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%% Cell type:code id: tags:
``` python
import numpy as np
from scipy import optimize
import matplotlib.pyplot as plt
```
%% Cell type:markdown id: tags:
Some necessary libraries are imported.
%% Cell type:code id: tags:
``` python
def f(x):
first_elm = np.dot(np.dot(x, Q), x)
second_elm = np.dot(b, x)
f_x = first_elm + second_elm + c
return f_x
```
%% Cell type:markdown id: tags:
Objective function has been defined in order to use the scipy.optimize later.
%% Cell type:code id: tags:
``` python
global Q
global b
global c
Q = np.loadtxt("Q.txt")
b = np.loadtxt("b.txt")
c = np.loadtxt("c.txt")
```
%% Cell type:markdown id: tags:
The matrices defining the objective function have been inputted as global variables.
%% Cell type:code id: tags:
``` python
print(b.size)
n = b.size
err_tol = 1e-5
alpha = 1
# recommended sigma [10^-5, 10^-1]
sigma = 1e-1
# recommended beta [1/10, 1/2]
beta = 0.4
x = np.random.uniform(-1e4, 1e4, n)
gradient = np.add(np.dot(2, np.dot(Q, x)), b)
alpha_arr = np.array([])
f_x_arr = np.array([])
x_arr = np.array([])
```
%% Output
5
%% Cell type:markdown id: tags:
Initial guess is created and initial gradient is calculated.
%% Cell type:code id: tags:
``` python
while np.linalg.norm(gradient) >= err_tol:
# Calculation of f(x_k)
first_elm = np.dot(np.dot(x, Q), x)
second_elm = np.dot(b, x)
f_x = first_elm + second_elm + c
alpha_arr = np.append(alpha_arr, alpha)
f_x_arr = np.append(f_x_arr, f_x)
x_arr = np.append(x_arr, x)
# Calculation of f(x_k - alpha*gradient(f(x)))
x_ = np.subtract(x, alpha*gradient)
first_elm_ = np.dot(np.dot(x_, Q), x_)
second_elm_ = np.dot(b, x_)
f_x_ = first_elm_ + second_elm_ + c
# Armijo's rule for selecting step size
while f_x_ > (f_x - alpha*sigma*np.linalg.norm(gradient)*np.linalg.norm(gradient)):
alpha = alpha*beta
# Calculation of f(x_k - alpha*gradient(f(x)))
x_ = np.subtract(x, alpha * gradient)
first_elm_ = np.dot(np.dot(x_, Q), x_)
second_elm_ = np.dot(b, x_)
f_x_ = first_elm_ + second_elm_ + c
# x and gradient are updated
x = x_
gradient = np.add(np.dot(2, np.dot(Q, x)), b)
print(alpha)
print(x)
print(f_x)
plt.plot(alpha_arr)
plt.title("Alpha values over iterations")
plt.show()
plt.plot(f_x_arr)
plt.title("Function values over iterations")
plt.show()
```
%% Output
0.06400000000000002
[-0.05708123 0.68024703 -0.0796436 -0.93631147 -0.16706706]
0.6922831525384057
%% Cell type:code id: tags:
``` python
# matrix inversion
x_min = np.dot(np.linalg.inv(Q), -b/2)
print(x_min)
print(f(x_min))
```
%% Output
[-0.05704387 0.68023741 -0.07965977 -0.93629827 -0.16708381]
0.692283152298834
%% Cell type:code id: tags:
``` python
# scipy optimization
x0 = np.random.uniform(-1e5, 1e5, n)
minimum = optimize.fmin(f, x0)
print(minimum)
```
%% Output
Optimization terminated successfully.
Current function value: 0.692283
Iterations: 467
Function evaluations: 765
[-0.05709283 0.68023614 -0.07964932 -0.93628897 -0.16705926]
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