Method selection (Nelder-Mead, BFGS, Powell) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of optimization with Nelder-Mead method

What is the value of the variable result.x after running this code?

SciPy

from scipy.optimize import minimize

def f(x):
    return (x[0] - 3)**2 + (x[1] + 1)**2

result = minimize(f, [0, 0], method='Nelder-Mead')
print(result.x)

A [3.0, -1.0]

B [0.0, 0.0]

C [3.5, -0.5]

D [2.999, -1.001]

💡 Hint

Nelder-Mead is a simplex method that approximates the minimum but may not reach exact values.

Nelder-Mead is a heuristic method that finds a minimum close to the true minimum but may not be exact. The function minimum is at (3, -1), but the output is an approximation close to it.

❓ Predict Output

intermediate

Output of optimization with BFGS method

What is the value of result.success after running this code?

SciPy

from scipy.optimize import minimize

def f(x):
    return (x[0] - 1)**4 + (x[1] - 2)**4

result = minimize(f, [0, 0], method='BFGS')
print(result.success)

A True

B False

C None

D Raises an exception

💡 Hint

BFGS is a gradient-based method that usually converges for smooth functions.

BFGS uses gradients and converges successfully on smooth functions like this quartic function, so result.success is True.

❓ data_output

advanced

Number of iterations for Powell method

How many iterations does the Powell method take to minimize this function?

SciPy

from scipy.optimize import minimize

def f(x):
    return (x[0] - 5)**2 + (x[1] + 3)**2

result = minimize(f, [0, 0], method='Powell')
print(result.nit)

A 5

B 10

C 20

D 1

💡 Hint

Powell is a direction set method that usually takes multiple iterations but fewer than 20 here.

Powell method typically converges in about 10 iterations for this simple quadratic function.

🧠 Conceptual

advanced

Choosing optimization method for noisy function

Which optimization method is best suited for minimizing a noisy function without gradient information?

A Powell

B BFGS

C Nelder-Mead

D Gradient Descent

💡 Hint

Consider methods that do not require gradient information.

Nelder-Mead does not require gradients and is robust for noisy functions, unlike BFGS or gradient descent which need gradients.

🚀 Application

expert

Selecting method for high-dimensional smooth function

You want to minimize a smooth, high-dimensional function with available gradient. Which method should you choose for fastest convergence?

A BFGS

B Powell

C Random Search

D Nelder-Mead

💡 Hint

Use gradient-based methods for smooth functions with gradients available.

BFGS uses gradient information and approximates Hessian, making it efficient for smooth, high-dimensional problems.