Minima

\[f(x,y)=(x^2+y-11)^2+(x+y^2-7)^2 \]

\[f(3.0, 2.0)=0.0\\ f(-2.8505118, 3.131312)=0.0\\ f(-3.779310, -3.283186)=0.0\\ f(3.584428, -1.84126)=0.0\\ \]

%matplotlib inline
import numpy as np, torch, torch.nn.functional as F
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# plot
def himmelblau(x):
    return (x[0] ** 2 + x[1] - 11) ** 2 + (x[0] + x[1] ** 2 - 7) ** 2

x = np.arange(-6, 6, 0.1)
y = np.arange(-6, 6, 0.1)
print("x, y range:", x.shape, y.shape)

x, y range: (120,) (120,)

X, Y = np.meshgrid(x, y)
print("X, Y maps:", X.shape, Y.shape)
Z = himmelblau([X, Y])

X, Y maps: (120, 120) (120, 120)

fig = plt.figure('himmelblau')
ax = fig.gca(projection = '3d')
ax.plot_surface(X, Y, Z)
ax.view_init(60, -30)
ax.set_xlabel('x')
ax.set_ylabel('y')
plt.show()

# Gradient Descent
#[1., 0] [-4, 0.] [4, 0.]
x = torch.tensor([0., 0.], requires_grad=True)
opt = torch.optim.Adam([x], lr=1e-3)
for step in range(20001):
    pred = himmelblau(x)
    opt.zero_grad()
    pred.backward()
    opt.step()
    if step % 2000 == 0:
        print("step {}: x = {}, f(x) = {}".format(step, x.tolist(), pred.item()))

step 0: x = [0.0009999999310821295, 0.0009999999310821295], f(x) = 170.0
step 2000: x = [2.3331806659698486, 1.9540694952011108], f(x) = 13.730916023254395
step 4000: x = [2.9820079803466797, 2.0270984172821045], f(x) = 0.014858869835734367
step 6000: x = [2.999983549118042, 2.0000221729278564], f(x) = 1.1074007488787174e-08
step 8000: x = [2.9999938011169434, 2.0000083446502686], f(x) = 1.5572823031106964e-09
step 10000: x = [2.999997854232788, 2.000002861022949], f(x) = 1.8189894035458565e-10
step 12000: x = [2.9999992847442627, 2.0000009536743164], f(x) = 1.6370904631912708e-11
step 14000: x = [2.999999761581421, 2.000000238418579], f(x) = 1.8189894035458565e-12
step 16000: x = [3.0, 2.0], f(x) = 0.0
step 18000: x = [3.0, 2.0], f(x) = 0.0
step 20000: x = [3.0, 2.0], f(x) = 0.0