import numpy as np
from scipy.optimize import minimize

# Define the objective function to minimize
def objective(x):
    '''
    Minimize f(x1, x2) = x1^2 + 2 * x2^2 (similar to minimizing the volume)
    x[0] is x1, x[1] is x2
    '''
    return (x[0]**2 + 2 * x[1]**2)


def constraint1(x):
    '''
    Define non-linear constraints (SciPy requires them as >= 0). 
    Algebraically rewrite your formulas to match this format.
    Minimum area requirement:
    x1 * x2 >= 4  should be reformatted as  x1 * x2 - 4 >= 0
    '''
    return (x[0] * x[1] - 4)

def constraint2(x):
    '''
    Parabolic boundary
    x1^2 + 2 * x2 <= 12 should be changed  12 - x1^2 - 2 * x2 >= 0
    
    '''
    return 12 - x[0]**2 - 2 * x[1]

# Setup configuration
initial_guess = np.array([0.0, 1.0])
bounds = ((0.5, None), (0.5, None))

# Combine multiple constraints into a list of dictionaries
constraints = [
    {'type': 'ineq', 'fun': constraint1},
    {'type': 'ineq', 'fun': constraint2}
]

# Run the SLSQP optimizer
result = minimize(
    fun=objective,
    x0=initial_guess,
    method='SLSQP',
    bounds=bounds,
    constraints=constraints
)

# Print the optimized output
if result.success:
    print(f"Optimal x1: {result.x[0]:.3f}")
    print(f"Optimal x2: {result.x[1]:.3f}")
    print(f"Minimum Objective Value: {result.fun:.2f}")
else:
    print("Optimization failed:", result.message)