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# (which is just using Newton on a Lagrangian)
# In[2]:
import numpy as np
import numpy.linalg as la
# Here's an objective function $f$ and a constraint $g(x)=0$:
# In[3]:
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:
# In[4]:
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`:
# In[5]:
vec = np.zeros(3)
# Implement Newton and run this cell in place a number of times (Ctrl-Enter):
# In[18]:
vec = vec - la.solve(hess_L(vec), grad_L(vec))
vec
# Let's first check that we satisfy the constraint:
# In[21]:
g(vec)
# Next, let's look at a plot:
# In[22]:
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")
# In[ ]: