CODEWITHSUNDEEP
pythonmatrixoptimizationeigenvalue
Yeti
Yeti

Reputation: 615

Minimise (optimise) particular eigenvalue in Python

I have written a simple Python code that reads arrays from a file. All the array elements contain basic algebraic operations (e.g. y*w+2*(w+y)). These are read in, and then all the array elements are evaluated using the eval command (I know the security issues with using eval, but this script will only ever be used by me, and I hopefully won't do something stupid).

These arrays are then used to solve a general eigenvalue problem and eigenvalue and vectors are obtained. My question is how do I minimise (optimise) a particular eigenvalue by varying certain parameters within the matrix? Here is a basic example:

import numpy as np
from scipy.linalg import eig

y=10
w=3.45

x = np.array([[y+w,-2],[-2,w-2*(w+y)]])

eigenVal,eigenVec=eig(x)

print eigenVal[0]

The matrix is not the best example, but is just a minimal example. The idea is to vary the two parameters y and w to minimise a particular eigenvalue (e.g. the first eigenvalue, eigenVal[0]). I have written some basic optimisation algorithms in C++ before, but wanted to explore the Python optimisaton library. These matrices can get very large, so what is the best way to go about this problem?

Upvotes: 0

Views: 674

Answers (1)

Arya McCarthy
Arya McCarthy

Reputation: 8829

You want to use scipy.optimize.minimize.

import numpy as np
from scipy.linalg import eig
from scipy.optimize import minimize

def my_func(x):
    y, w = x
    arr = np.array([[y+w,-2],[-2,w-2*(w+y)]])
    ev, ew=eig(arr)
    return ev[0]

x0 = np.array([10, 3.45])  # Initial guess

minimize(my_func, x0)
# ComplexWarning: Casting complex values to real discards the imaginary part
#       fun: -127806953.0230245
#  hess_inv: array([[1, 0],
#        [0, 1]])
#       jac: array([ 0.,  0.])
#   message: 'Optimization terminated successfully.'
#      nfev: 60
#       nit: 1
#      njev: 15
#    status: 0
#   success: True
#         x: array([-63809761.27752077, -63997191.74550374])

This ComplexError is worrying; you should use the bounds parameter or the constraints parameter. I don't know the constraints for your problem, so I'm skipping that part. A decent constraint would be that the matrix should be totally positive, which guarantees positive, real eigenvalues. But that's not possible, given that you have a constant -2 in your matrix.

You can access the result of the optimization as:

a = minimize(my_func, x0)
print(a.x)
# array([-63809761.27752077, -63997191.74550374])

Upvotes: 1

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