- 问题描述
对以下优化问题
选取初始点，分别用以下方法求解
（1）最速下降法；
（2）Newton法或修正Newton法；
（3）共轭梯度法。
- 基本原理
（1）最速下降法
图1 最速下降法流程图
（2）Newton法
图2 Newton法流程图
（3）共轭梯度法
图3 共轭梯度法流程图
- 实验结果
（1）最速下降法
| 迭代 次数 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
| 梯度 模值 |
|
5.4210 |
1.6680 |
0.9532 |
0.2933 |
0.1676 |
0.0516 |
0.0295 |
0.0091 |
| 搜索 方向 |
|
[9.00, 3.00] |
[-1.71, 5.14] |
[1.58, 0.53] |
[-0.30, 0.90] |
[0.28, 0.09] |
[-0.05, 0.16] |
[0.05, 0.02] |
[-0.01, 0.03] |
| 当前 迭代点 |
(1.00, 1.00) |
(7.43, 3.14) |
(6.77, 5.12) |
(7.90, 5.50) |
(7.78, 5.85) |
(7.98, 5.91) |
(7.96, 5.97) |
(8.00, 5.98) |
(7.99, 6.00) |
| 当前迭代点值 |
47.00 |
14.8571 |
9.2057 |
8.2120 |
8.0373 |
8.0066 |
8.0012 |
8.0002 |
8.0000 |
表1 最速下降法迭代过程
图4 最速下降法迭代过程图
（2）Newton法
| 迭代次数 |
1 |
2 |
| 梯度模值 |
|
0.0000 |
| 搜索方向 |
|
[7.00,5.00] |
| 当前迭代点 |
(1.00,1.00) |
(8.00,6.00) |
| 当前迭代点值 |
47.00 |
8.0000 |
表2 Newton法迭代过程
图5 Newton法迭代过程图
（3）共轭梯度法
| 迭代次数 |
1 |
2 |
3 |
| 梯度模值 |
|
5.4210 |
0.0000 |
| 搜索方向 |
|
[9.00,3.00] |
[1.22,6.12] |
| 当前迭代点 |
(1.00,1.00) |
(7.43,3.14) |
(8.00,6.00) |
| 当前迭代点值 |
47.00 |
14.8571 |
8.0000 |
表3 共轭梯度法迭代过程
图6 共轭梯度法迭代过程图
对比结果可得，三种算法均得到同一个极值点（8, 6）。
- 代码展示
import matplotlib.pyplot as plt
from sympy import *
import numpy as np
from mpl_toolkits.mplot3d import Axes3D
t = symbols('t')
# 优化目标函数
def fun1():
x1, x2 = symbols('x1 x2')
y = np.power(x1, 2) + np.power(x2, 2) - x1*x2 -10 * x1 - 4 *x2 +60
return y
def fun2(x1, x2):
return np.power(x1, 2) + np.power(x2, 2) - x1 * x2 - 10 * x1 - 4 * x2 + 60
# 计算当前梯度
def cal_gradient_cur(X_cur):
x1, x2 = symbols('x1 x2')
f = fun1()
g = [diff(f, x1), diff(f, x2)]
g[0] = g[0].evalf(subs={x1:X_cur[0], x2:X_cur[1]})
g[1] = g[1].evalf(subs={x1:X_cur[0], x2:X_cur[1]})
return np.array(g)
# 计算lambda, X1: 上一轮迭代点, X2: 本次迭代点
def cal_lambda(X1, X2):
g1 = np.array(cal_gradient_cur(X1))
g2 = np.array(cal_gradient_cur(X2))
g1_norm_2 = np.sum(g1**2, axis=0)
g2_norm_2 = np.sum(g2**2, axis=0)
lamda = g2_norm_2 / g1_norm_2
return lamda
# 更新迭代点X
def update_X(X, P):
return np.array(X + t*P)
# 更新迭代点X
def update_X_cur(X, t, P):
return np.array(X + t*P)
# 计算最优步长
def cal_best_t(X_cur):
x1, x2 = symbols('x1 x2')
f = fun1()
f_t = f.subs({x1: X_cur[0], x2: X_cur[1]})
return solve(diff(f_t, t), t)
# 计算梯度模值
def cal_g_norm_cur(X):
g_cur = np.array(cal_gradient_cur(X), dtype=np.float32)
return np.sqrt(np.sum(g_cur**2, axis=0))
def draw(X0):
plt.figure()
ax = plt.axes(projection='3d')
xx = np.arange(-20, 20, 0.1)
yy = np.arange(-20, 20, 0.1)
x1, x2 = np.meshgrid(xx, yy)
Z = fun2(x1, x2)
ax.plot_surface(x1, x2, Z, cmap='rainbow', alpha=0.5)
X = np.array([X0[0]])
Y = np.array([X0[1]])
X, Y = np.meshgrid(X, Y)
Z = fun2(X, Y)
print("初始点：(%0.2f,%0.2f,%0.2f)" % (X, Y, Z))
ax.scatter(X, Y, Z, c='k', s=20)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
# ax.legend()
# ax.contour(X,Y,Z, zdim='z',offset=-2，cmap='rainbow) #等高线图，要设置offset，为Z的最小值
return ax
class C_gradient(object):
def __init__(self, X0):
self.X0 = X0
# 更新搜索方向
def cal_P(self, g_cur, P1, lamda):
P = -1 * g_cur + lamda*P1
return np.array(P)
def search(self):
X1 = self.X0
g_norm_cur = cal_g_norm_cur(X1) # 计算梯度模值
count = 0
result = []
if(g_norm_cur <= 0.01):
print("极值点为({:.2f},{:.2f})".format(X1[0], X1[1]))
x1, x2 = symbols('x1 x2')
f = fun1()
min_value = f.evalf(subs={x1: X1[0], x2: X1[1]})
print("极小值为{:.4f}".format(min_value))
else:
P = -1 * cal_gradient_cur(X1) # 计算当前负梯度方向
while True:
X2 = update_X(X1, P)
t_cur = cal_best_t(X2)
X2 = update_X_cur(X1, t_cur, P)
g_cur = cal_gradient_cur(X2)
g_norm_cur = cal_g_norm_cur(X2)
x1, x2 = symbols('x1 x2')
f = fun1()
min_value = f.evalf(subs={x1: X2[0], x2: X2[1]})
result.append([float(X2[0]), float(X2[1]), float(min_value)])
print("当前梯度模值为{:.4f}，搜索方向为[{:.2f},{:.2f}]".format(g_norm_cur, P[0], P[1]), end=" ")
print("极值点为({:.2f},{:.2f}), 极小值为{:.4f}".format(result[count][0], result[count][1], result[count][2]))
if(g_norm_cur <= 0.01):
return np.array(result)
else:
lamda = cal_lambda(X1, X2)
P = self.cal_P(g_cur, P, lamda)
X1 = X2
count += 1
def C_gradient_main():
print("当前搜索方法为共轭梯度法")
X0 = np.array([1, 1])
ax = draw(X0)
cg = C_gradient(X0)
cg.ax = ax
result = cg.search()
ax.scatter(np.array([result[:, 0]]), np.array([result[:, 1]]), np.array([result[:, 2]]), c='k', s=20)
plt.show()
class steepest_gradient(object):
def __init__(self, X0):
self.X0 = X0
def search(self):
X1 = self.X0
result = []
count = 0
while True:
P = -1 * cal_gradient_cur(X1) # 计算当前负梯度方向
X2 = update_X(X1, P)
t_cur = cal_best_t(X2)
X2 = update_X_cur(X1, t_cur, P)
g_norm_cur = cal_g_norm_cur(X2)
x1, x2 = symbols('x1 x2')
f = fun1()
min_value = f.evalf(subs={x1: X2[0], x2: X2[1]})
result.append([float(X2[0]), float(X2[1]), float(min_value)])
print("当前梯度模值为{:.4f}，搜索方向为[{:.2f},{:.2f}]".format(g_norm_cur, P[0], P[1]), end=" ")
print("极值点为({:.2f},{:.2f}), 极小值为{:.4f}".format(result[count][0], result[count][1], result[count][2]))
if(g_norm_cur <= 0.01):
return np.array(result)
else:
X1 = X2
count += 1
def steepest_gradient_main():
print("当前搜索方法为最速下降法")
X0 = np.array([1, 1])
ax = draw(X0)
a = steepest_gradient(X0)
a.ax = ax
result = a.search()
ax.scatter(np.array([result[:, 0]]), np.array([result[:, 1]]), np.array([result[:, 2]]), c='k', s=20)
plt.show()
class Newton(object):
def __init__(self, X0):
self.X0 = X0
def cal_hesse(self):
return np.array([[2, -1], [-1, 2]])
def search(self):
X1 = self.X0
count = 0
result = []
while True:
g = cal_gradient_cur(X1)
g = g.reshape((1, 2)).T
h = np.linalg.inv(self.cal_hesse())
P = -1 * np.dot(h, g).ravel()
X2 = update_X(X1, P)
t_cur = cal_best_t(X2)
X2 = update_X_cur(X1, t_cur, P)
x1, x2 = symbols('x1 x2')
f = fun1()
min_value = f.evalf(subs={x1: X2[0], x2: X2[1]})
g_norm_cur = cal_g_norm_cur(X2)
result.append([float(X2[0]), float(X2[1]), float(min_value)])
print("当前梯度模值为{:.4f}，搜索方向为[{:.2f},{:.2f}]".format(g_norm_cur, P[0], P[1]), end=" ")
print("极值点为({:.2f},{:.2f}), 极小值为{:.4f}".format(result[count][0], result[count][1], result[count][2]))
if(g_norm_cur <= 0.01):
return np.array(result)
else:
X1 = X2
count += 1
def newton_main():
print("当前搜索方法为newton法")
X0 = np.array([1, 1])
ax = draw(X0)
b = Newton(X0)
result = b.search()
ax.scatter(np.array([result[:, 0]]), np.array([result[:, 1]]), np.array([result[:, 2]]), c='k', s=20)
plt.show()
if __name__ == '__main__':
steepest_gradient_main()
newton_main()
C_gradient_main()