type':'ineq','fun': f2}),
bounds=((0,None), (0,None)),
method='SLSQP')
x = sol.x
assert_allclose(f1(x), 0, atol=1e-8)
assert_(f2(x) >= -1e-8)
assert_(sol.success, sol)
def test_regression_5743(self):
# SLSQP must not indicate success for this problem,
# which is infeasible.
x = [1, 2]
sol = minimize(
lambda x: x[0]**2 + x[1]**2,
x,
constraints=({'type':'eq','fun': lambda x: x[0]+x[1]-1},
{'type':'ineq','fun': lambda x: x[0]-2}),
bounds=((0,None), (0,None)),
method='SLSQP')
assert_(not sol.success, sol)
def test_gh_6676(self):
def func(x):
return (x[0] - 1)**2 + 2*(x[1] - 1)**2 + 0.5*(x[2] - 1)**2
sol = minimize(func, [0, 0, 0], method='SLSQP')
assert_(sol.jac.shape == (3,))
def test_invalid_bounds(self):
# Raise correct error when lower bound is greater than upper bound.
# See Github issue 6875.
bounds_list = [
((1, 2), (2, 1)),
((2, 1), (1, 2)),
((2, 1), (2, 1)),
((np.inf, 0), (np.inf, 0)),
((1, -np.inf), (0, 1)),
]
for bounds in bounds_list:
with assert_raises(ValueError):
minimize(self.fun, [-1.0, 1.0], bounds=bounds, method='SLSQP')
def test_bounds_clipping(self):
#
# SLSQP returns bogus results for initial guess out of bounds, gh-6859
#
def f(x):
return (x[0] - 1)**2
sol = minimize(f, [10], method='slsqp', bounds=[(None, 0)])
assert_(sol.success)
assert_allclose(sol.x, 0, atol=1e-10)
sol = minimize(f, [-10], method='slsqp', bounds=[(2, None)])
assert_(sol.success)
assert_allclose(sol.x, 2, atol=1e-10)
sol = minimize(f, [-10], method='slsqp', bounds=[(None, 0)])
assert_(sol.success)
assert_allclose(sol.x, 0, atol=1e-10)
sol = minimize(f, [10], method='slsqp', bounds=[(2, None)])
assert_(sol.success)
assert_allclose(sol.x, 2, atol=1e-10)
sol = minimize(f, [-0.5], method='slsqp', bounds=[(-1, 0)])
assert_(sol.success)
assert_allclose(sol.x, 0, atol=1e-10)
sol = minimize(f, [10], method='slsqp', bounds=[(-1, 0)])
assert_(sol.success)
assert_allclose(sol.x, 0, atol=1e-10)
def test_infeasible_initial(self):
# Check SLSQP behavior with infeasible initial point
def f(x):
x, = x
return x*x - 2*x + 1
cons_u = [{'type': 'ineq', 'fun': lambda x: 0 - x}]
cons_l = [{'type': 'ineq', 'fun': lambda x: x - 2}]
cons_ul = [{'type': 'ineq', 'fun': lambda x: 0 - x},
{'type': 'ineq', 'fun': lambda x: x + 1}]
sol = minimize(f, [10], method='slsqp', constraints=cons_u)
assert_(sol.success)
assert_allclose(sol.x, 0, atol=1e-10)
sol = minimize(f, [-10], method='slsqp', constraints=cons_l)
assert_(sol.success)
assert_allclose(sol.x, 2, atol=1e-10)
sol = minimize(f, [-10], method='slsqp', constraints=cons_u)
assert_(sol.success)
assert_allclose(sol.x, 0, atol=1e-10)
sol = minimize(f, [10], method='slsqp', constraints=cons_l)
assert_(sol.success)
assert_allclose(sol.x, 2, atol=1e-10)
sol = minimize(f, [-0.5], method='slsqp', constraints=cons_ul)
assert_(sol.success)
assert_allclose(sol.x, 0, atol=1e-10)
sol = minimize(f, [10], method='slsqp', constraints=cons_ul)
assert_(sol.success)
assert_allclose(sol.x, 0, atol=1e-10)
def test_inconsistent_inequalities(self):
# gh-7618
def cost(x):
return -1 * x[0] + 4 * x[1]
def ineqcons1(x):
return x[1] - x[0] - 1
def ineqcons2(x):
return x[0] - x[1]
# The inequalities are inconsistent, so no solution can exist:
#
# x1 >= x0 + 1
# x0 >= x1
x0 = (1,5)
bounds = ((-5, 5), (-5, 5))
cons = (dict(type='ineq', fun=ineqcons1), dict(type='ineq', fun=ineqcons2))
res = minimize(cost, x0, method='SLSQP', bounds=bounds, constraints=cons)
assert_(not res.success)
def test_new_bounds_type(self):
f = lambda x: x[0]**2 + x[1]**2
bounds = Bounds([1, 0], [np.inf, np.inf])
sol = minimize(f, [0, 0], method='slsqp', bounds=bounds)
assert_(sol.success)
assert_allclose(sol.x, [1, 0])
def test_nested_minimization(self):
class NestedProblem():
def __init__(self):
self.F_outer_count = 0
def F_outer(self, x):
self.F_outer_count += 1
if self.F_outer_count > 1000:
raise Exception("Nested minimization failed to terminate.")
inner_res = minimize(self.F_inner, (3, 4), method="SLSQP")
assert_(inner_res.success)
assert_allclose(inner_res.x, [1, 1])
return x[0]**2 + x[1]**2 + x[2]**2
def F_inner(self, x):
return (x[0] - 1)**2 + (x[1] - 1)**2
def solve(self):
outer_res = minimize(self.F_outer, (5, 5, 5), method="SLSQP")
assert_(outer_res.success)
assert_allclose(outer_res.x, [0, 0, 0])
problem = NestedProblem()
problem.solve()
def test_gh1758(self):
# the test suggested in gh1758
# https://nlopt.readthedocs.io/en/latest/NLopt_Tutorial/
# implement two equality constraints, in R^2.
def fun(x):
return np.sqrt(x[1])
def f_eqcon(x):
""" Equality constraint """
return x[1] - (2 * x[0]) ** 3
def f_eqcon2(x):
""" Equality constraint """
return x[1] - (-x[0] + 1) ** 3
c1 = {'type': 'eq', 'fun': f_eqcon}
c2 = {'type': 'eq', 'fun': f_eqcon2}
res = minimize(fun, [8, 0.25], method='SLSQP',
constraints=[c1, c2], bounds=[(-0.5, 1), (0, 8)])
np.testing.assert_allclose(res.fun, 0.5443310539518)
np.testing.assert_allclose(res.x, [0.33333333, 0.2962963])
assert res.success
def test_gh9640(self):
np.random.seed(10)
cons = ({'type': 'ineq', 'fun': lambda x: -x[0] - x[1] - 3},
{'type': 'ineq', 'fun': lambda x: x[1] + x[2] - 2})
bnds = ((-2, 2), (-2, 2), (-2, 2))
target = lambda x: 1
x0 = [-1.8869783504471584, -0.640096352696244, -0.8174212253407696]
res = minimize(target, x0, method='SLSQP', bounds=bnds, constraints=cons,
options={'disp':False, 'maxiter':10000})
# The problem is infeasible, so it cannot succeed
assert not res.success