Find minimum and maximum values of a function

Question

I have a function and I would like to find its maximum and minimum values. My function is this:

def function(x, y):
    exp = (math.pow(x, 2) + math.pow(y, 2)) * -1
    return math.exp(exp) * math.cos(x * y) * math.sin(x * y)

I have an interval for x [-1, 1] and y [-1, 1]. I would like to find a way, limited to this interval, to discover the max and min values of this function.

Answer 1

Using, for instance, scipy's fmin (which contains an implementation of the Nelder-Mead algorithm), you can try this:

import numpy as np
from scipy.optimize import fmin
import math

def f(x):
    exp = (math.pow(x[0], 2) + math.pow(x[1], 2)) * -1
    return math.exp(exp) * math.cos(x[0] * x[1]) * math.sin(x[0] * x[1])

fmin(f,np.array([0,0]))

which yields the following output:

Optimization terminated successfully.
     Current function value: -0.161198
     Iterations: 60
     Function evaluations: 113 
array([ 0.62665701, -0.62663095])

Please keep in mind that:

1) with scipy you need to convert your function into a function accepting an array (I showed how to do it in the example above);

2) fmin uses, like most of its pairs, an iterative algorithm, therefore you must provide a starting point (in my example, I provided (0,0)). You can provide different starting points to obtain different minima/maxima.