Hello all.
I am a relatively new user of python and scipy and I have been trying
out scipy's optimization facilities. I am using scipy version 0.6.0,
as distributed with Ubuntu 8.04.
My exploration has centered around the minimization of x*x*y, subject
to the equality constraint 2*x*x+y*y=3. In my experience, this
problem is solved by introducing a Lagrange multiplier and minimizing
the Lagrangian:
L = x*x*y - lambda * ( 2*x*x+y*y-3 )
I have had no problem finding the desired solution via Newton-Raphson
using the function and its first and second derivatives:
import scipy.optimize as opt
import numpy
import numpy.linalg as l
def f(r):
x,y,lam=r
return x*x*y -lam*(2*x*x+y*y-3)
def g(r):
x,y,lam=r
return numpy.array([2*x*y-4*lam*x, x*x-2*lam*y, -(2*x*x+y*y-3)])
def h(r):
x,y,lam=r
return numpy.mat([[2.*y-4.*lam, 2.*x,
-4.*x],[2.*x,-2.*lam,-2.*y],[-4.*x,-2.*y,0.]])
def NR(f, g, h, x0, tol=1e-5, maxit=100):
"Find a local extremum of f (a root of g) using Newton-Raphson"
x1 = numpy.asarray(x0)
f1 = f(x1)
for i in range(0,maxit):
dx = l.solve(h(x1),g(x1))
ldx = numpy.sqrt(numpy.dot(dx,dx))
x2 = x1-dx
f2 = f(x2)
if(ldx < tol): # x is close enough
df = numpy.abs(f1-f2)
if(df < tol): # f is close enough
return x2, f2, df, ldx, i
x1=x2
f1=f2
return x2, f2, df, ldx, i
print NR(f,g,h,[-2.,2.,3.],tol=1e-10)
My Newton-Raphson iteration converges in 5 iterations, but I have had
no success using any of the functions in scipy.optimize, for example:
print opt.fmin_bfgs(f=f, x0=[-2.,2.,3.], fprime=g)
print opt.fmin_ncg(f=f, x0=[-2.,2.,3.], fprime=g, fhess=h)
neither of which converges.
I am beginning to suspect some fundamental misunderstanding on my
part. Could someone throw me a bone?
Best regards
Gísli