梯度下降法：以三元函数f(x,y,z) = x^2 + y^2 + z^2 + 2x + 4y - 6z 为例，求其极值及极值点 python代码实现：
from sympy import * import time def gradient_descent(): x = symbols('x') y = symbols('y') z = symbols('z') w = x**2 + y**2 + z**2 + 2*x + 4*y - 6*z x1 = 20 y1 = 20 z1 = 20 alpha = 0.03 dx = diff(w, x).subs({x: x1, y: y1, z: z1}) dy = diff(w, y).subs({x: x1, y: y1, z: z1}) dz = diff(w, z).subs({x: x1, y: y1, z: z1}) last_result = (w.subs({x: x1, y: y1, z: z1})) # 先上升一步以进行后面的收敛判断 x1 -= alpha * dx y1 -= alpha * dy z1 -= alpha * dz this_result = (w.subs({x: x1, y: y1, z: z1})) time_start = time.time() while abs(this_result - last_result) > 0.0001: # 收敛条件：|f(x^k)-f(x^k+1)|<& (&取参数，代表下降已经不明显即收敛) last_result = (w.subs({x: x1, y: y1, z: z1})) dx = diff(w, x).subs({x: x1, y: y1, z: z1}) dy = diff(w, y).subs({x: x1, y: y1, z: z1}) dz = diff(w, z).subs({x: x1, y: y1, z: z1}) x1 -= alpha * dx y1 -= alpha * dy z1 -= alpha * dz this_result = (w.subs({x: x1, y: y1, z: z1})) time_end = time.time() print("优化结果：x="+str(x1)) print("\t\ty= " + str(y1)) print("\t\tz= " + str(z1)) print("\t\tmin f(x,y) = " + str(this_result)) print('总时间：', time_end - time_start) gradient_descent()最后的输出：
优化结果：x=-0.988232354405425 y= -1.98767199032949 z= 3.00952618929085 min f(x,y) = -13.9996631667427 总时间： 0.7430148124694824实际根据多元函数极值求法求得的极值点： 注:A为三元函数的Hessian矩阵，A矩阵正定说明（-1，-2，3）为极小值点
