原文链接: Python 优化 回溯下降算法

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使用sympy构造表达式，实现回溯下降算法

画出函数图像，先使用暴力搜索，找到最小值约为2.5左右

然后选定初始点，开始进行回溯搜索，下降方向为负梯度方向

下降的误差与步数大致呈现下面的状态

完整代码

from matplotlib import pyplot as plt
import numpy as np
from mpl_toolkits.mplot3d import Axes3D
from sympy import *
import math# 构造函数表达式
x1 = symbols('x1')
x2 = symbols('x2')
fx = E ** (x1 + 3 * x2 - 0.1) + E ** (x1 - 3 * x2 - 0.1) + E ** (-x1 - 0.1)
# 求偏导数
dx = diff(fx, x1)
dy = diff(fx, x2)# 显示函数图像
def show():fig = plt.figure()ax = Axes3D(fig)max_x = 1X = np.linspace(-max_x, max_x, 100)Y = np.linspace(-max_x, max_x, 100)Z = np.array([np.array([float(fx.evalf(subs={x1: i, x2: j}))for j in Y])for i in X])X, Y = np.meshgrid(X, Y)ax.plot_surface(X, Y, Z, rstride=3, cstride=3, cmap='rainbow')plt.show()# 给定a,b求回溯直线下降
def down(a, b):# 初始值x = np.array([1, 1])t = 1vals = [fx.evalf(subs={x1: x[0], x2: x[1]})]d = [-dx.evalf(subs={x1: x[0], x2: x[1]}), -dy.evalf(subs={x1: x[0], x2: x[1]})]while (d[0] ** 2 + d[1] ** 2) > 1:d = [-dx.evalf(subs={x1: x[0], x2: x[1]}), -dy.evalf(subs={x1: x[0], x2: x[1]})]xk = np.array(x) + np.array(d) * twhile fx.evalf(subs={x1: xk[0], x2: xk[1]}) > \fx.evalf(subs={x1: x[0], x2: x[1]}) - \a * t * (d[0] ** 2 + d[1] ** 2):xk = x + np.array(d) * tt = b * tx = x + np.array(d) * tvals.append(fx.evalf(subs={x1: x[0], x2: x[1]}))return vals# 暴力求解最小值
def test():minf = 1 << 64minx1 = 0minx2 = 0for i in np.linspace(-10, 10, 1000):for j in np.linspace(-10, 10, 1000):f = math.e ** (i + 3 * j - 0.1) + math.e ** (i - 3 * j - 0.1) + math.e ** (-i - 0.1)if f < minf:minx1 = iminx2 = jminf = fprint(minf, minx1, minx2)# 保存图像
def saveImg(a, b, arr):print(a, b, len(arr))plt.figure()plt.plot([i for i in range(len(arr))], [i for i in arr])plt.xlabel('k')plt.ylabel('fx')plt.title(f"a={a}\nb={b}")plt.savefig(f'./imgs/{a}_{b}.png')plt.close()#  根据不同的ab，保存图片
def fun():for i in np.linspace(0.1, 0.4, 4):for j in np.linspace(0.1, 0.9, 9):saveImg(i, j, down(i, j))test()
# fun()# show()