求解Rosenbrock函数最小值
- 定义a=1，b=100
图像如：
采用梯度下降：
x n + 1 = x n ? α ? f ( x n ) , n ≥ 0 x_{n+1}=x_n-\alpha \nabla f(x_n),\quad n\geq0 xn+1?=xn??α?f(xn?),n≥0
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import ticker
def f(x, y):
return (1 - x) ** 2 + 100 * (y - x * x) ** 2
def H(x, y):
return np.matrix([[1200 * x * x - 400 * y + 2, -400 * x],
[-400 * x, 200]])
def grad(x, y):
return np.matrix([[2 * x - 2 + 400 * x * (x * x - y)],
[200 * (y - x * x)]])
def delta_grad(x, y):
g = grad(x, y)
alpha = 0.002
delta = alpha * g
return delta
# ----- 绘制等高线 -----
# 数据数目
n = 256
# 定义x, y
x = np.linspace(-1, 1.1, n)
y = np.linspace(-0.1, 1.1, n)
# 生成网格数据
X, Y = np.meshgrid(x, y)
plt.figure()
# 填充等高线的颜色, 8是等高线分为几部分
plt.contourf(X, Y, f(X, Y), 5, alpha=0, cmap=plt.cm.hot)
# 绘制等高线
C = plt.contour(X, Y, f(X, Y), 8, locator=ticker.LogLocator(), colors='black', linewidth=0.01)
# 绘制等高线数据
plt.clabel(C, inline=True, fontsize=10)
# ---------------------
x = np.matrix([[-0.2],
[0.4]])
tol = 0.00001
xv = [x[0, 0]]
yv = [x[1, 0]]
plt.plot(x[0, 0], x[1, 0], marker='o')
for t in range(6000):
delta = delta_grad(x[0, 0], x[1, 0])
if abs(delta[0, 0]) < tol and abs(delta[1, 0]) < tol:
break
x = x - delta
xv.append(x[0, 0])
yv.append(x[1, 0])
plt.plot(xv, yv, label='track')
# plt.plot(xv, yv, label='track', marker='o')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Gradient for Rosenbrock Function')
plt.legend()
plt.show()
- 牛顿法
x n + 1 = x n ? [ H f ( x n ) ? 1 ? f ( x n ) , n ≥ 0. x_{n+1}=x_n-[\bold{H}f(x_n)^{-1}\nabla f(x_n),\quad n\geq0. xn+1?=xn??[Hf(xn?)?1?f(xn?),n≥0.
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import ticker
def f(x, y):
return (1 - x) ** 2 + 100 * (y - x * x) ** 2
def H(x, y):
return np.matrix([[1200 * x * x - 400 * y + 2, -400 * x],
[-400 * x, 200]])
def grad(x, y):
return np.matrix([[2 * x - 2 + 400 * x * (x * x - y)],
[200 * (y - x * x)]])
def delta_newton(x, y):
alpha = 1.0
delta = alpha * H(x, y).I * grad(x, y)
return delta
# ----- 绘制等高线 -----
# 数据数目
n = 256
# 定义x, y
x = np.linspace(-1, 1.1, n)
y = np.linspace(-1, 1.1, n)
# 生成网格数据
X, Y = np.meshgrid(x, y)
plt.figure()
# 填充等高线的颜色, 8是等高线分为几部分
plt.contourf(X, Y, f(X, Y), 5, alpha=0, cmap=plt.cm.hot)
# 绘制等高线
C = plt.contour(X, Y, f(X, Y), 8, locator=ticker.LogLocator(), colors='black', linewidth=0.01)
# 绘制等高线数据
plt.clabel(C, inline=True, fontsize=10)
# ---------------------
x = np.matrix([[-0.3],
[0.4]])
tol = 0.00001
xv = [x[0, 0]]
yv = [x[1, 0]]
plt.plot(x[0, 0], x[1, 0], marker='o')
for t in range(100):
delta = delta_newton(x[0, 0], x[1, 0])
if abs(delta[0, 0]) < tol and abs(delta[1, 0]) < tol:
break
x = x - delta
xv.append(x[0, 0])
yv.append(x[1, 0])
plt.plot(xv, yv, label='track')
# plt.plot(xv, yv, label='track', marker='o')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Newton\'s Method for Rosenbrock Function')
plt.legend()
plt.show()
引用自: SpringHerald.