Optimization Problems

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import minimize_scalar

def optimization_problem():
    """
    Problem: Maximize area of rectangle with fixed perimeter
    
    Given: Perimeter = 16
    Find: Dimensions (width, height) that maximize area
    
    Constraint: 2w + 2h = 16  =>  h = 8 - w
    Objective: A(w) = w * h = w(8 - w) = 8w - w²
    """
    
    # Define area function (to maximize)
    def area(w):
        return w * (8 - w)
    
    # Define negative area (to minimize, equivalent to maximizing area)
    def neg_area(w):
        return -area(w)
    
    # Method 1: Using calculus (derivative = 0)
    # A(w) = 8w - w²
    # A'(w) = 8 - 2w
    # Critical point: 8 - 2w = 0  =>  w = 4
    
    w_critical = 4
    h_critical = 8 - w_critical
    max_area_analytical = area(w_critical)
    
    print("=== Analytical Solution (Calculus) ===")
    print(f"Critical point: w = {w_critical}, h = {h_critical}")
    print(f"Maximum area: {max_area_analytical}")
    
    # Verify it's a maximum (second derivative test)
    # A''(w) = -2 < 0, so it's a maximum ✓
    print(f"Second derivative: A''(w) = -2 (< 0, confirms maximum)")
    
    # Method 2: Using numerical optimization
    result = minimize_scalar(neg_area, bounds=(0, 8), method='bounded')
    w_optimal = result.x
    h_optimal = 8 - w_optimal
    max_area_numerical = -result.fun
    
    print("\n=== Numerical Solution (scipy) ===")
    print(f"Optimal width: {w_optimal:.4f}")
    print(f"Optimal height: {h_optimal:.4f}")
    print(f"Maximum area: {max_area_numerical:.4f}")
    
    # Visualize
    w_values = np.linspace(0, 8, 100)
    area_values = [area(w) for w in w_values]
    
    plt.figure(figsize=(10, 6))
    plt.plot(w_values, area_values, 'b-', linewidth=2, label='A(w) = w(8-w)')
    plt.plot(w_critical, max_area_analytical, 'ro', markersize=10, 
             label=f'Maximum: ({w_critical}, {max_area_analytical})')
    plt.axvline(w_critical, color='r', linestyle='--', alpha=0.5)
    plt.xlabel('Width (w)')
    plt.ylabel('Area (A)')
    plt.title('Optimization: Maximize Rectangle Area')
    plt.legend()
    plt.grid(True)
    plt.show()
    
    return w_critical, h_critical, max_area_analytical

# Run optimization
width, height, area = optimization_problem()
print(f"\nResult: {width} × {height} rectangle with area {area}")