问题
尝试找到函数的全局最优点（阅读Python for Finance 2nd edition - Chapter 11. Mathematical Tools）。
def fm(p):
x, y = p
return (np.sin(x) + 0.05 * x ** 2
+ np.sin(y) + 0.05 * y ** 2)
scipy.optimize.basinhopping说它找到了全局最小值。
使用盆地跳跃算法找到函数的全局最小值
但是，它看起来没有找到全局最优点。为什么会这样，如何使它找到全局最优？
import numpy as np
import scipy.optimize as sco
from pylab import plt, mpl
from mpl_toolkits.mplot3d import Axes3D
plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'
%matplotlib notebook
x = np.linspace(-10, 10, 50)
y = np.linspace(-10, 10, 50)
X, Y = np.meshgrid(x, y)
Z = fm((X, Y))
optima = sco.basinhopping(
fo,
(-10, 10),
stepsize=0.1
)
# Optimal
optima['x'] = np.append(optima['x'], fm((optima['x'][0], optima['x'][1])))
optima
结果
fun: 3.5447966927667616
lowest_optimization_result: fun: 3.5447966927667616
hess_inv: array([[9.01401735e-01, 1.68119491e-03],
[1.68119491e-03, 2.84089686e+00]])
jac: array([2.98023224e-08, 2.98023224e-08])
message: 'Optimization terminated successfully.'
nfev: 24
nit: 5
njev: 6
status: 0
success: True
x: array([-1.42755175, 9.67888407])
message: ['requested number of basinhopping iterations completed successfully']
minimization_failures: 0
nfev: 2516
nit: 100
njev: 629
x: array([-1.42755175, 9.67888407, 3.54479669])
阴谋：
fig = plt.figure(figsize=(10, 6))
ax = fig.gca(projection='3d')
# Function
surf = ax.plot_surface(
X, Y, Z,
rstride=2,
cstride=2,
cmap='coolwarm',
linewidth=0.5,
antialiased=True
)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('f(x, y)')
# Optima
ax.plot(
[optima['x'][0]], [optima['x'][1]], [optima['x'][2]],
color='r', marker='X', markersize=15
)
fig.colorbar(surf, shrink=0.5, aspect=5)
蛮横的
If used [scipy.optimize.brute][5], it may be able to find near point.
OX = []
OY = []
output = True
def fo(p):
x, y = p
z = np.sin(x) + 0.05 * x ** 2 + np.sin(y) + 0.05 * y ** 2
if output == True:
#print('%8.4f | %8.4f | %8.4f' % (x, y, z))
OX.append(x)
OY.append(y)
return z
optima = sco.brute(
fo,
(
(-10, 10.1, 2), # Step X from -10 to 10.1 by interval 2
(-10, 10.1, 2) # Step Y from -10 to 10.1 by interval 2
),
finish=None
)
OZ = fm((np.array(OX), np.array(OY)))
# Optimal
optima = np.append(optima, fm((optima[0], optima[1])))
fig = plt.figure(figsize=(10, 6))
ax = fig.gca(projection='3d')
# Function
surf = ax.plot_surface(
X, Y, Z,
rstride=2,
cstride=2,
cmap='coolwarm',
linewidth=0.5,
antialiased=True
)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('f(x, y)')
# Brute computation tack
ax.plot(np.array(OX), np.array(OY), np.array(OZ), ls="--", color='k', linewidth=0.5)
# Optima
ax.plot(
[optima[0]], [optima[1]], [optima[2]],
color='r', marker='X', markersize=15
)
fig.colorbar(surf, shrink=0.5, aspect=5)
上海办事处
scipy.optimize.shgo似乎也有效。
使用 SHG 优化查找函数的全局最小值。
optima = sco.shgo(
fo,
[(-10, 10), (-10, 10)]
)
# Optimal
optima['x'] = np.append(optima['x'], fm((optima['x'][0], optima['x'][1])))
fig = plt.figure(figsize=(10, 6))
ax = fig.gca(projection='3d')
# Function
surf = ax.plot_surface(
X, Y, Z,
rstride=2,
cstride=2,
cmap='coolwarm',
linewidth=0.5,
antialiased=True
)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('f(x, y)')
# Optima
ax.plot(
[optima['x'][0]], [optima['x'][1]], [optima['x'][2]],
color='r', marker='X', markersize=15
)
fig.colorbar(surf, shrink=0.5, aspect=5)
澄清
即使它是“全局优化”，是否有条件或限制需要考虑以确保他们找到全局最优？
全局优化
binhopping(func, x0[, niter, T, stepsize, ...])
使用盆地跳跃算法找到函数的全局最小值
brute(func, range[, args, Ns, full_output, ...])
最小化一个函数通过蛮力在给定的范围内。
different_evolution(func, bounds[, args, ...])
找到多元函数的全局最小值。
shgo(func, bounds[, args, constraints, n, …])
使用 SHG 优化找到函数的全局最小值。
dual_annealing(func, bounds[, args, ...])
使用双重退火求函数的全局最小值。