18. Finding minima of functions

At this point you’re quite skilled at finding local minima of functions analytically, so now we’ll explore how to do so numerically.

Summary of commands

In this exercise, we will demonstrate the following:

• scipy.optimize.fmin(func, x0) - Minimize a function func given an initial guess x0.

This is part of the SciPy package, which we have yet to see in this class! Whereas NumPy provides the foundational array structure, SciPy contains a lot of helpful routines that operate on such structures, such as optimization, statistics, integration, etc.

Part (a) - Functions of a single variable

Find the minimum of the function \(f(x) = 1 - xe^{-2x}\), first analytically and then numerically.

The analytical derivative is

\[ f'(x) = e^{-2x} (2x - 1) = 0 \implies x^* = \dfrac{1}{2} \]

The numerical solution is interesting. You’ll notice the function signature is fmin(func, x0), where:

• x0 is your initial guess, one for each variable.

• func is an actual function (“callable”), not simply an array of numbers! This means we need to define a new function that returns the appropriate computation on \(x\). In Python, the syntax for defining your own function is:

def my_func_name(arg1, arg2, ...):
    do something
    return something   # optional

• The def at the start is required, as is the colon at the end of the first line.

• Just like variables, function names should be descriptive.

• Arguments are optional.

• Returning a value is generally optional, but here we want to return the expression for the analytical function.

from scipy.optimize import fmin    # take note of syntax when importing specific functions!
import numpy as np
import matplotlib.pyplot as plt

def my_function(x):
    return 1 - x * np.exp(-2 * x)  # that's it!

minimum = fmin(my_function, 0)     # initial guess of 0
display(minimum)                   # many attributes!

# plot it for good measure
x = np.linspace(0, 5)
fig, ax = plt.subplots()
ax.plot(x, my_function(x))
plt.show()

Optimization terminated successfully.
         Current function value: 0.816060
         Iterations: 23
         Function evaluations: 46

array([0.5])

Part (b) - Functions of several variables

Find the minimum of the function \(f(x,y) = \cos(xy) \sin(x)\) over the domain \(0 < x < \pi\), \(0 < y < \pi\). Start by plotting the function to approximately locate the minimum and use these approximate values as your initial guess.

Important

fmin() can be kinda quirky. If you have two variables x and y, instead of inputting them as separate arguments, instead group them into a list, i.e., u = [x, y]. Then your x0 will be a list as well for the initial guesses.

# define the function - note the grouping of x,y into a list u
def new_func(u):
    return np.cos(u[0] * u[1]) * np.sin(u[0])

# plot it
x = np.linspace(0, np.pi)
X, Y = np.meshgrid(x, x)
Z = new_func([X, Y])

fig = plt.figure()
ax = fig.add_subplot(projection='3d')
ax.plot_surface(X, Y, Z, cmap='plasma', antialiased=False)
ax.set(xlabel='x', ylabel='y')
plt.show()

# minimum appears to be near (1.5, 2)
minimum = fmin(new_func, [1.5, 2])
display(minimum)

Optimization terminated successfully.
         Current function value: -1.000000
         Iterations: 25
         Function evaluations: 49

array([1.5708235 , 1.99997996])