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| 1 | +## ---- python Convex Optimization visualization the process |
| 2 | + |
| 3 | +# Visualizing the process of **Convex Optimization** typically involves showing how an algorithm (such as gradient descent) iterates toward the minimum of a convex function. Here's a general way to visualize this process in Python using `matplotlib` and `numpy`. |
| 4 | + |
| 5 | +# ### Steps: |
| 6 | +# 1. **Generate a convex function**: For simplicity, use a quadratic function (e.g., \( f(x) = x^2 \)). |
| 7 | +# 2. **Apply an optimization algorithm**: Use gradient descent as the optimization algorithm. |
| 8 | +# 3. **Visualize the process**: Plot the function and show how the optimization algorithm converges to the minimum. |
| 9 | + |
| 10 | +# Here's how you can implement it: |
| 11 | + |
| 12 | +# ### Python Code: |
| 13 | + |
| 14 | +# ```python |
| 15 | +import numpy as np |
| 16 | +import matplotlib.pyplot as plt |
| 17 | + |
| 18 | +# Convex function (quadratic function) |
| 19 | +def f(x): |
| 20 | + return x ** 2 |
| 21 | + |
| 22 | +# Derivative of the convex function (gradient) |
| 23 | +def grad_f(x): |
| 24 | + return 2 * x |
| 25 | + |
| 26 | +# Gradient descent algorithm |
| 27 | +def gradient_descent(starting_point, learning_rate, num_iterations): |
| 28 | + x = starting_point |
| 29 | + trajectory = [x] |
| 30 | + for _ in range(num_iterations): |
| 31 | + x = x - learning_rate * grad_f(x) |
| 32 | + trajectory.append(x) |
| 33 | + return np.array(trajectory) |
| 34 | + |
| 35 | +# Visualization of the optimization process |
| 36 | +def visualize_optimization(trajectory): |
| 37 | + # Define x values and their corresponding function values |
| 38 | + x_vals = np.linspace(-3, 3, 400) |
| 39 | + y_vals = f(x_vals) |
| 40 | + |
| 41 | + plt.figure(figsize=(8, 6)) |
| 42 | + plt.plot(x_vals, y_vals, label=r'$f(x) = x^2$', color='blue') |
| 43 | + |
| 44 | + # Plot the trajectory of the optimization algorithm |
| 45 | + for i, x in enumerate(trajectory): |
| 46 | + plt.plot(x, f(x), 'ro') # Mark the point |
| 47 | + plt.text(x, f(x), f'Iter {i}', fontsize=10) |
| 48 | + if i > 0: |
| 49 | + plt.arrow(trajectory[i-1], f(trajectory[i-1]), |
| 50 | + trajectory[i] - trajectory[i-1], |
| 51 | + f(trajectory[i]) - f(trajectory[i-1]), |
| 52 | + head_width=0.1, head_length=0.1, fc='green', ec='green') |
| 53 | + |
| 54 | + plt.xlabel('x') |
| 55 | + plt.ylabel('f(x)') |
| 56 | + plt.title('Convex Optimization using Gradient Descent') |
| 57 | + plt.legend() |
| 58 | + plt.grid(True) |
| 59 | + plt.show() |
| 60 | + |
| 61 | +# Parameters for gradient descent |
| 62 | +starting_point = 2.5 |
| 63 | +learning_rate = 0.1 |
| 64 | +num_iterations = 10 |
| 65 | + |
| 66 | +# Get the trajectory of the optimization process |
| 67 | +trajectory = gradient_descent(starting_point, learning_rate, num_iterations) |
| 68 | + |
| 69 | +# Visualize the optimization process |
| 70 | +visualize_optimization(trajectory) |
| 71 | +# ``` |
| 72 | + |
| 73 | +# ### Explanation: |
| 74 | +# 1. **Convex function**: \( f(x) = x^2 \), which is a simple convex function. |
| 75 | +# 2. **Gradient descent**: The optimization algorithm is performed over a set number of iterations, updating \( x \) based on the gradient (which is \( 2x \)). |
| 76 | +# 3. **Visualization**: The function is plotted, and each iteration of the optimization is marked. The movement of the points toward the minimum is visualized with arrows. |
| 77 | + |
| 78 | +# This code will generate a visualization where you can see the steps of the gradient descent as it moves towards the minimum at \( x = 0 \). You can adjust the learning rate and the number of iterations to observe different behaviors of the optimization process. |
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