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optimizer.py
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Wed, May 1, 08:01

optimizer.py
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import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
import scipy.optimize
# Global variables (in capital letters)
# Initial guess to minimizer
GUESS = np.array([4,4])
# This array will contain the steps of the minimizer on its way to the minimum
STEPS = GUESS.reshape(-1,2)
# This function fills the STEPS array, and is used as input to scipy.optimize.minimize
def my_callback(xk):
global STEPS
STEPS = np.vstack((STEPS, np.array(xk)))
# This function returns the minimum value of the function func given as input
# The second input is the initial guess of the minimum
def find_minimum(func, guess):
result = scipy.optimize.minimize(func, x0 = guess, method='BFGS', callback=my_callback)
return result.x
# This function implements S(x) as given in the exercise sheet
def quadratic_function(x):
# create column vector of input array
x.reshape(-1, 1)
# create matrix and vector coefficients of S(x) given in exercise sheet
mat = np.array([[4, 0], [1, 3]])
vec = np.array([0, 1]).transpose()
# return S(x)
return (x.transpose() @ mat) @ x - x.transpose() @ vec
# This function plots the S(x) surface and includes the path taken by the minimizer
def plotting():
fig = plt.figure(figsize=(14,8))
axe = fig.add_subplot(111, projection="3d")
num_of_points = 100
xmax = ymax = -10
xmin = ymin = 10
X, Y = np.meshgrid(np.linspace(xmin,xmax,num_of_points), np.linspace(ymin,ymax,num_of_points))
Z = np.zeros_like(X)
for i in range(0,num_of_points):
for j in range(0, num_of_points):
Z[i,j] = quadratic_function( np.array([X[i,j], Y[i,j]]) )
sp = axe.plot_surface(X, Y, Z, cmap=cm.coolwarm)
axe.view_init(60, 100)
fig.colorbar(sp)
step_value = np.zeros(STEPS.shape[0])
for i in range(0, STEPS.shape[0]):
step_value[i] = quadratic_function( np.array([STEPS[i, 0], STEPS[i, 1]]) )
axe.plot(STEPS[:,0], STEPS[:,1], step_value, 'r*--')
axe.set_xlabel('x', fontsize=15)
axe.set_ylabel('y', fontsize=15)
axe.set_zlabel(' S(x,y)', fontsize=15)
axe.zaxis.set_rotate_label(False)
plt.savefig("plot.png")
plt.show()
result = find_minimum(quadratic_function, GUESS)
print("The (approximated) minimum value is found at position (x,y) = ", result)
print("The (approximated) minimum value is ", quadratic_function(result))
plotting()

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