Optimization tutorials (TD)

The Rosenbrock function

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

$$ f(x, y) = (1-x)^2 + 100 (y-x^2)^2 $$



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%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

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

Curve fitting

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



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