#337 Minimization on an 2D quadratic problem using the gradient descent method in python

ex337.py

import numpy as np
import matplotlib.pyplot as plt

def get_next_point(a0, b0, gamma):
    # Get the next point $(a_1, b_1)$ starting from the current point $(a_0,b_0)$
    # using exact line search. The close-form of $t_{exact}$ is 
    # $\dfrac{a_0^2+\gamma^2 b_0^2}{a_0^2+\gamma^3 b_0^2}$.
    r = gamma
    t = (a0**2 + r**2 * b0**2) / (a0**2 + r**3 * b0**2)
    a1 = (1 - t)*a0
    b1 = (1 - t*r)*b0
    return a1, b1

def plot_quadratic_convergence():
    # plot the convergence of $x$ during the minimization on a quadratic 
    # objective function $f(x_1, x_2)=\frac{1}{2}(x_1^2+\gamma x_2^2)$,
    # where $\gamma=10$, starting from $x_0=(\gamma, 3)$
    gamma = 10
    a0 = gamma
    b0 = 3
    num_iteration = 100
    x = np.zeros((num_iteration, 1))
    y = np.zeros((num_iteration, 1))
    for iteration in range(num_iteration):
        print("({}, {})".format(a0, b0))
        x[iteration] = a0
        y[iteration] = b0
        a1, b1 = get_next_point(a0, b0, gamma)
        a0, b0 = a1, b1
    fig, ax = plt.subplots()
    ax.set_xlim([-gamma * 1, gamma * 1])
    ax.set_ylim([-1 * 4, 1 * 4])
    ax.plot(x, y, marker="o",markerfacecolor='red',
                markeredgecolor='red',
                markeredgewidth=2)
    delta = gamma * 1.0 / 100
    x = np.arange(-gamma * 1, gamma * 1, delta)
    y = np.arange(-4.0, 4.0, delta)
    X, Y = np.meshgrid(x, y)
    Z = 0.5 * (X ** 2 + gamma * Y ** 2)
    levels = np.exp(np.arange(-2, 10, 0.25))
    CS = ax.contour(X, Y, Z, levels=levels, vmin=0,vmax=100)
    ax.set_aspect(aspect='2')
    ax.clabel(CS, fontsize=6, inline=True)
    ax.set_xlabel(r'$x_1$')
    ax.set_ylabel(r'$x_2$')
    ax.set_title(r'$f(x)=\frac{1}{2}\left(x_1^2+\gamma x_2^2\right),\gamma=10,x^{0}=(\gamma, 3)$')
    plt.savefig('ex337.png', dpi=100)
    plt.show()        
    
if __name__ == "__main__":
    plot_quadratic_convergence() 

Last Updated on 2023-01-15 by gantovnik