(which is just using Newton on a Lagrangian)
import numpy as np
import numpy.linalg as la
Here's an objective function $f$ and a constraint $g(x)=0$:
def f(vec):
x = vec[0]
y = vec[1]
return (x-2)**4 + 2*(y-1)**2
def g(vec):
x = vec[0]
y = vec[1]
return x + 4*y - 3
Now define the Lagrangian, its gradient, and its Hessian:
def L(vec):
lam = vec[2]
return f(vec) + lam * g(vec)
def grad_L(vec):
x = vec[0]
y = vec[1]
lam = vec[2]
return np.array([
4*(x-2)**3 + lam,
4*(y-1) + 4*lam,
x+4*y-3
])
def hess_L(vec):
x = vec[0]
y = vec[1]
lam = vec[2]
return np.array([
[12*(x-2)**2, 0, 1],
[0, 4, 4],
[1, 4, 0]
])
At this point, we only need to find an unconstrained minimum of the Lagrangian!
Let's fix a starting vector vec
:
vec = np.zeros(3)
Implement Newton and run this cell in place a number of times (Ctrl-Enter):
vec = vec - la.solve(hess_L(vec), grad_L(vec))
vec
Let's first check that we satisfy the constraint:
g(vec)
Next, let's look at a plot:
import matplotlib.pyplot as pt
x, y = np.mgrid[0:4:30j, -3:5:30j]
pt.contour(x, y, f(np.array([x,y])), 20)
pt.contour(x, y, g(np.array([x,y])), levels=[0])
pt.plot(vec[0], vec[1], "o")