565 lines
18 KiB
Python
565 lines
18 KiB
Python
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from __future__ import division, print_function, absolute_import
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import numpy as np
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import pytest
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from scipy.linalg import block_diag
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from scipy.sparse import csc_matrix
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from numpy.testing import (TestCase, assert_array_almost_equal,
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assert_array_less)
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from pytest import raises
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from scipy.optimize import (NonlinearConstraint,
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LinearConstraint,
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Bounds,
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minimize,
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BFGS,
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SR1)
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from scipy._lib._numpy_compat import suppress_warnings
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class Maratos:
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"""Problem 15.4 from Nocedal and Wright
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The following optimization problem:
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minimize 2*(x[0]**2 + x[1]**2 - 1) - x[0]
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Subject to: x[0]**2 + x[1]**2 - 1 = 0
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"""
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def __init__(self, degrees=60, constr_jac=None, constr_hess=None):
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rads = degrees/180*np.pi
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self.x0 = [np.cos(rads), np.sin(rads)]
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self.x_opt = np.array([1.0, 0.0])
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self.constr_jac = constr_jac
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self.constr_hess = constr_hess
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self.bounds = None
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def fun(self, x):
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return 2*(x[0]**2 + x[1]**2 - 1) - x[0]
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def grad(self, x):
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return np.array([4*x[0]-1, 4*x[1]])
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def hess(self, x):
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return 4*np.eye(2)
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@property
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def constr(self):
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def fun(x):
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return x[0]**2 + x[1]**2
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if self.constr_jac is None:
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def jac(x):
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return [[2*x[0], 2*x[1]]]
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else:
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jac = self.constr_jac
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if self.constr_hess is None:
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def hess(x, v):
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return 2*v[0]*np.eye(2)
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else:
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hess = self.constr_hess
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return NonlinearConstraint(fun, 1, 1, jac, hess)
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class MaratosTestArgs:
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"""Problem 15.4 from Nocedal and Wright
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The following optimization problem:
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minimize 2*(x[0]**2 + x[1]**2 - 1) - x[0]
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Subject to: x[0]**2 + x[1]**2 - 1 = 0
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"""
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def __init__(self, a, b, degrees=60, constr_jac=None, constr_hess=None):
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rads = degrees/180*np.pi
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self.x0 = [np.cos(rads), np.sin(rads)]
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self.x_opt = np.array([1.0, 0.0])
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self.constr_jac = constr_jac
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self.constr_hess = constr_hess
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self.a = a
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self.b = b
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self.bounds = None
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def _test_args(self, a, b):
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if self.a != a or self.b != b:
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raise ValueError()
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def fun(self, x, a, b):
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self._test_args(a, b)
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return 2*(x[0]**2 + x[1]**2 - 1) - x[0]
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def grad(self, x, a, b):
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self._test_args(a, b)
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return np.array([4*x[0]-1, 4*x[1]])
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def hess(self, x, a, b):
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self._test_args(a, b)
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return 4*np.eye(2)
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@property
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def constr(self):
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def fun(x):
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return x[0]**2 + x[1]**2
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if self.constr_jac is None:
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def jac(x):
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return [[4*x[0], 4*x[1]]]
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else:
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jac = self.constr_jac
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if self.constr_hess is None:
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def hess(x, v):
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return 2*v[0]*np.eye(2)
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else:
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hess = self.constr_hess
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return NonlinearConstraint(fun, 1, 1, jac, hess)
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class MaratosGradInFunc:
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"""Problem 15.4 from Nocedal and Wright
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The following optimization problem:
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minimize 2*(x[0]**2 + x[1]**2 - 1) - x[0]
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Subject to: x[0]**2 + x[1]**2 - 1 = 0
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"""
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def __init__(self, degrees=60, constr_jac=None, constr_hess=None):
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rads = degrees/180*np.pi
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self.x0 = [np.cos(rads), np.sin(rads)]
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self.x_opt = np.array([1.0, 0.0])
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self.constr_jac = constr_jac
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self.constr_hess = constr_hess
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self.bounds = None
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def fun(self, x):
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return (2*(x[0]**2 + x[1]**2 - 1) - x[0],
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np.array([4*x[0]-1, 4*x[1]]))
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@property
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def grad(self):
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return True
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def hess(self, x):
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return 4*np.eye(2)
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@property
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def constr(self):
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def fun(x):
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return x[0]**2 + x[1]**2
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if self.constr_jac is None:
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def jac(x):
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return [[4*x[0], 4*x[1]]]
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else:
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jac = self.constr_jac
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if self.constr_hess is None:
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def hess(x, v):
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return 2*v[0]*np.eye(2)
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else:
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hess = self.constr_hess
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return NonlinearConstraint(fun, 1, 1, jac, hess)
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class HyperbolicIneq:
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"""Problem 15.1 from Nocedal and Wright
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The following optimization problem:
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minimize 1/2*(x[0] - 2)**2 + 1/2*(x[1] - 1/2)**2
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Subject to: 1/(x[0] + 1) - x[1] >= 1/4
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x[0] >= 0
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x[1] >= 0
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"""
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def __init__(self, constr_jac=None, constr_hess=None):
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self.x0 = [0, 0]
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self.x_opt = [1.952823, 0.088659]
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self.constr_jac = constr_jac
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self.constr_hess = constr_hess
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self.bounds = Bounds(0, np.inf)
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def fun(self, x):
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return 1/2*(x[0] - 2)**2 + 1/2*(x[1] - 1/2)**2
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def grad(self, x):
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return [x[0] - 2, x[1] - 1/2]
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def hess(self, x):
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return np.eye(2)
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@property
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def constr(self):
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def fun(x):
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return 1/(x[0] + 1) - x[1]
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if self.constr_jac is None:
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def jac(x):
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return [[-1/(x[0] + 1)**2, -1]]
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else:
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jac = self.constr_jac
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if self.constr_hess is None:
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def hess(x, v):
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return 2*v[0]*np.array([[1/(x[0] + 1)**3, 0],
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[0, 0]])
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else:
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hess = self.constr_hess
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return NonlinearConstraint(fun, 0.25, np.inf, jac, hess)
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class Rosenbrock:
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"""Rosenbrock function.
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The following optimization problem:
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minimize sum(100.0*(x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0)
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"""
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def __init__(self, n=2, random_state=0):
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rng = np.random.RandomState(random_state)
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self.x0 = rng.uniform(-1, 1, n)
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self.x_opt = np.ones(n)
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self.bounds = None
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def fun(self, x):
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x = np.asarray(x)
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r = np.sum(100.0 * (x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0,
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axis=0)
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return r
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def grad(self, x):
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x = np.asarray(x)
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xm = x[1:-1]
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xm_m1 = x[:-2]
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xm_p1 = x[2:]
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der = np.zeros_like(x)
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der[1:-1] = (200 * (xm - xm_m1**2) -
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400 * (xm_p1 - xm**2) * xm - 2 * (1 - xm))
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der[0] = -400 * x[0] * (x[1] - x[0]**2) - 2 * (1 - x[0])
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der[-1] = 200 * (x[-1] - x[-2]**2)
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return der
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def hess(self, x):
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x = np.atleast_1d(x)
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H = np.diag(-400 * x[:-1], 1) - np.diag(400 * x[:-1], -1)
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diagonal = np.zeros(len(x), dtype=x.dtype)
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diagonal[0] = 1200 * x[0]**2 - 400 * x[1] + 2
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diagonal[-1] = 200
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diagonal[1:-1] = 202 + 1200 * x[1:-1]**2 - 400 * x[2:]
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H = H + np.diag(diagonal)
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return H
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@property
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def constr(self):
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return ()
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class IneqRosenbrock(Rosenbrock):
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"""Rosenbrock subject to inequality constraints.
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The following optimization problem:
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minimize sum(100.0*(x[1] - x[0]**2)**2.0 + (1 - x[0])**2)
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subject to: x[0] + 2 x[1] <= 1
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Taken from matlab ``fmincon`` documentation.
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"""
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def __init__(self, random_state=0):
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Rosenbrock.__init__(self, 2, random_state)
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self.x0 = [-1, -0.5]
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self.x_opt = [0.5022, 0.2489]
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self.bounds = None
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@property
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def constr(self):
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A = [[1, 2]]
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b = 1
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return LinearConstraint(A, -np.inf, b)
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class EqIneqRosenbrock(Rosenbrock):
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"""Rosenbrock subject to equality and inequality constraints.
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The following optimization problem:
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minimize sum(100.0*(x[1] - x[0]**2)**2.0 + (1 - x[0])**2)
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subject to: x[0] + 2 x[1] <= 1
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2 x[0] + x[1] = 1
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Taken from matlab ``fimincon`` documentation.
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"""
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def __init__(self, random_state=0):
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Rosenbrock.__init__(self, 2, random_state)
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self.x0 = [-1, -0.5]
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self.x_opt = [0.41494, 0.17011]
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self.bounds = None
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@property
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def constr(self):
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A_ineq = [[1, 2]]
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b_ineq = 1
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A_eq = [[2, 1]]
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b_eq = 1
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return (LinearConstraint(A_ineq, -np.inf, b_ineq),
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LinearConstraint(A_eq, b_eq, b_eq))
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class Elec:
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"""Distribution of electrons on a sphere.
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Problem no 2 from COPS collection [2]_. Find
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the equilibrium state distribution (of minimal
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potential) of the electrons positioned on a
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conducting sphere.
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References
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----------
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.. [1] E. D. Dolan, J. J. Mor\'{e}, and T. S. Munson,
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"Benchmarking optimization software with COPS 3.0.",
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Argonne National Lab., Argonne, IL (US), 2004.
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"""
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def __init__(self, n_electrons=200, random_state=0,
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constr_jac=None, constr_hess=None):
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self.n_electrons = n_electrons
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self.rng = np.random.RandomState(random_state)
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# Initial Guess
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phi = self.rng.uniform(0, 2 * np.pi, self.n_electrons)
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theta = self.rng.uniform(-np.pi, np.pi, self.n_electrons)
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x = np.cos(theta) * np.cos(phi)
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y = np.cos(theta) * np.sin(phi)
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z = np.sin(theta)
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self.x0 = np.hstack((x, y, z))
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self.x_opt = None
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self.constr_jac = constr_jac
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self.constr_hess = constr_hess
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self.bounds = None
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def _get_cordinates(self, x):
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x_coord = x[:self.n_electrons]
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y_coord = x[self.n_electrons:2 * self.n_electrons]
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z_coord = x[2 * self.n_electrons:]
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return x_coord, y_coord, z_coord
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def _compute_coordinate_deltas(self, x):
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x_coord, y_coord, z_coord = self._get_cordinates(x)
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dx = x_coord[:, None] - x_coord
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dy = y_coord[:, None] - y_coord
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dz = z_coord[:, None] - z_coord
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return dx, dy, dz
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def fun(self, x):
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dx, dy, dz = self._compute_coordinate_deltas(x)
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with np.errstate(divide='ignore'):
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dm1 = (dx**2 + dy**2 + dz**2) ** -0.5
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dm1[np.diag_indices_from(dm1)] = 0
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return 0.5 * np.sum(dm1)
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def grad(self, x):
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dx, dy, dz = self._compute_coordinate_deltas(x)
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with np.errstate(divide='ignore'):
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dm3 = (dx**2 + dy**2 + dz**2) ** -1.5
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dm3[np.diag_indices_from(dm3)] = 0
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grad_x = -np.sum(dx * dm3, axis=1)
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|
grad_y = -np.sum(dy * dm3, axis=1)
|
||
|
|
grad_z = -np.sum(dz * dm3, axis=1)
|
||
|
|
|
||
|
|
return np.hstack((grad_x, grad_y, grad_z))
|
||
|
|
|
||
|
|
def hess(self, x):
|
||
|
|
dx, dy, dz = self._compute_coordinate_deltas(x)
|
||
|
|
d = (dx**2 + dy**2 + dz**2) ** 0.5
|
||
|
|
|
||
|
|
with np.errstate(divide='ignore'):
|
||
|
|
dm3 = d ** -3
|
||
|
|
dm5 = d ** -5
|
||
|
|
|
||
|
|
i = np.arange(self.n_electrons)
|
||
|
|
dm3[i, i] = 0
|
||
|
|
dm5[i, i] = 0
|
||
|
|
|
||
|
|
Hxx = dm3 - 3 * dx**2 * dm5
|
||
|
|
Hxx[i, i] = -np.sum(Hxx, axis=1)
|
||
|
|
|
||
|
|
Hxy = -3 * dx * dy * dm5
|
||
|
|
Hxy[i, i] = -np.sum(Hxy, axis=1)
|
||
|
|
|
||
|
|
Hxz = -3 * dx * dz * dm5
|
||
|
|
Hxz[i, i] = -np.sum(Hxz, axis=1)
|
||
|
|
|
||
|
|
Hyy = dm3 - 3 * dy**2 * dm5
|
||
|
|
Hyy[i, i] = -np.sum(Hyy, axis=1)
|
||
|
|
|
||
|
|
Hyz = -3 * dy * dz * dm5
|
||
|
|
Hyz[i, i] = -np.sum(Hyz, axis=1)
|
||
|
|
|
||
|
|
Hzz = dm3 - 3 * dz**2 * dm5
|
||
|
|
Hzz[i, i] = -np.sum(Hzz, axis=1)
|
||
|
|
|
||
|
|
H = np.vstack((
|
||
|
|
np.hstack((Hxx, Hxy, Hxz)),
|
||
|
|
np.hstack((Hxy, Hyy, Hyz)),
|
||
|
|
np.hstack((Hxz, Hyz, Hzz))
|
||
|
|
))
|
||
|
|
|
||
|
|
return H
|
||
|
|
|
||
|
|
@property
|
||
|
|
def constr(self):
|
||
|
|
def fun(x):
|
||
|
|
x_coord, y_coord, z_coord = self._get_cordinates(x)
|
||
|
|
return x_coord**2 + y_coord**2 + z_coord**2 - 1
|
||
|
|
|
||
|
|
if self.constr_jac is None:
|
||
|
|
def jac(x):
|
||
|
|
x_coord, y_coord, z_coord = self._get_cordinates(x)
|
||
|
|
Jx = 2 * np.diag(x_coord)
|
||
|
|
Jy = 2 * np.diag(y_coord)
|
||
|
|
Jz = 2 * np.diag(z_coord)
|
||
|
|
return csc_matrix(np.hstack((Jx, Jy, Jz)))
|
||
|
|
else:
|
||
|
|
jac = self.constr_jac
|
||
|
|
|
||
|
|
if self.constr_hess is None:
|
||
|
|
def hess(x, v):
|
||
|
|
D = 2 * np.diag(v)
|
||
|
|
return block_diag(D, D, D)
|
||
|
|
else:
|
||
|
|
hess = self.constr_hess
|
||
|
|
|
||
|
|
return NonlinearConstraint(fun, -np.inf, 0, jac, hess)
|
||
|
|
|
||
|
|
|
||
|
|
class TestTrustRegionConstr(TestCase):
|
||
|
|
|
||
|
|
def test_list_of_problems(self):
|
||
|
|
list_of_problems = [Maratos(),
|
||
|
|
Maratos(constr_hess='2-point'),
|
||
|
|
Maratos(constr_hess=SR1()),
|
||
|
|
Maratos(constr_jac='2-point', constr_hess=SR1()),
|
||
|
|
MaratosGradInFunc(),
|
||
|
|
HyperbolicIneq(),
|
||
|
|
HyperbolicIneq(constr_hess='3-point'),
|
||
|
|
HyperbolicIneq(constr_hess=BFGS()),
|
||
|
|
HyperbolicIneq(constr_jac='3-point',
|
||
|
|
constr_hess=BFGS()),
|
||
|
|
Rosenbrock(),
|
||
|
|
IneqRosenbrock(),
|
||
|
|
EqIneqRosenbrock(),
|
||
|
|
Elec(n_electrons=2),
|
||
|
|
Elec(n_electrons=2, constr_hess='2-point'),
|
||
|
|
Elec(n_electrons=2, constr_hess=SR1()),
|
||
|
|
Elec(n_electrons=2, constr_jac='3-point',
|
||
|
|
constr_hess=SR1())]
|
||
|
|
|
||
|
|
for prob in list_of_problems:
|
||
|
|
for grad in (prob.grad, '3-point', False):
|
||
|
|
for hess in (prob.hess,
|
||
|
|
'3-point',
|
||
|
|
SR1(),
|
||
|
|
BFGS(exception_strategy='damp_update'),
|
||
|
|
BFGS(exception_strategy='skip_update')):
|
||
|
|
|
||
|
|
# Remove exceptions
|
||
|
|
if grad in ('2-point', '3-point', 'cs', False) and \
|
||
|
|
hess in ('2-point', '3-point', 'cs'):
|
||
|
|
continue
|
||
|
|
if prob.grad is True and grad in ('3-point', False):
|
||
|
|
continue
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(UserWarning, "delta_grad == 0.0")
|
||
|
|
result = minimize(prob.fun, prob.x0,
|
||
|
|
method='trust-constr',
|
||
|
|
jac=grad, hess=hess,
|
||
|
|
bounds=prob.bounds,
|
||
|
|
constraints=prob.constr)
|
||
|
|
|
||
|
|
if prob.x_opt is not None:
|
||
|
|
assert_array_almost_equal(result.x, prob.x_opt, decimal=5)
|
||
|
|
# gtol
|
||
|
|
if result.status == 1:
|
||
|
|
assert_array_less(result.optimality, 1e-8)
|
||
|
|
# xtol
|
||
|
|
if result.status == 2:
|
||
|
|
assert_array_less(result.tr_radius, 1e-8)
|
||
|
|
|
||
|
|
if result.method == "tr_interior_point":
|
||
|
|
assert_array_less(result.barrier_parameter, 1e-8)
|
||
|
|
# max iter
|
||
|
|
if result.status in (0, 3):
|
||
|
|
raise RuntimeError("Invalid termination condition.")
|
||
|
|
|
||
|
|
def test_no_constraints(self):
|
||
|
|
prob = Rosenbrock()
|
||
|
|
result = minimize(prob.fun, prob.x0,
|
||
|
|
method='trust-constr',
|
||
|
|
jac=prob.grad, hess=prob.hess)
|
||
|
|
result1 = minimize(prob.fun, prob.x0,
|
||
|
|
method='L-BFGS-B',
|
||
|
|
jac='2-point')
|
||
|
|
with pytest.warns(UserWarning):
|
||
|
|
result2 = minimize(prob.fun, prob.x0,
|
||
|
|
method='L-BFGS-B',
|
||
|
|
jac='3-point')
|
||
|
|
assert_array_almost_equal(result.x, prob.x_opt, decimal=5)
|
||
|
|
assert_array_almost_equal(result1.x, prob.x_opt, decimal=5)
|
||
|
|
assert_array_almost_equal(result2.x, prob.x_opt, decimal=5)
|
||
|
|
|
||
|
|
def test_hessp(self):
|
||
|
|
prob = Maratos()
|
||
|
|
|
||
|
|
def hessp(x, p):
|
||
|
|
H = prob.hess(x)
|
||
|
|
return H.dot(p)
|
||
|
|
|
||
|
|
result = minimize(prob.fun, prob.x0,
|
||
|
|
method='trust-constr',
|
||
|
|
jac=prob.grad, hessp=hessp,
|
||
|
|
bounds=prob.bounds,
|
||
|
|
constraints=prob.constr)
|
||
|
|
|
||
|
|
if prob.x_opt is not None:
|
||
|
|
assert_array_almost_equal(result.x, prob.x_opt, decimal=2)
|
||
|
|
|
||
|
|
# gtol
|
||
|
|
if result.status == 1:
|
||
|
|
assert_array_less(result.optimality, 1e-8)
|
||
|
|
# xtol
|
||
|
|
if result.status == 2:
|
||
|
|
assert_array_less(result.tr_radius, 1e-8)
|
||
|
|
|
||
|
|
if result.method == "tr_interior_point":
|
||
|
|
assert_array_less(result.barrier_parameter, 1e-8)
|
||
|
|
# max iter
|
||
|
|
if result.status in (0, 3):
|
||
|
|
raise RuntimeError("Invalid termination condition.")
|
||
|
|
|
||
|
|
def test_args(self):
|
||
|
|
prob = MaratosTestArgs("a", 234)
|
||
|
|
|
||
|
|
result = minimize(prob.fun, prob.x0, ("a", 234),
|
||
|
|
method='trust-constr',
|
||
|
|
jac=prob.grad, hess=prob.hess,
|
||
|
|
bounds=prob.bounds,
|
||
|
|
constraints=prob.constr)
|
||
|
|
|
||
|
|
if prob.x_opt is not None:
|
||
|
|
assert_array_almost_equal(result.x, prob.x_opt, decimal=2)
|
||
|
|
|
||
|
|
# gtol
|
||
|
|
if result.status == 1:
|
||
|
|
assert_array_less(result.optimality, 1e-8)
|
||
|
|
# xtol
|
||
|
|
if result.status == 2:
|
||
|
|
assert_array_less(result.tr_radius, 1e-8)
|
||
|
|
|
||
|
|
if result.method == "tr_interior_point":
|
||
|
|
assert_array_less(result.barrier_parameter, 1e-8)
|
||
|
|
# max iter
|
||
|
|
if result.status in (0, 3):
|
||
|
|
raise RuntimeError("Invalid termination condition.")
|
||
|
|
|
||
|
|
def test_raise_exception(self):
|
||
|
|
prob = Maratos()
|
||
|
|
|
||
|
|
raises(ValueError, minimize, prob.fun, prob.x0, method='trust-constr',
|
||
|
|
jac='2-point', hess='2-point', constraints=prob.constr)
|