def case_6():
'''Find stationary point of Easom's function. Stationary points include (pi, pi), (pi/2, pi/2).'''
def f(variables):
x, y = variables
return -np.cos(x) * np.cos(y) * np.exp(-((x - np.pi)**2 +
(y - np.pi)**2))
f_string = 'f(x, y) = -\cos(x)\cos(y)\exp(-((x-\pi)^2 + (y-\pi)^2))'
x0 = 2.5
y0 = 2.5
init_vars = [x0, y0]
solution, xy_path, f_path = opt.BFGS(f, init_vars, iters=10000)
opt.plot_results(f,
xy_path,
f_path,
f_string,
threedim=True,
bfgs=True,
hide=True)
xn, yn = solution.val
assert (np.allclose(xn, np.pi) and np.allclose(yn, np.pi))
plt.close('all')
def case_7():
'''Find stationary point of Himmelblau's function.'''
def f(variables):
x, y = variables
return (x**2 + y - 11)**2 + (x + y**2 - 7)**2
f_string = 'f(x, y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2'
x0 = 5
y0 = 5
init_vars = [x0, y0]
solution, xy_path, f_path = opt.BFGS(f, init_vars, iters=10000)
opt.plot_results(f,
xy_path,
f_path,
f_string,
x_lims=(-10, 10),
y_lims=(-10, 10),
threedim=True,
animate=True,
bfgs=True,
hide=True)
xn, yn = solution.val
assert ((np.allclose(xn, 3) and np.allclose(yn, 2))
or (np.allclose(xn, -2.805118) and np.allclose(yn, 3.131312))
or (np.allclose(xn, -3.779310) and np.allclose(yn, -3.283186))
or (np.allclose(xn, 3.584428) and np.allclose(yn, -1.848126)))
plt.close('all')
def case_5():
'''Find stationary point of bowl function: f(x, y) = x^2 + y^2 + 2.
The global minimum is 2 when (x, y) = (0, 0).
'''
def f(variables):
x, y = variables
return x**2 + y**2 + 2
f_string = 'f(x, y) = x^2 + y^2 + 2'
for x_val, y_val in [[2., 3.], [-2., 5.]]:
x0 = x_val
y0 = y_val
init_vars = [x0, y0]
solution, xy_path, f_path = opt.BFGS(f, init_vars)
xn, yn = solution.val
opt.plot_results(f,
xy_path,
f_path,
f_string,
x_lims=(-7.5, 7.5),
y_lims=(-7.5, 7.5),
threedim=True,
animate=True,
bfgs=True,
hide=True)
minimum = [0, 0]
assert np.allclose([xn, yn], minimum)
der = [0, 0]
assert np.allclose(solution.der, der)
plt.close('all')
def case_3():
'''Find stationary point of f(x) = sin(x).'''
def f(x):
return np.sin(x)
f_string = 'f(x) = sin(x)'
for x_val in [-3, -1, 1, 3]:
# Test case when 1D input is a list
x0 = [da.Var(x_val)]
solution, x_path, f_path = opt.BFGS(f, x0)
opt.plot_results(f,
x_path,
f_path,
f_string,
x_lims=(-2 * np.pi, 2 * np.pi),
y_lims=(-1.5, 1.5),
bfgs=True,
hide=True)
# BFGS finds a stationary point, which are every pi offset from pi/2
first = solution.val - (np.pi / 2)
multiple_of_one_half_pi = ((abs(solution.val) - (np.pi / 2)) %
(np.pi))
np.testing.assert_array_almost_equal(multiple_of_one_half_pi, [0])
np.testing.assert_array_almost_equal(solution.der, [0])
plt.close('all')
def case_1():
'''Simple quadratic function with stationary point at x = 0.'''
def f(x):
return x**2
f_string = 'f(x) = x^2'
for x_val in [-4, -2, 0, 2, 4]:
x0 = da.Var(x_val)
solution, x_path, f_path = opt.BFGS(f, x0)
opt.plot_results(f, x_path, f_path, f_string, bfgs=True, hide=True)
assert np.allclose(solution.val, [0])
assert np.allclose(solution.der, [0])
plt.close('all')
def case_4():
'''Find stationary point of Rosenbrock function: f(x, y) = 4(y - x^2)^2 + (1 - x)^2.
The global minimum is 0 when (x, y) = (1, 1).
'''
def f(variables):
x, y = variables
return 4 * (y - (x**2))**2 + (1 - x)**2
f_string = 'f(x, y) = 4(y - x^2)^2 + (1 - x)^2'
for x_val, y_val in [[-6., -6], [2., 3.], [-2., 5.]]:
x0 = da.Var(x_val, [1, 0])
y0 = da.Var(y_val, [0, 1])
init_vars = [x0, y0]
solution, xy_path, f_path = opt.BFGS(f, init_vars, iters=25000)
xn, yn = solution.val
opt.plot_results(f,
xy_path,
f_path,
f_string,
x_lims=(-7.5, 7.5),
y_lims=(-7.5, 7.5),
threedim=True,
animate=True,
bfgs=True,
hide=True)
minimum = [1, 1]
assert np.allclose([xn, yn], minimum)
# dfdx = 2(200x^3 - 200xy + x - 1)
# dfdy = 200(y - x^2)
der_x = 2 * (8 * (xn**3) - 8 * xn * yn + xn - 1)
der_y = 8 * (yn - (xn**2))
assert np.allclose(solution.der, [der_x, der_y])
plt.close('all')
def case_2():
'''Stationary point (minimum) at x = 1.'''
def f(x):
return (x - 1)**2 + 3
f_string = 'f(x) = (x - 1)^2 + 3'
for x_val in [-3, -1, 1, 3, 5]:
# Test initial guess without using da.Var type
solution, x_path, f_path = opt.BFGS(f, x_val)
opt.plot_results(f,
x_path,
f_path,
f_string,
x_lims=(-3, 5),
y_lims=(2, 10),
animate=True,
bfgs=True,
hide=True)
assert np.allclose(solution.val, [1])
assert np.allclose(solution.der, [0])
plt.close('all')