Optimization tutorials (TD)

The Rosenbrock function

The Rosenbrock function is a classical benchmark for optimization algorithms. It is defined by the following equation:

\[f(x, y) = (1-x)^2 + 100 (y-x^2)^2\]

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

def Rosen(X):
    """
    Rosenbrock function
    """
    x, y = X
    return (1-x)**2 + 100. * (y-x**2)**2

x = np.linspace(-2., 2., 100)
y = np.linspace(-1., 3., 100)
X, Y = np.meshgrid(x,y)
Z = Rosen( (X, Y) )

fig = plt.figure(0)
plt.clf()
plt.contourf(X, Y, Z, 20)
plt.colorbar()
plt.contour(X, Y, Z, 20, colors = "black")
plt.grid()
plt.xlabel("x")
plt.ylabel("y")
plt.show()

Questions

1. Find the minimum of the function using brute force. Comment the accuracy and number of function evaluations.
2. Same question with the simplex (Nelder-Mead) algorithm.

Curve fitting

1. Chose a mathematical function \(y = f(x, a, b)\) and code it.
2. Chose target values of \(a\) and \(b\) that you will try to find back using optimization.
3. Evaluate it on a grid of \(x\) values.
4. Add some noise to the result.
5. Find back \(a\) and \(b\) using curve_fit